Optimal. Leaf size=210 \[ -\frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{2 x^2 \left (4 b^2 n^2+1\right )}-\frac {b n \sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x^2 \left (4 b^2 n^2+1\right )}-\frac {3 b^2 n^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{2 x^2 \left (4 b^4 n^4+5 b^2 n^2+1\right )}-\frac {3 b^3 n^3 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{2 x^2 \left (4 b^4 n^4+5 b^2 n^2+1\right )}-\frac {3 b^4 n^4}{4 x^2 \left (4 b^4 n^4+5 b^2 n^2+1\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 210, 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} \[ -\frac {3 b^2 n^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{2 x^2 \left (4 b^4 n^4+5 b^2 n^2+1\right )}-\frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{2 x^2 \left (4 b^2 n^2+1\right )}-\frac {b n \sin ^3\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x^2 \left (4 b^2 n^2+1\right )}-\frac {3 b^3 n^3 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{2 x^2 \left (4 b^4 n^4+5 b^2 n^2+1\right )}-\frac {3 b^4 n^4}{4 x^2 \left (4 b^4 n^4+5 b^2 n^2+1\right )} \]
Antiderivative was successfully verified.
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Rule 30
Rule 4487
Rubi steps
\begin {align*} \int \frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx &=-\frac {b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x^2}-\frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+4 b^2 n^2\right ) x^2}+\frac {\left (3 b^2 n^2\right ) \int \frac {\sin ^2\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx}{1+4 b^2 n^2}\\ &=-\frac {3 b^3 n^3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{2 \left (1+5 b^2 n^2+4 b^4 n^4\right ) x^2}-\frac {3 b^2 n^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+5 b^2 n^2+4 b^4 n^4\right ) x^2}-\frac {b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x^2}-\frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+4 b^2 n^2\right ) x^2}+\frac {\left (3 b^4 n^4\right ) \int \frac {1}{x^3} \, dx}{2 \left (1+5 b^2 n^2+4 b^4 n^4\right )}\\ &=-\frac {3 b^4 n^4}{4 \left (1+5 b^2 n^2+4 b^4 n^4\right ) x^2}-\frac {3 b^3 n^3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{2 \left (1+5 b^2 n^2+4 b^4 n^4\right ) x^2}-\frac {3 b^2 n^2 \sin ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+5 b^2 n^2+4 b^4 n^4\right ) x^2}-\frac {b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin ^3\left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x^2}-\frac {\sin ^4\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+4 b^2 n^2\right ) x^2}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 169, normalized size = 0.80 \[ -\frac {16 b^3 n^3 \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-2 b^3 n^3 \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )-4 \left (4 b^2 n^2+1\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\left (b^2 n^2+1\right ) \cos \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+4 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-2 b n \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+12 b^4 n^4+15 b^2 n^2+3}{16 x^2 \left (4 b^4 n^4+5 b^2 n^2+1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 163, normalized size = 0.78 \[ -\frac {3 \, b^{4} n^{4} + 2 \, {\left (b^{2} n^{2} + 1\right )} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4} + 8 \, b^{2} n^{2} - 2 \, {\left (5 \, b^{2} n^{2} + 2\right )} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - 2 \, {\left (2 \, {\left (b^{3} n^{3} + b n\right )} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3} - {\left (5 \, b^{3} n^{3} + 2 \, b n\right )} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )} \sin \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + 2}{4 \, {\left (4 \, b^{4} n^{4} + 5 \, b^{2} n^{2} + 1\right )} x^{2}} \]
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 \[ \int \frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{4}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 1082, normalized size = 5.15 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^4}{x^3} \,d x \]
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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