Optimal. Leaf size=150 \[ \frac{2 (e x)^{m+1} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2} \text{Hypergeometric2F1}\left (\frac{3}{2},-\frac{-3 b d n+2 i m+2 i}{4 b d n},-\frac{-7 b d n+2 i m+2 i}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (3 i b d n+2 m+2) \sin ^{\frac{3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \]
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Rubi [A] time = 0.113057, antiderivative size = 145, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {4493, 4491, 364} \[ \frac{2 (e x)^{m+1} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2} \, _2F_1\left (\frac{3}{2},\frac{1}{4} \left (3-\frac{2 i (m+1)}{b d n}\right );-\frac{2 i m-7 b d n+2 i}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (3 i b d n+2 m+2) \sin ^{\frac{3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \]
Antiderivative was successfully verified.
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Rule 4493
Rule 4491
Rule 364
Rubi steps
\begin{align*} \int \frac{(e x)^m}{\sin ^{\frac{3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )} \, dx &=\frac{\left ((e x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+\frac{1+m}{n}}}{\sin ^{\frac{3}{2}}(d (a+b \log (x)))} \, dx,x,c x^n\right )}{e n}\\ &=\frac{\left ((e x)^{1+m} \left (c x^n\right )^{-\frac{3}{2} i b d-\frac{1+m}{n}} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+\frac{3 i b d}{2}+\frac{1+m}{n}}}{\left (1-e^{2 i a d} x^{2 i b d}\right )^{3/2}} \, dx,x,c x^n\right )}{e n \sin ^{\frac{3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}\\ &=\frac{2 (e x)^{1+m} \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{3/2} \, _2F_1\left (\frac{3}{2},\frac{1}{4} \left (3-\frac{2 i (1+m)}{b d n}\right );-\frac{2 i+2 i m-7 b d n}{4 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{e (2+2 m+3 i b d n) \sin ^{\frac{3}{2}}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}\\ \end{align*}
Mathematica [B] time = 5.17482, size = 544, normalized size = 3.63 \[ \frac{(e x)^m \left (b^2 d^2 n^2+4 m^2+8 m+4\right ) x^{1+i b d n} \sqrt{2-2 e^{2 i a d} \left (c x^n\right )^{2 i b d}} \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{i \left (\frac{3}{2} i b d n+m+1\right )}{2 b d n},-\frac{-7 b d n+2 i m+2 i}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )+\frac{(e x)^m (3 b d n-2 i m-2 i) x^{1-i b d n} \left ((b d n-2 i m-2 i) \sqrt{2-2 e^{2 i a d} \left (c x^n\right )^{2 i b d}} \sqrt{\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )} \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{b d n+2 i m+2 i}{4 b d n},-\frac{-3 b d n+2 i m+2 i}{4 b d n},e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )-2 x^{i b d n} \sqrt{-i e^{-i a d} \left (c x^n\right )^{-i b d} \left (-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )} (b d n \cos (b d n \log (x))-2 (m+1) \sin (b d n \log (x)))\right )}{\sqrt{\sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}}}{b d n (3 b d n-2 i m-2 i) \sqrt{-i e^{-i a d} \left (c x^n\right )^{-i b d} \left (-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )} \left (2 (m+1) \sin \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+b d n \cos \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.261, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex \right ) ^{m} \left ( \sin \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{-{\frac{3}{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{\left (e x\right )^{m}}{\sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{\frac{3}{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{\left (e x\right )^{m}}{\sin \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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