Optimal. Leaf size=32 \[ -\frac {d^x \cos (x)}{1+\log ^2(d)}+\frac {d^x \log (d) \sin (x)}{1+\log ^2(d)} \]
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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4517}
\begin {gather*} \frac {d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac {d^x \cos (x)}{\log ^2(d)+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 4517
Rubi steps
\begin {align*} \int d^x \sin (x) \, dx &=-\frac {d^x \cos (x)}{1+\log ^2(d)}+\frac {d^x \log (d) \sin (x)}{1+\log ^2(d)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 22, normalized size = 0.69 \begin {gather*} \frac {d^x (-\cos (x)+\log (d) \sin (x))}{1+\log ^2(d)} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.83, size = 88, normalized size = 2.75 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\left (-I x \text {Cos}\left [x\right ]+x \text {Sin}\left [x\right ]-\text {Cos}\left [x\right ]\right ) E^{-I x}}{2},d\text {==}E^{-I}\right \},\left \{\frac {\left (I x \text {Cos}\left [x\right ]+x \text {Sin}\left [x\right ]-\text {Cos}\left [x\right ]\right ) E^{I x}}{2},d\text {==}E^I\right \}\right \},-\frac {\text {Cos}\left [x\right ] d^x}{1+\text {Log}\left [d\right ]^2}+\frac {\text {Log}\left [d\right ] d^x \text {Sin}\left [x\right ]}{1+\text {Log}\left [d\right ]^2}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.03, size = 33, normalized size = 1.03
method | result | size |
risch | \(-\frac {d^{x} \cos \left (x \right )}{1+\ln \left (d \right )^{2}}+\frac {d^{x} \ln \left (d \right ) \sin \left (x \right )}{1+\ln \left (d \right )^{2}}\) | \(33\) |
norman | \(\frac {\frac {{\mathrm e}^{x \ln \left (d \right )} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{1+\ln \left (d \right )^{2}}-\frac {{\mathrm e}^{x \ln \left (d \right )}}{1+\ln \left (d \right )^{2}}+\frac {2 \ln \left (d \right ) {\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{1+\ln \left (d \right )^{2}}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 25, normalized size = 0.78 \begin {gather*} \frac {d^{x} \log \left (d\right ) \sin \left (x\right ) - d^{x} \cos \left (x\right )}{\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 = 0.31, size = 22, normalized size = 0.69 \begin {gather*} \frac {{\left (\log \left (d\right ) \sin \left (x\right ) - \cos \left (x\right )\right )} d^{x}}{\log \left (d\right )^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.29, size = 104, normalized size = 3.25 \begin {gather*} \begin {cases} \frac {x e^{- i x} \sin {\left (x \right )}}{2} - \frac {i x e^{- i x} \cos {\left (x \right )}}{2} - \frac {e^{- i x} \cos {\left (x \right )}}{2} & \text {for}\: d = e^{- i} \\\frac {x e^{i x} \sin {\left (x \right )}}{2} + \frac {i x e^{i x} \cos {\left (x \right )}}{2} - \frac {e^{i x} \cos {\left (x \right )}}{2} & \text {for}\: d = e^{i} \\\frac {d^{x} \log {\left (d \right )} \sin {\left (x \right )}}{\log {\left (d \right )}^{2} + 1} - \frac {d^{x} \cos {\left (x \right )}}{\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 = 0.00, size = 381, normalized size = 11.91 \begin {gather*} \mathrm {e}^{\ln \left |d\right |\cdot x} \left (-\frac {2 \left (-2 \pi \mathrm {sign}\left (d\right )+2 \pi +4\right ) \cos \left (\frac {\pi x \mathrm {sign}\left (d\right )-\pi x-2 x}{2}\right )}{\left (4 \ln \left |d\right |\right )^{2}+\left (2 \pi \mathrm {sign}\left (d\right )-2 \pi -4\right )^{2}}-\frac {4\cdot 2 \ln \left |d\right |\cdot \sin \left (\frac {\pi x \mathrm {sign}\left (d\right )-\pi x-2 x}{2}\right )}{\left (4 \ln \left |d\right |\right )^{2}+\left (2 \pi \mathrm {sign}\left (d\right )-2 \pi -4\right )^{2}}\right )+\mathrm {e}^{\ln \left |d\right |\cdot x} \left (\frac {2 \left (-2 \pi \mathrm {sign}\left (d\right )+2 \pi -4\right ) \cos \left (\frac {\pi x \mathrm {sign}\left (d\right )-\pi x+2 x}{2}\right )}{\left (4 \ln \left |d\right |\right )^{2}+\left (2 \pi \mathrm {sign}\left (d\right )-2 \pi +4\right )^{2}}+\frac {4\cdot 2 \ln \left |d\right |\cdot \sin \left (\frac {\pi x \mathrm {sign}\left (d\right )-\pi x+2 x}{2}\right )}{\left (4 \ln \left |d\right |\right )^{2}+\left (2 \pi \mathrm {sign}\left (d\right )-2 \pi +4\right )^{2}}\right )+\frac {1}{2} \mathrm {e}^{\ln \left |d\right |\cdot x} \left (\frac {2 \mathrm {i} \mathrm {e}^{\frac {1}{2} \mathrm {i} \left (\pi x \mathrm {sign}\left (d\right )-\pi x+2 x\right )}}{4 \ln \left |d\right |+\pi \cdot 2 \mathrm {i} \mathrm {sign}\left (d\right )-\pi \cdot 2 \mathrm {i}+4 \mathrm {i}}+\frac {2 \mathrm {i} \mathrm {e}^{-\frac {1}{2} \mathrm {i} \left (\pi x \mathrm {sign}\left (d\right )-\pi x+2 x\right )}}{4 \ln \left |d\right |-\pi \cdot 2 \mathrm {i} \mathrm {sign}\left (d\right )+\pi \cdot 2 \mathrm {i}-4 \mathrm {i}}\right )+\frac {1}{2} \mathrm {e}^{\ln \left |d\right |\cdot x} \left (-\frac {2 \mathrm {i} \mathrm {e}^{\frac {1}{2} \mathrm {i} \left (\pi x \mathrm {sign}\left (d\right )-\pi x-2 x\right )}}{4 \ln \left |d\right |+\pi \cdot 2 \mathrm {i} \mathrm {sign}\left (d\right )-\pi \cdot 2 \mathrm {i}-4 \mathrm {i}}-\frac {2 \mathrm {i} \mathrm {e}^{-\frac {1}{2} \mathrm {i} \left (\pi x \mathrm {sign}\left (d\right )-\pi x-2 x\right )}}{4 \ln \left |d\right |-\pi \cdot 2 \mathrm {i} \mathrm {sign}\left (d\right )+\pi \cdot 2 \mathrm {i}+4 \mathrm {i}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 22, normalized size = 0.69 \begin {gather*} -\frac {d^x\,\left (\cos \left (x\right )-\ln \left (d\right )\,\sin \left (x\right )\right )}{{\ln \left (d\right )}^2+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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