Optimal. Leaf size=88 \[ \frac {x \sin ^2\left (a+b \log \left (c x^n\right )\right )}{4 b^2 n^2+1}-\frac {2 b n x \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{4 b^2 n^2+1}+\frac {2 b^2 n^2 x}{4 b^2 n^2+1} \]
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Rubi [A] time = 0.02, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4477, 8} \[ \frac {x \sin ^2\left (a+b \log \left (c x^n\right )\right )}{4 b^2 n^2+1}-\frac {2 b n x \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{4 b^2 n^2+1}+\frac {2 b^2 n^2 x}{4 b^2 n^2+1} \]
Antiderivative was successfully verified.
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Rule 8
Rule 4477
Rubi steps
\begin {align*} \int \sin ^2\left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {2 b n x \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+4 b^2 n^2}+\frac {x \sin ^2\left (a+b \log \left (c x^n\right )\right )}{1+4 b^2 n^2}+\frac {\left (2 b^2 n^2\right ) \int 1 \, dx}{1+4 b^2 n^2}\\ &=\frac {2 b^2 n^2 x}{1+4 b^2 n^2}-\frac {2 b n x \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+4 b^2 n^2}+\frac {x \sin ^2\left (a+b \log \left (c x^n\right )\right )}{1+4 b^2 n^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 56, normalized size = 0.64 \[ \frac {x \left (-2 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-\cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+4 b^2 n^2+1\right )}{8 b^2 n^2+2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 73, normalized size = 0.83 \[ -\frac {2 \, b n x \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sin \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + x \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} - {\left (2 \, b^{2} n^{2} + 1\right )} x}{4 \, b^{2} n^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 786, normalized size = 8.93 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \sin ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 280, normalized size = 3.18 \[ -\frac {{\left (2 \, {\left (b \cos \left (2 \, b \log \relax (c)\right ) \sin \left (4 \, b \log \relax (c)\right ) - b \cos \left (4 \, b \log \relax (c)\right ) \sin \left (2 \, b \log \relax (c)\right ) + b \sin \left (2 \, b \log \relax (c)\right )\right )} n + \cos \left (4 \, b \log \relax (c)\right ) \cos \left (2 \, b \log \relax (c)\right ) + \sin \left (4 \, b \log \relax (c)\right ) \sin \left (2 \, b \log \relax (c)\right ) + \cos \left (2 \, b \log \relax (c)\right )\right )} x \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) + {\left (2 \, {\left (b \cos \left (4 \, b \log \relax (c)\right ) \cos \left (2 \, b \log \relax (c)\right ) + b \sin \left (4 \, b \log \relax (c)\right ) \sin \left (2 \, b \log \relax (c)\right ) + b \cos \left (2 \, b \log \relax (c)\right )\right )} n - \cos \left (2 \, b \log \relax (c)\right ) \sin \left (4 \, b \log \relax (c)\right ) + \cos \left (4 \, b \log \relax (c)\right ) \sin \left (2 \, b \log \relax (c)\right ) - \sin \left (2 \, b \log \relax (c)\right )\right )} x \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) - 2 \, {\left (4 \, {\left (b^{2} \cos \left (2 \, b \log \relax (c)\right )^{2} + b^{2} \sin \left (2 \, b \log \relax (c)\right )^{2}\right )} n^{2} + \cos \left (2 \, b \log \relax (c)\right )^{2} + \sin \left (2 \, b \log \relax (c)\right )^{2}\right )} x}{4 \, {\left (4 \, {\left (b^{2} \cos \left (2 \, b \log \relax (c)\right )^{2} + b^{2} \sin \left (2 \, b \log \relax (c)\right )^{2}\right )} n^{2} + \cos \left (2 \, b \log \relax (c)\right )^{2} + \sin \left (2 \, b \log \relax (c)\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.47, size = 56, normalized size = 0.64 \[ \frac {x\,\left (2\,{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^2+4\,b^2\,n^2-2\,b\,n\,\sin \left (2\,a+2\,b\,\ln \left (c\,x^n\right )\right )\right )}{8\,b^2\,n^2+2} \]
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 \[ \begin {cases} \int \sin ^{2}{\left (a - \frac {i \log {\left (c x^{n} \right )}}{2 n} \right )}\, dx & \text {for}\: b = - \frac {i}{2 n} \\\int \sin ^{2}{\left (a + \frac {i \log {\left (c x^{n} \right )}}{2 n} \right )}\, dx & \text {for}\: b = \frac {i}{2 n} \\\frac {2 b^{2} n^{2} x \sin ^{2}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{4 b^{2} n^{2} + 1} + \frac {2 b^{2} n^{2} x \cos ^{2}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{4 b^{2} n^{2} + 1} - \frac {2 b n x \sin {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )} \cos {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{4 b^{2} n^{2} + 1} + \frac {x \sin ^{2}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{4 b^{2} n^{2} + 1} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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