Optimal. Leaf size=109 \[ \frac{2 x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \text{Hypergeometric2F1}\left (\frac{5}{2},\frac{1}{4} \left (5-\frac{2 i}{b n}\right ),\frac{1}{4} \left (9-\frac{2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sec ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )}{2+5 i b n} \]
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Rubi [A] time = 0.0722975, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4503, 4507, 364} \[ \frac{2 x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \, _2F_1\left (\frac{5}{2},\frac{1}{4} \left (5-\frac{2 i}{b n}\right );\frac{1}{4} \left (9-\frac{2 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sec ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )}{2+5 i b n} \]
Antiderivative was successfully verified.
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Rule 4503
Rule 4507
Rule 364
Rubi steps
\begin{align*} \int \sec ^{\frac{5}{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 ^{\frac{5}{2}}(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x \left (c x^n\right )^{-\frac{5 i b}{2}-\frac{1}{n}} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \sec ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )\right ) \operatorname{Subst}\left (\int \frac{x^{-1+\frac{5 i b}{2}+\frac{1}{n}}}{\left (1+e^{2 i a} x^{2 i b}\right )^{5/2}} \, dx,x,c x^n\right )}{n}\\ &=\frac{2 x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \, _2F_1\left (\frac{5}{2},\frac{1}{4} \left (5-\frac{2 i}{b n}\right );\frac{1}{4} \left (9-\frac{2 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sec ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )}{2+5 i b n}\\ \end{align*}
Mathematica [A] time = 1.35077, size = 124, normalized size = 1.14 \[ \frac{2 x \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \left ((2-i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \text{Hypergeometric2F1}\left (1,\frac{3}{4}-\frac{i}{2 b n},\frac{5}{4}-\frac{i}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+b n \tan \left (a+b \log \left (c x^n\right )\right )-2\right )}{3 b^2 n^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.275, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sec \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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