Optimal. Leaf size=83 \[ \frac {x d^x \sin (x)}{\log ^2(d)+1}-\frac {2 d^x \log (d) \sin (x)}{\left (\log ^2(d)+1\right )^2}+\frac {x d^x \log (d) \cos (x)}{\log ^2(d)+1}-\frac {d^x \log ^2(d) \cos (x)}{\left (\log ^2(d)+1\right )^2}+\frac {d^x \cos (x)}{\left (\log ^2(d)+1\right )^2} \]
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Rubi [A] time = 0.05, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4433, 4466, 4432} \[ \frac {x d^x \sin (x)}{\log ^2(d)+1}-\frac {2 d^x \log (d) \sin (x)}{\left (\log ^2(d)+1\right )^2}+\frac {x d^x \log (d) \cos (x)}{\log ^2(d)+1}-\frac {d^x \log ^2(d) \cos (x)}{\left (\log ^2(d)+1\right )^2}+\frac {d^x \cos (x)}{\left (\log ^2(d)+1\right )^2} \]
Antiderivative was successfully verified.
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Rule 4432
Rule 4433
Rule 4466
Rubi steps
\begin {align*} \int d^x x \cos (x) \, dx &=\frac {d^x x \cos (x) \log (d)}{1+\log ^2(d)}+\frac {d^x x \sin (x)}{1+\log ^2(d)}-\int \left (\frac {d^x \cos (x) \log (d)}{1+\log ^2(d)}+\frac {d^x \sin (x)}{1+\log ^2(d)}\right ) \, dx\\ &=\frac {d^x x \cos (x) \log (d)}{1+\log ^2(d)}+\frac {d^x x \sin (x)}{1+\log ^2(d)}-\frac {\int d^x \sin (x) \, dx}{1+\log ^2(d)}-\frac {\log (d) \int d^x \cos (x) \, dx}{1+\log ^2(d)}\\ &=\frac {d^x \cos (x)}{\left (1+\log ^2(d)\right )^2}-\frac {d^x \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x \cos (x) \log (d)}{1+\log ^2(d)}-\frac {2 d^x \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x \sin (x)}{1+\log ^2(d)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 49, normalized size = 0.59 \[ \frac {d^x \left (\sin (x) \left (x \log ^2(d)-2 \log (d)+x\right )+\cos (x) \left (x \log ^3(d)+x \log (d)-\log ^2(d)+1\right )\right )}{\left (\log ^2(d)+1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 58, normalized size = 0.70 \[ \frac {{\left (x \cos \relax (x) \log \relax (d)^{3} + x \cos \relax (x) \log \relax (d) - \cos \relax (x) \log \relax (d)^{2} + {\left (x \log \relax (d)^{2} + x - 2 \, \log \relax (d)\right )} \sin \relax (x) + \cos \relax (x)\right )} d^{x}}{\log \relax (d)^{4} + 2 \, \log \relax (d)^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 1.29, size = 1165, normalized size = 14.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 142, normalized size = 1.71 \[ \frac {-\frac {x \,{\mathrm e}^{x \ln \relax (d )} \ln \relax (d ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{\ln \relax (d )^{2}+1}+\frac {x \,{\mathrm e}^{x \ln \relax (d )} \ln \relax (d )}{\ln \relax (d )^{2}+1}+\frac {2 x \,{\mathrm e}^{x \ln \relax (d )} \tan \left (\frac {x}{2}\right )}{\ln \relax (d )^{2}+1}-\frac {4 \,{\mathrm e}^{x \ln \relax (d )} \ln \relax (d ) \tan \left (\frac {x}{2}\right )}{\left (\ln \relax (d )^{2}+1\right )^{2}}+\frac {\left (\ln \relax (d )^{2}-1\right ) {\mathrm e}^{x \ln \relax (d )} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{\left (\ln \relax (d )^{2}+1\right )^{2}}-\frac {\left (\ln \relax (d )^{2}-1\right ) {\mathrm e}^{x \ln \relax (d )}}{\left (\ln \relax (d )^{2}+1\right )^{2}}}{\tan ^{2}\left (\frac {x}{2}\right )+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 58, normalized size = 0.70 \[ \frac {{\left ({\left (\log \relax (d)^{3} + \log \relax (d)\right )} x - \log \relax (d)^{2} + 1\right )} d^{x} \cos \relax (x) + {\left ({\left (\log \relax (d)^{2} + 1\right )} x - 2 \, \log \relax (d)\right )} d^{x} \sin \relax (x)}{\log \relax (d)^{4} + 2 \, \log \relax (d)^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 55, normalized size = 0.66 \[ \frac {d^x\,\left (\cos \relax (x)-2\,\ln \relax (d)\,\sin \relax (x)-{\ln \relax (d)}^2\,\cos \relax (x)+x\,\sin \relax (x)+x\,\ln \relax (d)\,\cos \relax (x)+x\,{\ln \relax (d)}^3\,\cos \relax (x)+x\,{\ln \relax (d)}^2\,\sin \relax (x)\right )}{{\left ({\ln \relax (d)}^2+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.24, size = 304, normalized size = 3.66 \[ \begin {cases} \frac {i x^{2} e^{- i x} \sin {\relax (x )}}{4} + \frac {x^{2} e^{- i x} \cos {\relax (x )}}{4} + \frac {x e^{- i x} \sin {\relax (x )}}{4} + \frac {i x e^{- i x} \cos {\relax (x )}}{4} + \frac {e^{- i x} \cos {\relax (x )}}{4} & \text {for}\: d = e^{- i} \\- \frac {i x^{2} e^{i x} \sin {\relax (x )}}{4} + \frac {x^{2} e^{i x} \cos {\relax (x )}}{4} + \frac {x e^{i x} \sin {\relax (x )}}{4} - \frac {i x e^{i x} \cos {\relax (x )}}{4} + \frac {e^{i x} \cos {\relax (x )}}{4} & \text {for}\: d = e^{i} \\\frac {d^{x} x \log {\relax (d )}^{3} \cos {\relax (x )}}{\log {\relax (d )}^{4} + 2 \log {\relax (d )}^{2} + 1} + \frac {d^{x} x \log {\relax (d )}^{2} \sin {\relax (x )}}{\log {\relax (d )}^{4} + 2 \log {\relax (d )}^{2} + 1} + \frac {d^{x} x \log {\relax (d )} \cos {\relax (x )}}{\log {\relax (d )}^{4} + 2 \log {\relax (d )}^{2} + 1} + \frac {d^{x} x \sin {\relax (x )}}{\log {\relax (d )}^{4} + 2 \log {\relax (d )}^{2} + 1} - \frac {d^{x} \log {\relax (d )}^{2} \cos {\relax (x )}}{\log {\relax (d )}^{4} + 2 \log {\relax (d )}^{2} + 1} - \frac {2 d^{x} \log {\relax (d )} \sin {\relax (x )}}{\log {\relax (d )}^{4} + 2 \log {\relax (d )}^{2} + 1} + \frac {d^{x} \cos {\relax (x )}}{\log {\relax (d )}^{4} + 2 \log {\relax (d )}^{2} + 1} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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