Optimal. Leaf size=102 \[ \frac{4 e^{2 i a} x^{m+1} \left (c x^n\right )^{2 i b} \text{Hypergeometric2F1}\left (2,-\frac{-2 b n+i (m+1)}{2 b n},-\frac{-4 b n+i (m+1)}{2 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2 i b n+m+1} \]
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Rubi [A] time = 0.0846519, antiderivative size = 102, 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} \[ \frac{4 e^{2 i a} x^{m+1} \left (c x^n\right )^{2 i b} \, _2F_1\left (2,-\frac{i (m+1)-2 b n}{2 b n};-\frac{i (m+1)-4 b n}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2 i b n+m+1} \]
Antiderivative was successfully verified.
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Rule 4509
Rule 4505
Rule 364
Rubi steps
\begin{align*} \int x^m \sec ^2\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1+m}{n}} \sec ^2(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (4 e^{2 i a} x^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+2 i b+\frac{1+m}{n}}}{\left (1+e^{2 i a} x^{2 i b}\right )^2} \, dx,x,c x^n\right )}{n}\\ &=\frac{4 e^{2 i a} x^{1+m} \left (c x^n\right )^{2 i b} \, _2F_1\left (2,-\frac{i (1+m)-2 b n}{2 b n};-\frac{i (1+m)-4 b n}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+m+2 i b n}\\ \end{align*}
Mathematica [B] time = 17.5407, size = 482, normalized size = 4.73 \[ \frac{x^{m+1} \sin (b n \log (x)) \sec \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \sec \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )+b n \log (x)\right )}{b n}-\frac{(m+1) \sec \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \left (\frac{x^{m+1} \sin (b n \log (x)) \sec \left (a+b \log \left (c x^n\right )\right )}{m+1}-\frac{i \cos \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \exp \left (-\frac{(2 m+1) \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{b n}\right ) \left ((2 i b n+m+1) \left (-\exp \left (\frac{2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right )\right ) \text{Hypergeometric2F1}\left (1,-\frac{i (m+1)}{2 b n},1-\frac{i (m+1)}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+(m+1) \exp \left (\frac{a (2 i b n+2 m+1)}{b n}+\frac{(2 i b n+2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )}{n}+\log (x) (2 i b n+m+1)\right ) \text{Hypergeometric2F1}\left (1,-\frac{i (2 i b n+m+1)}{2 b n},-\frac{i (4 i b n+m+1)}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )-i (2 i b n+m+1) \tan \left (a+b \log \left (c x^n\right )\right ) \exp \left (\frac{2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right )\right )}{(m+1) (2 i b n+m+1)}\right )}{b n} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.779, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \sec \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}\, 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 (x^{m} \sec \left (b \log \left (c x^{n}\right ) + a\right )^{2}, 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 x^{m} \sec ^{2}{\left (a + b \log{\left (c x^{n} \right )} \right )}\, 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 x^{m} \sec \left (b \log \left (c x^{n}\right ) + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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