Optimal. Leaf size=202 \[ \frac {3 x^3 \sin ^4\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+9}-\frac {4 b n x^3 \sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+9}+\frac {36 b^2 n^2 x^3 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+180 b^2 n^2+81}-\frac {24 b^3 n^3 x^3 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+180 b^2 n^2+81}+\frac {8 b^4 n^4 x^3}{64 b^4 n^4+180 b^2 n^2+81} \]
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Rubi [A] time = 0.08, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4487, 30} \[ \frac {36 b^2 n^2 x^3 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+180 b^2 n^2+81}+\frac {3 x^3 \sin ^4\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+9}-\frac {4 b n x^3 \sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+9}-\frac {24 b^3 n^3 x^3 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+180 b^2 n^2+81}+\frac {8 b^4 n^4 x^3}{64 b^4 n^4+180 b^2 n^2+81} \]
Antiderivative was successfully verified.
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Rule 30
Rule 4487
Rubi steps
\begin {align*} \int x^2 \sin ^4\left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {4 b n x^3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{9+16 b^2 n^2}+\frac {3 x^3 \sin ^4\left (a+b \log \left (c x^n\right )\right )}{9+16 b^2 n^2}+\frac {\left (12 b^2 n^2\right ) \int x^2 \sin ^2\left (a+b \log \left (c x^n\right )\right ) \, dx}{9+16 b^2 n^2}\\ &=-\frac {24 b^3 n^3 x^3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{81+180 b^2 n^2+64 b^4 n^4}+\frac {36 b^2 n^2 x^3 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{81+180 b^2 n^2+64 b^4 n^4}-\frac {4 b n x^3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{9+16 b^2 n^2}+\frac {3 x^3 \sin ^4\left (a+b \log \left (c x^n\right )\right )}{9+16 b^2 n^2}+\frac {\left (24 b^4 n^4\right ) \int x^2 \, dx}{81+180 b^2 n^2+64 b^4 n^4}\\ &=\frac {8 b^4 n^4 x^3}{81+180 b^2 n^2+64 b^4 n^4}-\frac {24 b^3 n^3 x^3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{81+180 b^2 n^2+64 b^4 n^4}+\frac {36 b^2 n^2 x^3 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{81+180 b^2 n^2+64 b^4 n^4}-\frac {4 b n x^3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{9+16 b^2 n^2}+\frac {3 x^3 \sin ^4\left (a+b \log \left (c x^n\right )\right )}{9+16 b^2 n^2}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 171, normalized size = 0.85 \[ \frac {x^3 \left (-128 b^3 n^3 \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+16 b^3 n^3 \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )-12 \left (16 b^2 n^2+9\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+3 \left (4 b^2 n^2+9\right ) \cos \left (4 \left (a+b \log \left (c x^n\right )\right )\right )-72 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+36 b n \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+64 b^4 n^4+180 b^2 n^2+81\right )}{8 \left (64 b^4 n^4+180 b^2 n^2+81\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 178, normalized size = 0.88 \[ \frac {3 \, {\left (4 \, b^{2} n^{2} + 9\right )} x^{3} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} - 6 \, {\left (10 \, b^{2} n^{2} + 9\right )} x^{3} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + {\left (8 \, b^{4} n^{4} + 48 \, b^{2} n^{2} + 27\right )} x^{3} + 4 \, {\left ({\left (4 \, b^{3} n^{3} + 9 \, b n\right )} x^{3} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} - {\left (10 \, b^{3} n^{3} + 9 \, b n\right )} x^{3} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )} \sin \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{64 \, b^{4} n^{4} + 180 \, b^{2} n^{2} + 81} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int x^{2} \left (\sin ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 1107, normalized size = 5.48 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.12, size = 127, normalized size = 0.63 \[ \frac {x^3}{8}-\frac {x^3\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}}\,1{}\mathrm {i}}{8\,b\,n+12{}\mathrm {i}}-\frac {x^3\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}}{12+b\,n\,8{}\mathrm {i}}+\frac {x^3\,{\mathrm {e}}^{-a\,4{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}}\,1{}\mathrm {i}}{64\,b\,n+48{}\mathrm {i}}+\frac {x^3\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}}{48+b\,n\,64{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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