Optimal. Leaf size=109 \[ \frac{2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \text{Hypergeometric2F1}\left (\frac{5}{2},\frac{1}{4} \left (5-\frac{2 i}{b n}\right ),\frac{1}{4} \left (9-\frac{2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+5 i b n) \sin ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.0738603, 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 = {4483, 4491, 364} \[ \frac{2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \, _2F_1\left (\frac{5}{2},\frac{1}{4} \left (5-\frac{2 i}{b n}\right );\frac{1}{4} \left (9-\frac{2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+5 i b n) \sin ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 4483
Rule 4491
Rule 364
Rubi steps
\begin{align*} \int \frac{1}{\sin ^{\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 \frac{x^{-1+\frac{1}{n}}}{\sin ^{\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}\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 \sin ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ &=\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 )}{(2+5 i b n) \sin ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}
Mathematica [A] time = 1.50711, size = 125, normalized size = 1.15 \[ \frac{2 x \left (i (b n+2 i) \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \text{Hypergeometric2F1}\left (1,\frac{3}{4}-\frac{i}{2 b n},\frac{5}{4}-\frac{i}{2 b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )-b n \cot \left (a+b \log \left (c x^n\right )\right )-2\right )}{3 b^2 n^2 \sqrt{\sin \left (a+b \log \left (c x^n\right )\right )}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.167, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \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 \frac{1}{\sin \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sin \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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