Optimal. Leaf size=102 \[ \frac{8 e^{3 i a} x^{m+1} \left (c x^n\right )^{3 i b} \text{Hypergeometric2F1}\left (3,-\frac{-3 b n+i (m+1)}{2 b n},-\frac{-5 b n+i (m+1)}{2 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3 i b n+m+1} \]
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Rubi [A] time = 0.0886081, 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} \[ \frac{8 e^{3 i a} x^{m+1} \left (c x^n\right )^{3 i b} \, _2F_1\left (3,-\frac{i (m+1)-3 b n}{2 b n};-\frac{i (m+1)-5 b n}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3 i b n+m+1} \]
Antiderivative was successfully verified.
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Rule 4509
Rule 4505
Rule 364
Rubi steps
\begin{align*} \int x^m \sec ^3\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 ^3(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (8 e^{3 i a} x^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+3 i b+\frac{1+m}{n}}}{\left (1+e^{2 i a} x^{2 i b}\right )^3} \, dx,x,c x^n\right )}{n}\\ &=\frac{8 e^{3 i a} x^{1+m} \left (c x^n\right )^{3 i b} \, _2F_1\left (3,-\frac{i (1+m)-3 b n}{2 b n};-\frac{i (1+m)-5 b n}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+m+3 i b n}\\ \end{align*}
Mathematica [A] time = 5.65811, size = 134, normalized size = 1.31 \[ \frac{x^{m+1} \left (-2 \sec \left (a+b \log \left (c x^n\right )\right ) \left (-b n \tan \left (a+b \log \left (c x^n\right )\right )+m+1\right )+4 e^{i a} (-i b n+m+1) \left (c x^n\right )^{i b} \text{Hypergeometric2F1}\left (1,\frac{1}{2}-\frac{i (m+1)}{2 b n},\frac{3}{2}-\frac{i (m+1)}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{4 b^2 n^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 2.368, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \sec \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{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 (x^{m} \sec \left (b \log \left (c x^{n}\right ) + a\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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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 )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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