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$$\int \; \frac{\tan(x) + \tan^3(x)} {e^{\sec^2(x)} + e^{-\sec^3(x)}} \, \mathrm{d}x$$

Analysis:

This integral is complex due to the combination of:

Rational function: $\tan(x)$ and $\tan^3(x)$ form a rational function where the degree of the numerator (3) isn't smaller than the denominator (1). Composite exponential terms: The denominator includes $e^{\sec^2(x)}$ and $e^{-\sec^3(x)}$, making it a composite function.

Challenges for Analytical Solution:

These factors make it difficult to find an exact analytical solution using standard integration techniques like integration by parts, u-substitution, or partial fractions.

What can I do please help me.

Ѕᴀᴀᴅ
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Nsnansn Jwjwj
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1 Answers1

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Evaluate: $$\int{\tan(x)+\tan^3{x}\over e^{\sec^2{x}}+e^{-\sec^3{x}}}\,\,dx$$

Substitute $\phi = \sec^2x\implies dx = {d\phi\over2\tan(x)\sec^2x}$

So we would have: $$ \frac{1}{2}\int {1+\tan^2x\over\phi(e^\phi+e^{-\phi^{3\over2}})}d\phi = \frac{1}{2}\int{d\phi\over{e^\phi+e^{-\sqrt{\phi^3}}}} $$ Which most likely is not elementary....

Masd
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