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 = 0.18, antiderivative size = 106, normalized size of antiderivative = 1.00, 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 12
Rule 2194
Rule 5474
Rule 6741
Rule 6742
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*}
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Mathematica [A] time = 0.66, 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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fricas [A] time = 0.49, size = 135, normalized size = 1.27 \[ -\frac {{\left (2 \, e^{2} f e^{\left (e x + d\right )} - {\left (i \, b^{2} c^{2} f e^{\left (2 \, e x + 2 \, d\right )} + 2 \, b^{2} c^{2} f e^{\left (e x + d\right )} - i \, b^{2} c^{2} f\right )} \log \relax (F)^{2} - {\left (-i \, b c e f e^{\left (2 \, e x + 2 \, d\right )} - i \, b c e f\right )} \log \relax (F)\right )} F^{b c x + a c}}{2 \, {\left (b^{3} c^{3} e^{\left (e x + d\right )} \log \relax (F)^{3} - b c e^{2} e^{\left (e x + d\right )} \log \relax (F)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 899, normalized size = 8.48 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 141, normalized size = 1.33 \[ \frac {f \left (-i \ln \relax (F )^{2} b^{2} c^{2} {\mathrm e}^{2 e x +2 d}+i \ln \relax (F )^{2} b^{2} c^{2}-2 \ln \relax (F )^{2} b^{2} c^{2} {\mathrm e}^{e x +d}+i \ln \relax (F ) b c e \,{\mathrm e}^{2 e x +2 d}+i \ln \relax (F ) b c e +2 e^{2} {\mathrm e}^{e x +d}\right ) {\mathrm e}^{-e x -d} F^{c \left (b x +a \right )}}{2 b c \ln \relax (F ) \left (e -b c \ln \relax (F )\right ) \left (e +b c \ln \relax (F )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 88, normalized size = 0.83 \[ \frac {1}{2} i \, f {\left (\frac {F^{a c} e^{\left (b c x \log \relax (F) + e x + d\right )}}{b c \log \relax (F) + e} - \frac {F^{a c} e^{\left (b c x \log \relax (F) - e x\right )}}{b c e^{d} \log \relax (F) - e e^{d}}\right )} + \frac {F^{b c x + a c} f}{b c \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.98, size = 88, normalized size = 0.83 \[ \frac {F^{c\,\left (a+b\,x\right )}\,f\,\left (e^2-b^2\,c^2\,{\ln \relax (F)}^2-b^2\,c^2\,\mathrm {sinh}\left (d+e\,x\right )\,{\ln \relax (F)}^2\,1{}\mathrm {i}+b\,c\,e\,\mathrm {cosh}\left (d+e\,x\right )\,\ln \relax (F)\,1{}\mathrm {i}\right )}{b\,c\,\ln \relax (F)\,\left (e^2-b^2\,c^2\,{\ln \relax (F)}^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.90, size = 400, normalized size = 3.77 \[ \begin {cases} f x + \frac {i f \cosh {\left (d + e x \right )}}{e} & \text {for}\: F = 1 \\\tilde {\infty } e^{2} f \left (e^{- \frac {e}{b c}}\right )^{a c} \left (e^{- \frac {e}{b c}}\right )^{b c x} \sinh {\left (d + e x \right )} + \tilde {\infty } e^{2} f \left (e^{- \frac {e}{b c}}\right )^{a c} \left (e^{- \frac {e}{b c}}\right )^{b c x} \cosh {\left (d + e x \right )} & \text {for}\: F = e^{- \frac {e}{b c}} \\\tilde {\infty } e^{2} f \left (e^{\frac {e}{b c}}\right )^{a c} \left (e^{\frac {e}{b c}}\right )^{b c x} \sinh {\left (d + e x \right )} + \tilde {\infty } e^{2} f \left (e^{\frac {e}{b c}}\right )^{a c} \left (e^{\frac {e}{b c}}\right )^{b c x} \cosh {\left (d + e x \right )} & \text {for}\: F = e^{\frac {e}{b c}} \\F^{a c} \left (f x + \frac {i f \cosh {\left (d + e x \right )}}{e}\right ) & \text {for}\: b = 0 \\f x + \frac {i f \cosh {\left (d + e x \right )}}{e} & \text {for}\: c = 0 \\- \frac {i F^{a c} F^{b c x} b^{2} c^{2} f \log {\relax (F )}^{2} \sinh {\left (d + e x \right )}}{- b^{3} c^{3} \log {\relax (F )}^{3} + b c e^{2} \log {\relax (F )}} - \frac {F^{a c} F^{b c x} b^{2} c^{2} f \log {\relax (F )}^{2}}{- b^{3} c^{3} \log {\relax (F )}^{3} + b c e^{2} \log {\relax (F )}} + \frac {i F^{a c} F^{b c x} b c e f \log {\relax (F )} \cosh {\left (d + e x \right )}}{- b^{3} c^{3} \log {\relax (F )}^{3} + b c e^{2} \log {\relax (F )}} + \frac {F^{a c} F^{b c x} e^{2} f}{- b^{3} c^{3} \log {\relax (F )}^{3} + b c e^{2} \log {\relax (F )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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