Optimal. Leaf size=73 \[ \frac {b c \log (F) \sin (d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+e^2}-\frac {e \cos (d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+e^2} \]
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Rubi [A] time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4432} \[ \frac {b c \log (F) \sin (d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+e^2}-\frac {e \cos (d+e x) F^{c (a+b x)}}{b^2 c^2 \log ^2(F)+e^2} \]
Antiderivative was successfully verified.
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Rule 4432
Rubi steps
\begin {align*} \int F^{c (a+b x)} \sin (d+e x) \, dx &=-\frac {e F^{c (a+b x)} \cos (d+e x)}{e^2+b^2 c^2 \log ^2(F)}+\frac {b c F^{c (a+b x)} \log (F) \sin (d+e x)}{e^2+b^2 c^2 \log ^2(F)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 48, normalized size = 0.66 \[ \frac {F^{c (a+b x)} (b c \log (F) \sin (d+e x)-e \cos (d+e x))}{b^2 c^2 \log ^2(F)+e^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 49, normalized size = 0.67 \[ \frac {{\left (b c \log \relax (F) \sin \left (e x + d\right ) - e \cos \left (e x + d\right )\right )} F^{b c x + a c}}{b^{2} c^{2} \log \relax (F)^{2} + e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.21, size = 652, normalized size = 8.93 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 130, normalized size = 1.78 \[ \frac {\frac {e \,{\mathrm e}^{c \left (b x +a \right ) \ln \relax (F )} \left (\tan ^{2}\left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{e^{2}+b^{2} c^{2} \ln \relax (F )^{2}}-\frac {e \,{\mathrm e}^{c \left (b x +a \right ) \ln \relax (F )}}{e^{2}+b^{2} c^{2} \ln \relax (F )^{2}}+\frac {2 b c \ln \relax (F ) {\mathrm e}^{c \left (b x +a \right ) \ln \relax (F )} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )}{e^{2}+b^{2} c^{2} \ln \relax (F )^{2}}}{1+\tan ^{2}\left (\frac {d}{2}+\frac {e x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.13, size = 194, normalized size = 2.66 \[ -\frac {{\left (F^{a c} b c \log \relax (F) \sin \relax (d) + F^{a c} e \cos \relax (d)\right )} F^{b c x} \cos \left (e x + 2 \, d\right ) - {\left (F^{a c} b c \log \relax (F) \sin \relax (d) - F^{a c} e \cos \relax (d)\right )} F^{b c x} \cos \left (e x\right ) - {\left (F^{a c} b c \cos \relax (d) \log \relax (F) - F^{a c} e \sin \relax (d)\right )} F^{b c x} \sin \left (e x + 2 \, d\right ) - {\left (F^{a c} b c \cos \relax (d) \log \relax (F) + F^{a c} e \sin \relax (d)\right )} F^{b c x} \sin \left (e x\right )}{2 \, {\left (b^{2} c^{2} \cos \relax (d)^{2} \log \relax (F)^{2} + b^{2} c^{2} \log \relax (F)^{2} \sin \relax (d)^{2} + {\left (\cos \relax (d)^{2} + \sin \relax (d)^{2}\right )} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.40, size = 50, normalized size = 0.68 \[ -\frac {F^{a\,c+b\,c\,x}\,\left (e\,\cos \left (d+e\,x\right )-b\,c\,\sin \left (d+e\,x\right )\,\ln \relax (F)\right )}{b^2\,c^2\,{\ln \relax (F)}^2+e^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.84, size = 326, normalized size = 4.47 \[ \begin {cases} \frac {\left (-1\right )^{a c} \left (-1\right )^{\frac {e x}{\pi }} x \sin {\left (d + e x \right )}}{2} + \frac {\left (-1\right )^{a c} \left (-1\right )^{\frac {e x}{\pi }} i x \cos {\left (d + e x \right )}}{2} - \frac {\left (-1\right )^{a c} \left (-1\right )^{\frac {e x}{\pi }} \cos {\left (d + e x \right )}}{2 e} & \text {for}\: F = -1 \wedge b = \frac {e}{\pi c} \\x \sin {\relax (d )} & \text {for}\: F = 1 \wedge e = 0 \\\tilde {\infty } e \left (e^{- \frac {i e}{b c}}\right )^{a c} \left (e^{- \frac {i e}{b c}}\right )^{b c x} \sin {\left (d + e x \right )} + \tilde {\infty } e \left (e^{- \frac {i e}{b c}}\right )^{a c} \left (e^{- \frac {i e}{b c}}\right )^{b c x} \cos {\left (d + e x \right )} & \text {for}\: F = e^{- \frac {i e}{b c}} \\\tilde {\infty } e \left (e^{\frac {i e}{b c}}\right )^{a c} \left (e^{\frac {i e}{b c}}\right )^{b c x} \sin {\left (d + e x \right )} + \tilde {\infty } e \left (e^{\frac {i e}{b c}}\right )^{a c} \left (e^{\frac {i e}{b c}}\right )^{b c x} \cos {\left (d + e x \right )} & \text {for}\: F = e^{\frac {i e}{b c}} \\\frac {F^{a c} F^{b c x} b c \log {\relax (F )} \sin {\left (d + e x \right )}}{b^{2} c^{2} \log {\relax (F )}^{2} + e^{2}} - \frac {F^{a c} F^{b c x} e \cos {\left (d + e x \right )}}{b^{2} c^{2} \log {\relax (F )}^{2} + e^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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