Optimal. Leaf size=111 \[ -\frac{2 \text{Hypergeometric2F1}\left (-\frac{3}{2},\frac{1}{4} \left (-3+\frac{4 i}{b n}\right ),\frac{1}{4} \left (1+\frac{4 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x^2 (4+3 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]
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Rubi [A] time = 0.0869713, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4493, 4491, 364} \[ -\frac{2 \, _2F_1\left (-\frac{3}{2},\frac{1}{4} \left (\frac{4 i}{b n}-3\right );\frac{1}{4} \left (1+\frac{4 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x^2 (4+3 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4493
Rule 4491
Rule 364
Rubi steps
\begin{align*} \int \frac{\sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx &=\frac{\left (c x^n\right )^{2/n} \operatorname{Subst}\left (\int x^{-1-\frac{2}{n}} \sin ^{\frac{3}{2}}(a+b \log (x)) \, dx,x,c x^n\right )}{n x^2}\\ &=\frac{\left (\left (c x^n\right )^{\frac{3 i b}{2}+\frac{2}{n}} \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )\right ) \operatorname{Subst}\left (\int x^{-1-\frac{3 i b}{2}-\frac{2}{n}} \left (1-e^{2 i a} x^{2 i b}\right )^{3/2} \, dx,x,c x^n\right )}{n x^2 \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}}\\ &=-\frac{2 \, _2F_1\left (-\frac{3}{2},\frac{1}{4} \left (-3+\frac{4 i}{b n}\right );\frac{1}{4} \left (1+\frac{4 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{(4+3 i b n) x^2 \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 1.19889, size = 168, normalized size = 1.51 \[ \frac{6 i b^2 n^2 \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \text{Hypergeometric2F1}\left (1,\frac{3}{4}+\frac{i}{b n},\frac{5}{4}+\frac{i}{b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )-(b n+4 i) \left (8 \sin ^2\left (a+b \log \left (c x^n\right )\right )+3 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right )}{x^2 (b n+4 i) (3 b n-4 i) (3 b n+4 i) \sqrt{\sin \left (a+b \log \left (c x^n\right )\right )}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.16, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( \sin \left ( a+b\ln \left ( c{x}^{n} \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{\sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{2}}}{x^{3}}\,{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{\sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{2}}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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