Optimal. Leaf size=161 \[ -\frac {6 d^x \cos (x) \log (d)}{\left (1+\log ^2(d)\right )^3}+\frac {2 d^x \cos (x) \log ^3(d)}{\left (1+\log ^2(d)\right )^3}+\frac {2 d^x x \cos (x)}{\left (1+\log ^2(d)\right )^2}-\frac {2 d^x x \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^2 \cos (x) \log (d)}{1+\log ^2(d)}-\frac {2 d^x \sin (x)}{\left (1+\log ^2(d)\right )^3}+\frac {6 d^x \log ^2(d) \sin (x)}{\left (1+\log ^2(d)\right )^3}-\frac {4 d^x x \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^2 \sin (x)}{1+\log ^2(d)} \]
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Rubi [A]
time = 0.13, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 5, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4518, 4554, 14,
4517, 4553} \begin {gather*} \frac {x^2 d^x \sin (x)}{\log ^2(d)+1}+\frac {x^2 d^x \log (d) \cos (x)}{\log ^2(d)+1}-\frac {4 x d^x \log (d) \sin (x)}{\left (\log ^2(d)+1\right )^2}+\frac {6 d^x \log ^2(d) \sin (x)}{\left (\log ^2(d)+1\right )^3}-\frac {2 d^x \sin (x)}{\left (\log ^2(d)+1\right )^3}-\frac {2 x d^x \log ^2(d) \cos (x)}{\left (\log ^2(d)+1\right )^2}+\frac {2 x d^x \cos (x)}{\left (\log ^2(d)+1\right )^2}-\frac {6 d^x \log (d) \cos (x)}{\left (\log ^2(d)+1\right )^3}+\frac {2 d^x \log ^3(d) \cos (x)}{\left (\log ^2(d)+1\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 4517
Rule 4518
Rule 4553
Rule 4554
Rubi steps
\begin {align*} \int d^x x^2 \cos (x) \, dx &=\frac {d^x x^2 \cos (x) \log (d)}{1+\log ^2(d)}+\frac {d^x x^2 \sin (x)}{1+\log ^2(d)}-2 \int x \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^2 \cos (x) \log (d)}{1+\log ^2(d)}+\frac {d^x x^2 \sin (x)}{1+\log ^2(d)}-2 \int \left (\frac {d^x x \cos (x) \log (d)}{1+\log ^2(d)}+\frac {d^x x \sin (x)}{1+\log ^2(d)}\right ) \, dx\\ &=\frac {d^x x^2 \cos (x) \log (d)}{1+\log ^2(d)}+\frac {d^x x^2 \sin (x)}{1+\log ^2(d)}-\frac {2 \int d^x x \sin (x) \, dx}{1+\log ^2(d)}-\frac {(2 \log (d)) \int d^x x \cos (x) \, dx}{1+\log ^2(d)}\\ &=\frac {2 d^x x \cos (x)}{\left (1+\log ^2(d)\right )^2}-\frac {2 d^x x \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^2 \cos (x) \log (d)}{1+\log ^2(d)}-\frac {4 d^x x \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^2 \sin (x)}{1+\log ^2(d)}+\frac {2 \int \left (-\frac {d^x \cos (x)}{1+\log ^2(d)}+\frac {d^x \log (d) \sin (x)}{1+\log ^2(d)}\right ) \, dx}{1+\log ^2(d)}+\frac {(2 \log (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}{1+\log ^2(d)}\\ &=\frac {2 d^x x \cos (x)}{\left (1+\log ^2(d)\right )^2}-\frac {2 d^x x \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^2 \cos (x) \log (d)}{1+\log ^2(d)}-\frac {4 d^x x \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^2 \sin (x)}{1+\log ^2(d)}-\frac {2 \int d^x \cos (x) \, dx}{\left (1+\log ^2(d)\right )^2}+2 \frac {(2 \log (d)) \int d^x \sin (x) \, dx}{\left (1+\log ^2(d)\right )^2}+\frac {\left (2 \log ^2(d)\right ) \int d^x \cos (x) \, dx}{\left (1+\log ^2(d)\right )^2}\\ &=-\frac {2 d^x \cos (x) \log (d)}{\left (1+\log ^2(d)\right )^3}+\frac {2 d^x \cos (x) \log ^3(d)}{\left (1+\log ^2(d)\right )^3}+\frac {2 d^x x \cos (x)}{\left (1+\log ^2(d)\right )^2}-\frac {2 d^x x \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^2 \cos (x) \log (d)}{1+\log ^2(d)}-\frac {2 d^x \sin (x)}{\left (1+\log ^2(d)\right )^3}+\frac {2 d^x \log ^2(d) \sin (x)}{\left (1+\log ^2(d)\right )^3}-\frac {4 d^x x \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^2 \sin (x)}{1+\log ^2(d)}+2 \left (-\frac {2 d^x \cos (x) \log (d)}{\left (1+\log ^2(d)\right )^3}+\frac {2 d^x \log ^2(d) \sin (x)}{\left (1+\log ^2(d)\right )^3}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 93, normalized size = 0.58 \begin {gather*} \frac {d^x \left (\cos (x) \left (2 x+\left (-6+x^2\right ) \log (d)+2 \left (1+x^2\right ) \log ^3(d)-2 x \log ^4(d)+x^2 \log ^5(d)\right )+\left (-2+x^2-4 x \log (d)+2 \left (3+x^2\right ) \log ^2(d)-4 x \log ^3(d)+x^2 \log ^4(d)\right ) \sin (x)\right )}{\left (1+\log ^2(d)\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.06, size = 100, normalized size = 0.62
method | result | size |
risch | \(\frac {\left (2+\ln \left (d \right )^{2} x^{2}+2 i \ln \left (d \right ) x^{2}-x^{2}-2 x \ln \left (d \right )-2 i x \right ) d^{x} {\mathrm e}^{i x}}{2 \left (\ln \left (d \right )+i\right )^{3}}+\frac {\left (2-2 x \ln \left (d \right )+2 i x +\ln \left (d \right )^{2} x^{2}-2 i \ln \left (d \right ) x^{2}-x^{2}\right ) d^{x} {\mathrm e}^{-i x}}{2 \left (\ln \left (d \right )-i\right )^{3}}\) | \(100\) |
norman | \(\frac {\frac {\ln \left (d \right ) x^{2} {\mathrm e}^{x \ln \left (d \right )}}{1+\ln \left (d \right )^{2}}+\frac {2 x^{2} {\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{1+\ln \left (d \right )^{2}}-\frac {2 \left (\ln \left (d \right )^{2}-1\right ) x \,{\mathrm e}^{x \ln \left (d \right )}}{\left (1+\ln \left (d \right )^{2}\right )^{2}}+\frac {4 \left (3 \ln \left (d \right )^{2}-1\right ) {\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{\left (1+\ln \left (d \right )^{2}\right )^{3}}+\frac {2 \ln \left (d \right ) \left (\ln \left (d \right )^{2}-3\right ) {\mathrm e}^{x \ln \left (d \right )}}{\left (1+\ln \left (d \right )^{2}\right )^{3}}-\frac {8 \ln \left (d \right ) x \,{\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{\left (1+\ln \left (d \right )^{2}\right )^{2}}-\frac {\ln \left (d \right ) x^{2} {\mathrm e}^{x \ln \left (d \right )} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{1+\ln \left (d \right )^{2}}+\frac {2 \left (\ln \left (d \right )^{2}-1\right ) x \,{\mathrm e}^{x \ln \left (d \right )} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{\left (1+\ln \left (d \right )^{2}\right )^{2}}-\frac {2 \ln \left (d \right ) \left (\ln \left (d \right )^{2}-3\right ) {\mathrm e}^{x \ln \left (d \right )} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{\left (1+\ln \left (d \right )^{2}\right )^{3}}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\) | \(231\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.08, size = 105, normalized size = 0.65 \begin {gather*} \frac {{\left ({\left (\log \left (d\right )^{5} + 2 \, \log \left (d\right )^{3} + \log \left (d\right )\right )} x^{2} + 2 \, \log \left (d\right )^{3} - 2 \, {\left (\log \left (d\right )^{4} - 1\right )} x - 6 \, \log \left (d\right )\right )} d^{x} \cos \left (x\right ) + {\left ({\left (\log \left (d\right )^{4} + 2 \, \log \left (d\right )^{2} + 1\right )} x^{2} - 4 \, {\left (\log \left (d\right )^{3} + \log \left (d\right )\right )} x + 6 \, \log \left (d\right )^{2} - 2\right )} d^{x} \sin \left (x\right )}{\log \left (d\right )^{6} + 3 \, \log \left (d\right )^{4} + 3 \, \log \left (d\right )^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.57, size = 111, normalized size = 0.69 \begin {gather*} \frac {{\left (x^{2} \cos \left (x\right ) \log \left (d\right )^{5} - 2 \, x \cos \left (x\right ) \log \left (d\right )^{4} + 2 \, {\left (x^{2} + 1\right )} \cos \left (x\right ) \log \left (d\right )^{3} + {\left (x^{2} - 6\right )} \cos \left (x\right ) \log \left (d\right ) + 2 \, x \cos \left (x\right ) + {\left (x^{2} \log \left (d\right )^{4} - 4 \, x \log \left (d\right )^{3} + 2 \, {\left (x^{2} + 3\right )} \log \left (d\right )^{2} + x^{2} - 4 \, x \log \left (d\right ) - 2\right )} \sin \left (x\right )\right )} d^{x}}{\log \left (d\right )^{6} + 3 \, \log \left (d\right )^{4} + 3 \, \log \left (d\right )^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.01, size = 665, normalized size = 4.13 \begin {gather*} \begin {cases} \frac {i x^{3} e^{- i x} \sin {\left (x \right )}}{6} + \frac {x^{3} e^{- i x} \cos {\left (x \right )}}{6} + \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 {e^{- i x} \sin {\left (x \right )}}{4} & \text {for}\: d = e^{- i} \\- \frac {i x^{3} e^{i x} \sin {\left (x \right )}}{6} + \frac {x^{3} e^{i x} \cos {\left (x \right )}}{6} + \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 {e^{i x} \sin {\left (x \right )}}{4} & \text {for}\: d = e^{i} \\\frac {d^{x} x^{2} \log {\left (d \right )}^{5} \cos {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} + \frac {d^{x} x^{2} \log {\left (d \right )}^{4} \sin {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} + \frac {2 d^{x} x^{2} \log {\left (d \right )}^{3} \cos {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} + \frac {2 d^{x} x^{2} \log {\left (d \right )}^{2} \sin {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} + \frac {d^{x} x^{2} \log {\left (d \right )} \cos {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} + \frac {d^{x} x^{2} \sin {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} - \frac {2 d^{x} x \log {\left (d \right )}^{4} \cos {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} - \frac {4 d^{x} x \log {\left (d \right )}^{3} \sin {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} - \frac {4 d^{x} x \log {\left (d \right )} \sin {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} + \frac {2 d^{x} x \cos {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} + \frac {2 d^{x} \log {\left (d \right )}^{3} \cos {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} + \frac {6 d^{x} \log {\left (d \right )}^{2} \sin {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} - \frac {6 d^{x} \log {\left (d \right )} \cos {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} - \frac {2 d^{x} \sin {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 1.09, size = 2631, normalized size = 16.34 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.35, size = 132, normalized size = 0.82 \begin {gather*} \frac {d^x\,\left (x^2\,\sin \left (x\right )-2\,\sin \left (x\right )+2\,x\,\cos \left (x\right )\right )+d^x\,{\ln \left (d\right )}^3\,\left (2\,\cos \left (x\right )+2\,x^2\,\cos \left (x\right )-4\,x\,\sin \left (x\right )\right )+d^x\,{\ln \left (d\right )}^2\,\left (6\,\sin \left (x\right )+2\,x^2\,\sin \left (x\right )\right )-d^x\,\ln \left (d\right )\,\left (6\,\cos \left (x\right )-x^2\,\cos \left (x\right )+4\,x\,\sin \left (x\right )\right )+d^x\,{\ln \left (d\right )}^4\,\left (x^2\,\sin \left (x\right )-2\,x\,\cos \left (x\right )\right )+d^x\,x^2\,{\ln \left (d\right )}^5\,\cos \left (x\right )}{{\left ({\ln \left (d\right )}^2+1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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