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 = 0.0162222, antiderivative size = 73, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.115778, 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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Maple [A] time = 0.015, size = 130, normalized size = 1.8 \begin{align*}{ \left ({\frac{e{{\rm e}^{c \left ( bx+a \right ) \ln \left ( F \right ) }}}{{e}^{2}+{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}} \left ( \tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{2}}-{\frac{e{{\rm e}^{c \left ( bx+a \right ) \ln \left ( F \right ) }}}{{e}^{2}+{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}}+2\,{\frac{bc\ln \left ( F \right ){{\rm e}^{c \left ( bx+a \right ) \ln \left ( F \right ) }}\tan \left ( d/2+1/2\,ex \right ) }{{e}^{2}+{b}^{2}{c}^{2} \left ( \ln \left ( F \right ) \right ) ^{2}}} \right ) \left ( 1+ \left ( \tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{2} \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.08017, size = 262, normalized size = 3.59 \begin{align*} -\frac{{\left (F^{a c} b c \log \left (F\right ) \sin \left (d\right ) + F^{a c} e \cos \left (d\right )\right )} F^{b c x} \cos \left (e x + 2 \, d\right ) -{\left (F^{a c} b c \log \left (F\right ) \sin \left (d\right ) - F^{a c} e \cos \left (d\right )\right )} F^{b c x} \cos \left (e x\right ) -{\left (F^{a c} b c \cos \left (d\right ) \log \left (F\right ) - F^{a c} e \sin \left (d\right )\right )} F^{b c x} \sin \left (e x + 2 \, d\right ) -{\left (F^{a c} b c \cos \left (d\right ) \log \left (F\right ) + F^{a c} e \sin \left (d\right )\right )} F^{b c x} \sin \left (e x\right )}{2 \,{\left (b^{2} c^{2} \cos \left (d\right )^{2} \log \left (F\right )^{2} + b^{2} c^{2} \log \left (F\right )^{2} \sin \left (d\right )^{2} +{\left (\cos \left (d\right )^{2} + \sin \left (d\right )^{2}\right )} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.480878, size = 115, normalized size = 1.58 \begin{align*} \frac{{\left (b c \log \left (F\right ) \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 \left (F\right )^{2} + e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 70.0082, size = 326, normalized size = 4.47 \begin{align*} \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{\left (d \right )} & \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{\left (F \right )} \sin{\left (d + e x \right )}}{b^{2} c^{2} \log{\left (F \right )}^{2} + e^{2}} - \frac{F^{a c} F^{b c x} e \cos{\left (d + e x \right )}}{b^{2} c^{2} \log{\left (F \right )}^{2} + e^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.22642, size = 880, normalized size = 12.05 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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