Optimal. Leaf size=85 \[ \frac{4 e^{2 i a} x \left (c x^n\right )^{2 i b} \text{Hypergeometric2F1}\left (2,\frac{1}{2} \left (2-\frac{i}{b n}\right ),\frac{1}{2} \left (4-\frac{i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+2 i b n} \]
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Rubi [A] time = 0.0625784, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {4503, 4505, 364} \[ \frac{4 e^{2 i a} x \left (c x^n\right )^{2 i b} \, _2F_1\left (2,\frac{1}{2} \left (2-\frac{i}{b n}\right );\frac{1}{2} \left (4-\frac{i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+2 i b n} \]
Antiderivative was successfully verified.
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Rule 4503
Rule 4505
Rule 364
Rubi steps
\begin{align*} \int \sec ^2\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \sec ^2(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (4 e^{2 i a} x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+2 i b+\frac{1}{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 \left (c x^n\right )^{2 i b} \, _2F_1\left (2,\frac{1}{2} \left (2-\frac{i}{b n}\right );\frac{1}{2} \left (4-\frac{i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+2 i b n}\\ \end{align*}
Mathematica [A] time = 6.40699, size = 147, normalized size = 1.73 \[ \frac{x \left (\frac{e^{2 i a} \left (c x^n\right )^{2 i b} \text{Hypergeometric2F1}\left (1,1-\frac{i}{2 b n},2-\frac{i}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b n-i}-i \text{Hypergeometric2F1}\left (1,-\frac{i}{2 b n},1-\frac{i}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+\tan \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.22, size = 0, normalized size = 0. \begin{align*} \int \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 (\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 \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 \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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