Optimal. Leaf size=79 \[ -\frac{2 e^{2 i a} \left (c x^n\right )^{2 i b} \text{Hypergeometric2F1}\left (2,1+\frac{i}{b n},2+\frac{i}{b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{x^2 (1-i b n)} \]
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Rubi [A] time = 0.0706627, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {4509, 4505, 364} \[ -\frac{2 e^{2 i a} \left (c x^n\right )^{2 i b} \, _2F_1\left (2,1+\frac{i}{b n};2+\frac{i}{b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{x^2 (1-i b n)} \]
Antiderivative was successfully verified.
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Rule 4509
Rule 4505
Rule 364
Rubi steps
\begin{align*} \int \frac{\sec ^2\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx &=\frac{\left (c x^n\right )^{2/n} \operatorname{Subst}\left (\int x^{-1-\frac{2}{n}} \sec ^2(a+b \log (x)) \, dx,x,c x^n\right )}{n x^2}\\ &=\frac{\left (4 e^{2 i a} \left (c x^n\right )^{2/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+2 i b-\frac{2}{n}}}{\left (1+e^{2 i a} x^{2 i b}\right )^2} \, dx,x,c x^n\right )}{n x^2}\\ &=-\frac{2 e^{2 i a} \left (c x^n\right )^{2 i b} \, _2F_1\left (2,1+\frac{i}{b n};2+\frac{i}{b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(1-i b n) x^2}\\ \end{align*}
Mathematica [A] time = 3.71769, size = 150, normalized size = 1.9 \[ \frac{(b n+i) \left (\tan \left (a+b \log \left (c x^n\right )\right )-i \text{Hypergeometric2F1}\left (1,\frac{i}{b n},1+\frac{i}{b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )\right )-e^{2 i a} \left (c x^n\right )^{2 i b} \text{Hypergeometric2F1}\left (1,1+\frac{i}{b n},2+\frac{i}{b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{b n x^2 (b n+i)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.615, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \sec \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}{{x}^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sec \left (b \log \left (c x^{n}\right ) + a\right )^{2}}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{2}{\left (a + b \log{\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (b \log \left (c x^{n}\right ) + a\right )^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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