Integrand size = 7, antiderivative size = 84 \[ \int d^x x \sin (x) \, dx=\frac {2 d^x \cos (x) \log (d)}{\left (1+\log ^2(d)\right )^2}-\frac {d^x x \cos (x)}{1+\log ^2(d)}+\frac {d^x \sin (x)}{\left (1+\log ^2(d)\right )^2}-\frac {d^x \log ^2(d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x \log (d) \sin (x)}{1+\log ^2(d)} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4517, 4553, 4518} \[ \int d^x x \sin (x) \, dx=\frac {x d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac {d^x \log ^2(d) \sin (x)}{\left (\log ^2(d)+1\right )^2}+\frac {d^x \sin (x)}{\left (\log ^2(d)+1\right )^2}-\frac {x d^x \cos (x)}{\log ^2(d)+1}+\frac {2 d^x \log (d) \cos (x)}{\left (\log ^2(d)+1\right )^2} \]
[In]
[Out]
Rule 4517
Rule 4518
Rule 4553
Rubi steps \begin{align*} \text {integral}& = -\frac {d^x x \cos (x)}{1+\log ^2(d)}+\frac {d^x x \log (d) \sin (x)}{1+\log ^2(d)}-\int \left (-\frac {d^x \cos (x)}{1+\log ^2(d)}+\frac {d^x \log (d) \sin (x)}{1+\log ^2(d)}\right ) \, dx \\ & = -\frac {d^x x \cos (x)}{1+\log ^2(d)}+\frac {d^x x \log (d) \sin (x)}{1+\log ^2(d)}+\frac {\int d^x \cos (x) \, dx}{1+\log ^2(d)}-\frac {\log (d) \int d^x \sin (x) \, dx}{1+\log ^2(d)} \\ & = \frac {2 d^x \cos (x) \log (d)}{\left (1+\log ^2(d)\right )^2}-\frac {d^x x \cos (x)}{1+\log ^2(d)}+\frac {d^x \sin (x)}{\left (1+\log ^2(d)\right )^2}-\frac {d^x \log ^2(d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x \log (d) \sin (x)}{1+\log ^2(d)} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.60 \[ \int d^x x \sin (x) \, dx=\frac {d^x \left (-\cos (x) \left (x-2 \log (d)+x \log ^2(d)\right )+\left (1+x \log (d)-\log ^2(d)+x \log ^3(d)\right ) \sin (x)\right )}{\left (1+\log ^2(d)\right )^2} \]
[In]
[Out]
Time = 0.19 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.67
method | result | size |
parallelrisch | \(\frac {d^{x} \left (\ln \left (d \right )^{3} x \sin \left (x \right )+\left (-x \cos \left (x \right )-\sin \left (x \right )\right ) \ln \left (d \right )^{2}+\left (x \sin \left (x \right )+2 \cos \left (x \right )\right ) \ln \left (d \right )-x \cos \left (x \right )+\sin \left (x \right )\right )}{\left (1+\ln \left (d \right )^{2}\right )^{2}}\) | \(56\) |
risch | \(-\frac {i \left (-1+x \ln \left (d \right )+i x \right ) d^{x} {\mathrm e}^{i x}}{2 \left (\ln \left (d \right )+i\right )^{2}}+\frac {i \left (-1+x \ln \left (d \right )-i x \right ) d^{x} {\mathrm e}^{-i x}}{2 \left (\ln \left (d \right )-i\right )^{2}}\) | \(58\) |
norman | \(\frac {\frac {x \,{\mathrm e}^{x \ln \left (d \right )} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{1+\ln \left (d \right )^{2}}+\frac {2 \ln \left (d \right ) {\mathrm e}^{x \ln \left (d \right )}}{\left (1+\ln \left (d \right )^{2}\right )^{2}}-\frac {x \,{\mathrm e}^{x \ln \left (d \right )}}{1+\ln \left (d \right )^{2}}-\frac {2 \ln \left (d \right ) {\mathrm e}^{x \ln \left (d \right )} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{\left (1+\ln \left (d \right )^{2}\right )^{2}}-\frac {2 \left (\ln \left (d \right )^{2}-1\right ) {\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{\left (1+\ln \left (d \right )^{2}\right )^{2}}+\frac {2 \ln \left (d \right ) x \,{\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{1+\ln \left (d \right )^{2}}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\) | \(137\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71 \[ \int d^x x \sin (x) \, dx=-\frac {{\left (x \cos \left (x\right ) \log \left (d\right )^{2} + x \cos \left (x\right ) - 2 \, \cos \left (x\right ) \log \left (d\right ) - {\left (x \log \left (d\right )^{3} + x \log \left (d\right ) - \log \left (d\right )^{2} + 1\right )} \sin \left (x\right )\right )} d^{x}}{\log \left (d\right )^{4} + 2 \, \log \left (d\right )^{2} + 1} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.56 (sec) , antiderivative size = 308, normalized size of antiderivative = 3.67 \[ \int d^x x \sin (x) \, dx=\begin {cases} \frac {x^{2} e^{- i x} \sin {\left (x \right )}}{4} - \frac {i x^{2} e^{- i x} \cos {\left (x \right )}}{4} + \frac {i x e^{- i x} \sin {\left (x \right )}}{4} - \frac {x e^{- i x} \cos {\left (x \right )}}{4} + \frac {i e^{- i x} \cos {\left (x \right )}}{4} & \text {for}\: d = e^{- i} \\\frac {x^{2} e^{i x} \sin {\left (x \right )}}{4} + \frac {i x^{2} e^{i x} \cos {\left (x \right )}}{4} - \frac {i x e^{i x} \sin {\left (x \right )}}{4} - \frac {x e^{i x} \cos {\left (x \right )}}{4} - \frac {i e^{i x} \cos {\left (x \right )}}{4} & \text {for}\: d = e^{i} \\\frac {d^{x} x \log {\left (d \right )}^{3} \sin {\left (x \right )}}{\log {\left (d \right )}^{4} + 2 \log {\left (d \right )}^{2} + 1} - \frac {d^{x} x \log {\left (d \right )}^{2} \cos {\left (x \right )}}{\log {\left (d \right )}^{4} + 2 \log {\left (d \right )}^{2} + 1} + \frac {d^{x} x \log {\left (d \right )} \sin {\left (x \right )}}{\log {\left (d \right )}^{4} + 2 \log {\left (d \right )}^{2} + 1} - \frac {d^{x} x \cos {\left (x \right )}}{\log {\left (d \right )}^{4} + 2 \log {\left (d \right )}^{2} + 1} - \frac {d^{x} \log {\left (d \right )}^{2} \sin {\left (x \right )}}{\log {\left (d \right )}^{4} + 2 \log {\left (d \right )}^{2} + 1} + \frac {2 d^{x} \log {\left (d \right )} \cos {\left (x \right )}}{\log {\left (d \right )}^{4} + 2 \log {\left (d \right )}^{2} + 1} + \frac {d^{x} \sin {\left (x \right )}}{\log {\left (d \right )}^{4} + 2 \log {\left (d \right )}^{2} + 1} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71 \[ \int d^x x \sin (x) \, dx=-\frac {{\left ({\left (\log \left (d\right )^{2} + 1\right )} x - 2 \, \log \left (d\right )\right )} d^{x} \cos \left (x\right ) - {\left ({\left (\log \left (d\right )^{3} + \log \left (d\right )\right )} x - \log \left (d\right )^{2} + 1\right )} d^{x} \sin \left (x\right )}{\log \left (d\right )^{4} + 2 \, \log \left (d\right )^{2} + 1} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 1156, normalized size of antiderivative = 13.76 \[ \int d^x x \sin (x) \, dx=\text {Too large to display} \]
[In]
[Out]
Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.68 \[ \int d^x x \sin (x) \, dx=\frac {d^x\,\left (\sin \left (x\right )+2\,\ln \left (d\right )\,\cos \left (x\right )-{\ln \left (d\right )}^2\,\sin \left (x\right )-x\,\cos \left (x\right )+x\,\ln \left (d\right )\,\sin \left (x\right )-x\,{\ln \left (d\right )}^2\,\cos \left (x\right )+x\,{\ln \left (d\right )}^3\,\sin \left (x\right )\right )}{{\left ({\ln \left (d\right )}^2+1\right )}^2} \]
[In]
[Out]