Optimal. Leaf size=31 \[ \frac {d^x \sin (x)}{\log ^2(d)+1}+\frac {d^x \log (d) \cos (x)}{\log ^2(d)+1} \]
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Rubi [A] time = 0.01, antiderivative size = 31, 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 = {4433} \[ \frac {d^x \sin (x)}{\log ^2(d)+1}+\frac {d^x \log (d) \cos (x)}{\log ^2(d)+1} \]
Antiderivative was successfully verified.
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Rule 4433
Rubi steps
\begin {align*} \int d^x \cos (x) \, dx &=\frac {d^x \cos (x) \log (d)}{1+\log ^2(d)}+\frac {d^x \sin (x)}{1+\log ^2(d)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 0.65 \[ \frac {d^x (\log (d) \cos (x)+\sin (x))}{\log ^2(d)+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 20, normalized size = 0.65 \[ \frac {{\left (\cos \relax (x) \log \relax (d) + \sin \relax (x)\right )} d^{x}}{\log \relax (d)^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 1.18, size = 329, normalized size = 10.61 \[ {\left | d \right |}^{x} {\left (\frac {2 \, \cos \left (\frac {1}{2} \, \pi x \mathrm {sgn}\relax (d) - \frac {1}{2} \, \pi x + x\right ) \log \left ({\left | d \right |}\right )}{{\left (\pi - \pi \mathrm {sgn}\relax (d) - 2\right )}^{2} + 4 \, \log \left ({\left | d \right |}\right )^{2}} - \frac {{\left (\pi - \pi \mathrm {sgn}\relax (d) - 2\right )} \sin \left (\frac {1}{2} \, \pi x \mathrm {sgn}\relax (d) - \frac {1}{2} \, \pi x + x\right )}{{\left (\pi - \pi \mathrm {sgn}\relax (d) - 2\right )}^{2} + 4 \, \log \left ({\left | d \right |}\right )^{2}}\right )} + {\left | d \right |}^{x} {\left (\frac {2 \, \cos \left (\frac {1}{2} \, \pi x \mathrm {sgn}\relax (d) - \frac {1}{2} \, \pi x - x\right ) \log \left ({\left | d \right |}\right )}{{\left (\pi - \pi \mathrm {sgn}\relax (d) + 2\right )}^{2} + 4 \, \log \left ({\left | d \right |}\right )^{2}} - \frac {{\left (\pi - \pi \mathrm {sgn}\relax (d) + 2\right )} \sin \left (\frac {1}{2} \, \pi x \mathrm {sgn}\relax (d) - \frac {1}{2} \, \pi x - x\right )}{{\left (\pi - \pi \mathrm {sgn}\relax (d) + 2\right )}^{2} + 4 \, \log \left ({\left | d \right |}\right )^{2}}\right )} - \frac {1}{2} i \, {\left | d \right |}^{x} {\left (-\frac {2 i \, e^{\left (\frac {1}{2} i \, \pi x \mathrm {sgn}\relax (d) - \frac {1}{2} i \, \pi x + i \, x\right )}}{-2 i \, \pi + 2 i \, \pi \mathrm {sgn}\relax (d) + 4 \, \log \left ({\left | d \right |}\right ) + 4 i} + \frac {2 i \, e^{\left (-\frac {1}{2} i \, \pi x \mathrm {sgn}\relax (d) + \frac {1}{2} i \, \pi x - i \, x\right )}}{2 i \, \pi - 2 i \, \pi \mathrm {sgn}\relax (d) + 4 \, \log \left ({\left | d \right |}\right ) - 4 i}\right )} - \frac {1}{2} i \, {\left | d \right |}^{x} {\left (-\frac {2 i \, e^{\left (\frac {1}{2} i \, \pi x \mathrm {sgn}\relax (d) - \frac {1}{2} i \, \pi x - i \, x\right )}}{-2 i \, \pi + 2 i \, \pi \mathrm {sgn}\relax (d) + 4 \, \log \left ({\left | d \right |}\right ) - 4 i} + \frac {2 i \, e^{\left (-\frac {1}{2} i \, \pi x \mathrm {sgn}\relax (d) + \frac {1}{2} i \, \pi x + i \, x\right )}}{2 i \, \pi - 2 i \, \pi \mathrm {sgn}\relax (d) + 4 \, \log \left ({\left | d \right |}\right ) + 4 i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 71, normalized size = 2.29 \[ \frac {-\frac {{\mathrm e}^{x \ln \relax (d )} \ln \relax (d ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{\ln \relax (d )^{2}+1}+\frac {{\mathrm e}^{x \ln \relax (d )} \ln \relax (d )}{\ln \relax (d )^{2}+1}+\frac {2 \,{\mathrm e}^{x \ln \relax (d )} \tan \left (\frac {x}{2}\right )}{\ln \relax (d )^{2}+1}}{\tan ^{2}\left (\frac {x}{2}\right )+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 24, normalized size = 0.77 \[ \frac {d^{x} \cos \relax (x) \log \relax (d) + d^{x} \sin \relax (x)}{\log \relax (d)^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 20, normalized size = 0.65 \[ \frac {d^x\,\left (\sin \relax (x)+\ln \relax (d)\,\cos \relax (x)\right )}{{\ln \relax (d)}^2+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.03, size = 107, normalized size = 3.45 \[ \begin {cases} \frac {i x e^{- i x} \sin {\relax (x )}}{2} + \frac {x e^{- i x} \cos {\relax (x )}}{2} + \frac {i e^{- i x} \cos {\relax (x )}}{2} & \text {for}\: d = e^{- i} \\- \frac {i x e^{i x} \sin {\relax (x )}}{2} + \frac {x e^{i x} \cos {\relax (x )}}{2} - \frac {i e^{i x} \cos {\relax (x )}}{2} & \text {for}\: d = e^{i} \\\frac {d^{x} \log {\relax (d )} \cos {\relax (x )}}{\log {\relax (d )}^{2} + 1} + \frac {d^{x} \sin {\relax (x )}}{\log {\relax (d )}^{2} + 1} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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