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 = 0.0455074, antiderivative size = 83, normalized size of antiderivative = 1., 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 4433
Rule 4466
Rule 4432
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*}
Mathematica [A] time = 0.0447757, 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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Maple [A] time = 0.02, size = 142, normalized size = 1.7 \begin{align*}{ \left ({\frac{x\ln \left ( d \right ){{\rm e}^{x\ln \left ( d \right ) }}}{1+ \left ( \ln \left ( d \right ) \right ) ^{2}}}+{\frac{ \left ( \left ( \ln \left ( d \right ) \right ) ^{2}-1 \right ){{\rm e}^{x\ln \left ( d \right ) }}}{ \left ( 1+ \left ( \ln \left ( d \right ) \right ) ^{2} \right ) ^{2}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{ \left ( \left ( \ln \left ( d \right ) \right ) ^{2}-1 \right ){{\rm e}^{x\ln \left ( d \right ) }}}{ \left ( 1+ \left ( \ln \left ( d \right ) \right ) ^{2} \right ) ^{2}}}-4\,{\frac{\ln \left ( d \right ){{\rm e}^{x\ln \left ( d \right ) }}\tan \left ( x/2 \right ) }{ \left ( 1+ \left ( \ln \left ( d \right ) \right ) ^{2} \right ) ^{2}}}+2\,{\frac{x{{\rm e}^{x\ln \left ( d \right ) }}\tan \left ( x/2 \right ) }{1+ \left ( \ln \left ( d \right ) \right ) ^{2}}}-{\frac{x\ln \left ( d \right ){{\rm e}^{x\ln \left ( d \right ) }}}{1+ \left ( \ln \left ( d \right ) \right ) ^{2}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.978334, size = 78, normalized size = 0.94 \begin{align*} \frac{{\left ({\left (\log \left (d\right )^{3} + \log \left (d\right )\right )} x - \log \left (d\right )^{2} + 1\right )} d^{x} \cos \left (x\right ) +{\left ({\left (\log \left (d\right )^{2} + 1\right )} x - 2 \, \log \left (d\right )\right )} d^{x} \sin \left (x\right )}{\log \left (d\right )^{4} + 2 \, \log \left (d\right )^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77531, size = 182, normalized size = 2.19 \begin{align*} \frac{{\left (x \cos \left (x\right ) \log \left (d\right )^{3} + x \cos \left (x\right ) \log \left (d\right ) - \cos \left (x\right ) \log \left (d\right )^{2} +{\left (x \log \left (d\right )^{2} + x - 2 \, \log \left (d\right )\right )} \sin \left (x\right ) + \cos \left (x\right )\right )} d^{x}}{\log \left (d\right )^{4} + 2 \, \log \left (d\right )^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.552, size = 304, normalized size = 3.66 \begin{align*} \begin{cases} \frac{i x^{2} e^{- i x} \sin{\left (x \right )}}{4} + \frac{x^{2} e^{- i x} \cos{\left (x \right )}}{4} + \frac{x e^{- i x} \sin{\left (x \right )}}{4} + \frac{i x e^{- i x} \cos{\left (x \right )}}{4} + \frac{e^{- i x} \cos{\left (x \right )}}{4} & \text{for}\: d = e^{- i} \\- \frac{i x^{2} e^{i x} \sin{\left (x \right )}}{4} + \frac{x^{2} e^{i x} \cos{\left (x \right )}}{4} + \frac{x e^{i x} \sin{\left (x \right )}}{4} - \frac{i x e^{i x} \cos{\left (x \right )}}{4} + \frac{e^{i x} \cos{\left (x \right )}}{4} & \text{for}\: d = e^{i} \\\frac{d^{x} x \log{\left (d \right )}^{3} \cos{\left (x \right )}}{\log{\left (d \right )}^{4} + 2 \log{\left (d \right )}^{2} + 1} + \frac{d^{x} x \log{\left (d \right )}^{2} \sin{\left (x \right )}}{\log{\left (d \right )}^{4} + 2 \log{\left (d \right )}^{2} + 1} + \frac{d^{x} x \log{\left (d \right )} \cos{\left (x \right )}}{\log{\left (d \right )}^{4} + 2 \log{\left (d \right )}^{2} + 1} + \frac{d^{x} x \sin{\left (x \right )}}{\log{\left (d \right )}^{4} + 2 \log{\left (d \right )}^{2} + 1} - \frac{d^{x} \log{\left (d \right )}^{2} \cos{\left (x \right )}}{\log{\left (d \right )}^{4} + 2 \log{\left (d \right )}^{2} + 1} - \frac{2 d^{x} \log{\left (d \right )} \sin{\left (x \right )}}{\log{\left (d \right )}^{4} + 2 \log{\left (d \right )}^{2} + 1} + \frac{d^{x} \cos{\left (x \right )}}{\log{\left (d \right )}^{4} + 2 \log{\left (d \right )}^{2} + 1} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.15571, size = 1573, normalized size = 18.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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