Optimal. Leaf size=57 \[ \frac {2 x^2 \sin \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+4}-\frac {b n x^2 \cos \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {4485} \[ \frac {2 x^2 \sin \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+4}-\frac {b n x^2 \cos \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4485
Rubi steps
\begin {align*} \int x \sin \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {b n x^2 \cos \left (a+b \log \left (c x^n\right )\right )}{4+b^2 n^2}+\frac {2 x^2 \sin \left (a+b \log \left (c x^n\right )\right )}{4+b^2 n^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 44, normalized size = 0.77 \[ -\frac {x^2 \left (b n \cos \left (a+b \log \left (c x^n\right )\right )-2 \sin \left (a+b \log \left (c x^n\right )\right )\right )}{b^2 n^2+4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.84, size = 49, normalized size = 0.86 \[ -\frac {b n x^{2} \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right ) - 2 \, x^{2} \sin \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{b^{2} n^{2} + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.65, size = 923, normalized size = 16.19 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int x \sin \left (a +b \ln \left (c \,x^{n}\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.36, size = 219, normalized size = 3.84 \[ -\frac {{\left ({\left (b \cos \left (2 \, b \log \relax (c)\right ) \cos \left (b \log \relax (c)\right ) + b \sin \left (2 \, b \log \relax (c)\right ) \sin \left (b \log \relax (c)\right ) + b \cos \left (b \log \relax (c)\right )\right )} n - 2 \, \cos \left (b \log \relax (c)\right ) \sin \left (2 \, b \log \relax (c)\right ) + 2 \, \cos \left (2 \, b \log \relax (c)\right ) \sin \left (b \log \relax (c)\right ) - 2 \, \sin \left (b \log \relax (c)\right )\right )} x^{2} \cos \left (b \log \left (x^{n}\right ) + a\right ) - {\left ({\left (b \cos \left (b \log \relax (c)\right ) \sin \left (2 \, b \log \relax (c)\right ) - b \cos \left (2 \, b \log \relax (c)\right ) \sin \left (b \log \relax (c)\right ) + b \sin \left (b \log \relax (c)\right )\right )} n + 2 \, \cos \left (2 \, b \log \relax (c)\right ) \cos \left (b \log \relax (c)\right ) + 2 \, \sin \left (2 \, b \log \relax (c)\right ) \sin \left (b \log \relax (c)\right ) + 2 \, \cos \left (b \log \relax (c)\right )\right )} x^{2} \sin \left (b \log \left (x^{n}\right ) + a\right )}{2 \, {\left ({\left (b^{2} \cos \left (b \log \relax (c)\right )^{2} + b^{2} \sin \left (b \log \relax (c)\right )^{2}\right )} n^{2} + 4 \, \cos \left (b \log \relax (c)\right )^{2} + 4 \, \sin \left (b \log \relax (c)\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.39, size = 44, normalized size = 0.77 \[ \frac {x^2\,\left (2\,\sin \left (a+b\,\ln \left (c\,x^n\right )\right )-b\,n\,\cos \left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{b^2\,n^2+4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \int x \sin {\left (a - \frac {2 i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = - \frac {2 i}{n} \\\int x \sin {\left (a + \frac {2 i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {2 i}{n} \\- \frac {b n x^{2} \cos {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{b^{2} n^{2} + 4} + \frac {2 x^{2} \sin {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{b^{2} n^{2} + 4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________