Integrand size = 6, antiderivative size = 32 \[ \int d^x \sin (x) \, dx=-\frac {d^x \cos (x)}{1+\log ^2(d)}+\frac {d^x \log (d) \sin (x)}{1+\log ^2(d)} \]
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Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4517} \[ \int d^x \sin (x) \, dx=\frac {d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac {d^x \cos (x)}{\log ^2(d)+1} \]
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Rule 4517
Rubi steps \begin{align*} \text {integral}& = -\frac {d^x \cos (x)}{1+\log ^2(d)}+\frac {d^x \log (d) \sin (x)}{1+\log ^2(d)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.69 \[ \int d^x \sin (x) \, dx=\frac {d^x (-\cos (x)+\log (d) \sin (x))}{1+\log ^2(d)} \]
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Time = 0.13 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72
method | result | size |
parallelrisch | \(\frac {d^{x} \left (\ln \left (d \right ) \sin \left (x \right )-\cos \left (x \right )\right )}{1+\ln \left (d \right )^{2}}\) | \(23\) |
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 | \(\frac {\frac {{\mathrm e}^{x \ln \left (d \right )} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{1+\ln \left (d \right )^{2}}-\frac {{\mathrm e}^{x \ln \left (d \right )}}{1+\ln \left (d \right )^{2}}+\frac {2 \ln \left (d \right ) {\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{1+\ln \left (d \right )^{2}}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\) | \(69\) |
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Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.69 \[ \int d^x \sin (x) \, dx=\frac {{\left (\log \left (d\right ) \sin \left (x\right ) - \cos \left (x\right )\right )} d^{x}}{\log \left (d\right )^{2} + 1} \]
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Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.25 \[ \int d^x \sin (x) \, dx=\begin {cases} \frac {x e^{- i x} \sin {\left (x \right )}}{2} - \frac {i x e^{- i x} \cos {\left (x \right )}}{2} - \frac {e^{- i x} \cos {\left (x \right )}}{2} & \text {for}\: d = e^{- i} \\\frac {x e^{i x} \sin {\left (x \right )}}{2} + \frac {i x e^{i x} \cos {\left (x \right )}}{2} - \frac {e^{i x} \cos {\left (x \right )}}{2} & \text {for}\: d = e^{i} \\\frac {d^{x} \log {\left (d \right )} \sin {\left (x \right )}}{\log {\left (d \right )}^{2} + 1} - \frac {d^{x} \cos {\left (x \right )}}{\log {\left (d \right )}^{2} + 1} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int d^x \sin (x) \, dx=\frac {d^{x} \log \left (d\right ) \sin \left (x\right ) - d^{x} \cos \left (x\right )}{\log \left (d\right )^{2} + 1} \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 328, normalized size of antiderivative = 10.25 \[ \int d^x \sin (x) \, dx={\left | d \right |}^{x} {\left (\frac {{\left (\pi - \pi \mathrm {sgn}\left (d\right ) - 2\right )} \cos \left (\frac {1}{2} \, \pi x \mathrm {sgn}\left (d\right ) - \frac {1}{2} \, \pi x + x\right )}{{\left (\pi - \pi \mathrm {sgn}\left (d\right ) - 2\right )}^{2} + 4 \, \log \left ({\left | d \right |}\right )^{2}} + \frac {2 \, \log \left ({\left | d \right |}\right ) \sin \left (\frac {1}{2} \, \pi x \mathrm {sgn}\left (d\right ) - \frac {1}{2} \, \pi x + x\right )}{{\left (\pi - \pi \mathrm {sgn}\left (d\right ) - 2\right )}^{2} + 4 \, \log \left ({\left | d \right |}\right )^{2}}\right )} - {\left | d \right |}^{x} {\left (\frac {{\left (\pi - \pi \mathrm {sgn}\left (d\right ) + 2\right )} \cos \left (\frac {1}{2} \, \pi x \mathrm {sgn}\left (d\right ) - \frac {1}{2} \, \pi x - x\right )}{{\left (\pi - \pi \mathrm {sgn}\left (d\right ) + 2\right )}^{2} + 4 \, \log \left ({\left | d \right |}\right )^{2}} + \frac {2 \, \log \left ({\left | d \right |}\right ) \sin \left (\frac {1}{2} \, \pi x \mathrm {sgn}\left (d\right ) - \frac {1}{2} \, \pi x - x\right )}{{\left (\pi - \pi \mathrm {sgn}\left (d\right ) + 2\right )}^{2} + 4 \, \log \left ({\left | d \right |}\right )^{2}}\right )} - {\left | d \right |}^{x} {\left (-\frac {i \, e^{\left (\frac {1}{2} i \, \pi x \mathrm {sgn}\left (d\right ) - \frac {1}{2} i \, \pi x + i \, x\right )}}{-2 i \, \pi + 2 i \, \pi \mathrm {sgn}\left (d\right ) + 4 \, \log \left ({\left | d \right |}\right ) + 4 i} - \frac {i \, e^{\left (-\frac {1}{2} i \, \pi x \mathrm {sgn}\left (d\right ) + \frac {1}{2} i \, \pi x - i \, x\right )}}{2 i \, \pi - 2 i \, \pi \mathrm {sgn}\left (d\right ) + 4 \, \log \left ({\left | d \right |}\right ) - 4 i}\right )} - {\left | d \right |}^{x} {\left (\frac {i \, e^{\left (\frac {1}{2} i \, \pi x \mathrm {sgn}\left (d\right ) - \frac {1}{2} i \, \pi x - i \, x\right )}}{-2 i \, \pi + 2 i \, \pi \mathrm {sgn}\left (d\right ) + 4 \, \log \left ({\left | d \right |}\right ) - 4 i} + \frac {i \, e^{\left (-\frac {1}{2} i \, \pi x \mathrm {sgn}\left (d\right ) + \frac {1}{2} i \, \pi x + i \, x\right )}}{2 i \, \pi - 2 i \, \pi \mathrm {sgn}\left (d\right ) + 4 \, \log \left ({\left | d \right |}\right ) + 4 i}\right )} \]
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Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.69 \[ \int d^x \sin (x) \, dx=-\frac {d^x\,\left (\cos \left (x\right )-\ln \left (d\right )\,\sin \left (x\right )\right )}{{\ln \left (d\right )}^2+1} \]
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