Optimal. Leaf size=98 \[ \frac {x^2 \cos ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (b^2 n^2+1\right )}+\frac {b n x^2 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{2 \left (b^2 n^2+1\right )}+\frac {b^2 n^2 x^2}{4 \left (b^2 n^2+1\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4488, 30} \[ \frac {x^2 \cos ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (b^2 n^2+1\right )}+\frac {b n x^2 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{2 \left (b^2 n^2+1\right )}+\frac {b^2 n^2 x^2}{4 \left (b^2 n^2+1\right )} \]
Antiderivative was successfully verified.
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Rule 30
Rule 4488
Rubi steps
\begin {align*} \int x \cos ^2\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {x^2 \cos ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+b^2 n^2\right )}+\frac {b n x^2 \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+b^2 n^2\right )}+\frac {\left (b^2 n^2\right ) \int x \, dx}{2 \left (1+b^2 n^2\right )}\\ &=\frac {b^2 n^2 x^2}{4 \left (1+b^2 n^2\right )}+\frac {x^2 \cos ^2\left (a+b \log \left (c x^n\right )\right )}{2 \left (1+b^2 n^2\right )}+\frac {b n x^2 \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+b^2 n^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 54, normalized size = 0.55 \[ \frac {x^2 \left (b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+b^2 n^2+1\right )}{4 b^2 n^2+4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 74, normalized size = 0.76 \[ \frac {b^{2} n^{2} x^{2} + 2 \, b n x^{2} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right ) \sin \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + 2 \, x^{2} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2}}{4 \, {\left (b^{2} n^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.51, size = 820, normalized size = 8.37 \[ \text {result too large to display} \]
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 \[ \int x \left (\cos ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.37, size = 282, normalized size = 2.88 \[ \frac {{\left ({\left (b \cos \left (2 \, b \log \relax (c)\right ) \sin \left (4 \, b \log \relax (c)\right ) - b \cos \left (4 \, b \log \relax (c)\right ) \sin \left (2 \, b \log \relax (c)\right ) + b \sin \left (2 \, b \log \relax (c)\right )\right )} n + \cos \left (4 \, b \log \relax (c)\right ) \cos \left (2 \, b \log \relax (c)\right ) + \sin \left (4 \, b \log \relax (c)\right ) \sin \left (2 \, b \log \relax (c)\right ) + \cos \left (2 \, b \log \relax (c)\right )\right )} x^{2} \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) + {\left ({\left (b \cos \left (4 \, b \log \relax (c)\right ) \cos \left (2 \, b \log \relax (c)\right ) + b \sin \left (4 \, b \log \relax (c)\right ) \sin \left (2 \, b \log \relax (c)\right ) + b \cos \left (2 \, b \log \relax (c)\right )\right )} n - \cos \left (2 \, b \log \relax (c)\right ) \sin \left (4 \, b \log \relax (c)\right ) + \cos \left (4 \, b \log \relax (c)\right ) \sin \left (2 \, b \log \relax (c)\right ) - \sin \left (2 \, b \log \relax (c)\right )\right )} x^{2} \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) + 2 \, {\left ({\left (b^{2} \cos \left (2 \, b \log \relax (c)\right )^{2} + b^{2} \sin \left (2 \, b \log \relax (c)\right )^{2}\right )} n^{2} + \cos \left (2 \, b \log \relax (c)\right )^{2} + \sin \left (2 \, b \log \relax (c)\right )^{2}\right )} x^{2}}{8 \, {\left ({\left (b^{2} \cos \left (2 \, b \log \relax (c)\right )^{2} + b^{2} \sin \left (2 \, b \log \relax (c)\right )^{2}\right )} n^{2} + \cos \left (2 \, b \log \relax (c)\right )^{2} + \sin \left (2 \, b \log \relax (c)\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.63, size = 66, normalized size = 0.67 \[ \frac {x^2}{4}+\frac {x^2\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}}\,1{}\mathrm {i}}{8\,b\,n+8{}\mathrm {i}}+\frac {x^2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}}{8+b\,n\,8{}\mathrm {i}} \]
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 \[ \begin {cases} \int x \cos ^{2}{\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = - \frac {i}{n} \\\int x \cos ^{2}{\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {i}{n} \\\frac {b^{2} n^{2} x^{2} \sin ^{2}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{4 b^{2} n^{2} + 4} + \frac {b^{2} n^{2} x^{2} \cos ^{2}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{4 b^{2} n^{2} + 4} + \frac {2 b n x^{2} \sin {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )} \cos {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{4 b^{2} n^{2} + 4} + \frac {2 x^{2} \cos ^{2}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{4 b^{2} n^{2} + 4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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