Optimal. Leaf size=111 \[ -\frac {2 \, _2F_1\left (-\frac {3}{2},\frac {1}{4} \left (\frac {4 i}{b n}-3\right );\frac {1}{4} \left (1+\frac {4 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x^2 (4+3 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]
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Rubi [A] time = 0.09, 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 = {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 364
Rule 4491
Rule 4493
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*}
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Mathematica [A] time = 1.20, size = 168, normalized size = 1.51 \[ \frac {6 i b^2 n^2 \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \, _2F_1\left (1,\frac {3}{4}+\frac {i}{b n};\frac {5}{4}+\frac {i}{b n};e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )-(b n+4 i) \left (8 \sin ^2\left (a+b \log \left (c x^n\right )\right )+3 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right )}{x^2 (b n+4 i) (3 b n-4 i) (3 b n+4 i) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, 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 \[ \int \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}}{x^3} \,d x \]
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 \[ \int \frac {\sin ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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