Optimal. Leaf size=106 \[ -\frac{i b c f \log (F) \sinh (d+e x) F^{a c+b c x}}{e^2-b^2 c^2 \log ^2(F)}+\frac{i e f \cosh (d+e x) F^{a c+b c x}}{e^2-b^2 c^2 \log ^2(F)}+\frac{f F^{a c+b c x}}{b c \log (F)} \]
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Rubi [A] time = 0.179155, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {6741, 12, 6742, 2194, 5474} \[ -\frac{i b c f \log (F) \sinh (d+e x) F^{a c+b c x}}{e^2-b^2 c^2 \log ^2(F)}+\frac{i e f \cosh (d+e x) F^{a c+b c x}}{e^2-b^2 c^2 \log ^2(F)}+\frac{f F^{a c+b c x}}{b c \log (F)} \]
Antiderivative was successfully verified.
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Rule 6741
Rule 12
Rule 6742
Rule 2194
Rule 5474
Rubi steps
\begin{align*} \int F^{c (a+b x)} (f+i f \sinh (d+e x)) \, dx &=\int f F^{a c+b c x} (1+i \sinh (d+e x)) \, dx\\ &=f \int F^{a c+b c x} (1+i \sinh (d+e x)) \, dx\\ &=f \int \left (F^{a c+b c x}+i F^{a c+b c x} \sinh (d+e x)\right ) \, dx\\ &=(i f) \int F^{a c+b c x} \sinh (d+e x) \, dx+f \int F^{a c+b c x} \, dx\\ &=\frac{f F^{a c+b c x}}{b c \log (F)}+\frac{i e f F^{a c+b c x} \cosh (d+e x)}{e^2-b^2 c^2 \log ^2(F)}-\frac{i b c f F^{a c+b c x} \log (F) \sinh (d+e x)}{e^2-b^2 c^2 \log ^2(F)}\\ \end{align*}
Mathematica [A] time = 0.612028, size = 93, normalized size = 0.88 \[ \frac{f F^{c (a+b x)} \left (i b^2 c^2 \log ^2(F) \sinh (d+e x)+b^2 c^2 \log ^2(F)-i b c e \log (F) \cosh (d+e x)-e^2\right )}{b c \log (F) (b c \log (F)-e) (b c \log (F)+e)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 141, normalized size = 1.3 \begin{align*}{\frac{f \left ( -i \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}{{\rm e}^{2\,ex+2\,d}}+i \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}-2\, \left ( \ln \left ( F \right ) \right ) ^{2}{b}^{2}{c}^{2}{{\rm e}^{ex+d}}+i\ln \left ( F \right ) bce{{\rm e}^{2\,ex+2\,d}}+i\ln \left ( F \right ) bce+2\,{e}^{2}{{\rm e}^{ex+d}} \right ){{\rm e}^{-ex-d}}{F}^{c \left ( bx+a \right ) }}{2\,bc\ln \left ( F \right ) \left ( e-bc\ln \left ( F \right ) \right ) \left ( e+bc\ln \left ( F \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03752, size = 119, normalized size = 1.12 \begin{align*} \frac{1}{2} i \, f{\left (\frac{F^{a c} e^{\left (b c x \log \left (F\right ) + e x + d\right )}}{b c \log \left (F\right ) + e} - \frac{F^{a c} e^{\left (b c x \log \left (F\right ) - e x\right )}}{b c e^{d} \log \left (F\right ) - e e^{d}}\right )} + \frac{F^{b c x + a c} f}{b c \log \left (F\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33784, size = 317, normalized size = 2.99 \begin{align*} -\frac{{\left (2 \, e^{2} f e^{\left (e x + d\right )} -{\left (i \, b^{2} c^{2} f e^{\left (2 \, e x + 2 \, d\right )} + 2 \, b^{2} c^{2} f e^{\left (e x + d\right )} - i \, b^{2} c^{2} f\right )} \log \left (F\right )^{2} -{\left (-i \, b c e f e^{\left (2 \, e x + 2 \, d\right )} - i \, b c e f\right )} \log \left (F\right )\right )} F^{b c x + a c}}{2 \,{\left (b^{3} c^{3} e^{\left (e x + d\right )} \log \left (F\right )^{3} - b c e^{2} e^{\left (e x + d\right )} \log \left (F\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.4982, size = 400, normalized size = 3.77 \begin{align*} \begin{cases} f x + \frac{i f \cosh{\left (d + e x \right )}}{e} & \text{for}\: F = 1 \\\tilde{\infty } e^{2} f \left (e^{- \frac{e}{b c}}\right )^{a c} \left (e^{- \frac{e}{b c}}\right )^{b c x} \sinh{\left (d + e x \right )} + \tilde{\infty } e^{2} f \left (e^{- \frac{e}{b c}}\right )^{a c} \left (e^{- \frac{e}{b c}}\right )^{b c x} \cosh{\left (d + e x \right )} & \text{for}\: F = e^{- \frac{e}{b c}} \\\tilde{\infty } e^{2} f \left (e^{\frac{e}{b c}}\right )^{a c} \left (e^{\frac{e}{b c}}\right )^{b c x} \sinh{\left (d + e x \right )} + \tilde{\infty } e^{2} f \left (e^{\frac{e}{b c}}\right )^{a c} \left (e^{\frac{e}{b c}}\right )^{b c x} \cosh{\left (d + e x \right )} & \text{for}\: F = e^{\frac{e}{b c}} \\F^{a c} \left (f x + \frac{i f \cosh{\left (d + e x \right )}}{e}\right ) & \text{for}\: b = 0 \\f x + \frac{i f \cosh{\left (d + e x \right )}}{e} & \text{for}\: c = 0 \\\frac{i F^{a c} F^{b c x} b^{2} c^{2} f \log{\left (F \right )}^{2} \sinh{\left (d + e x \right )}}{b^{3} c^{3} \log{\left (F \right )}^{3} - b c e^{2} \log{\left (F \right )}} + \frac{F^{a c} F^{b c x} b^{2} c^{2} f \log{\left (F \right )}^{2}}{b^{3} c^{3} \log{\left (F \right )}^{3} - b c e^{2} \log{\left (F \right )}} - \frac{i F^{a c} F^{b c x} b c e f \log{\left (F \right )} \cosh{\left (d + e x \right )}}{b^{3} c^{3} \log{\left (F \right )}^{3} - b c e^{2} \log{\left (F \right )}} - \frac{F^{a c} F^{b c x} e^{2} f}{b^{3} c^{3} \log{\left (F \right )}^{3} - b c e^{2} \log{\left (F \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25489, size = 1214, normalized size = 11.45 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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