Optimal. Leaf size=111 \[ -\frac {2 \, _2F_1\left (-\frac {3}{2},\frac {1}{4} \left (-3+\frac {4 i}{b n}\right );\frac {1}{4} \left (1+\frac {4 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{(4+3 i b n) x^2 \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 111, normalized size of antiderivative = 1.00, 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 = {4581, 4579,
371} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 4579
Rule 4581
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} \text {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 ) \text {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*}
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Mathematica [A]
time = 1.17, size = 216, normalized size = 1.95 \begin {gather*} \frac {6 b^2 \sqrt {2-2 e^{2 i \left (a+b \log \left (c x^n\right )\right )}} n^2 \, _2F_1\left (\frac {1}{2},\frac {1}{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 )}{\sqrt {-i e^{-i \left (a+b \log \left (c x^n\right )\right )} \left (-1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )} (4+3 i b n) (4 i+b n) (4 i+3 b n) x^2}-\frac {2 \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \left (3 b n \cos \left (a+b \log \left (c x^n\right )\right )+4 \sin \left (a+b \log \left (c x^n\right )\right )\right )}{\left (16+9 b^2 n^2\right ) x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sin ^{\frac {3}{2}}\left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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