Optimal. Leaf size=73 \[ -\frac {2 i f \text {Li}_2\left (-i e^{c+d x}\right )}{a d^2}-\frac {2 i (e+f x) \log \left (1+i e^{c+d x}\right )}{a d}+\frac {i (e+f x)^2}{2 a f} \]
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Rubi [A] time = 0.11, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5559, 2190, 2279, 2391} \[ -\frac {2 i f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^2}-\frac {2 i (e+f x) \log \left (1+i e^{c+d x}\right )}{a d}+\frac {i (e+f x)^2}{2 a f} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 5559
Rubi steps
\begin {align*} \int \frac {(e+f x) \cosh (c+d x)}{a+i a \sinh (c+d x)} \, dx &=\frac {i (e+f x)^2}{2 a f}+2 \int \frac {e^{c+d x} (e+f x)}{a+i a e^{c+d x}} \, dx\\ &=\frac {i (e+f x)^2}{2 a f}-\frac {2 i (e+f x) \log \left (1+i e^{c+d x}\right )}{a d}+\frac {(2 i f) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a d}\\ &=\frac {i (e+f x)^2}{2 a f}-\frac {2 i (e+f x) \log \left (1+i e^{c+d x}\right )}{a d}+\frac {(2 i f) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^2}\\ &=\frac {i (e+f x)^2}{2 a f}-\frac {2 i (e+f x) \log \left (1+i e^{c+d x}\right )}{a d}-\frac {2 i f \text {Li}_2\left (-i e^{c+d x}\right )}{a d^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 66, normalized size = 0.90 \[ \frac {i \left (d (e+f x) \left (d (e+f x)-4 f \log \left (1+i e^{c+d x}\right )\right )-4 f^2 \text {Li}_2\left (-i e^{c+d x}\right )\right )}{2 a d^2 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 89, normalized size = 1.22 \[ \frac {i \, d^{2} f x^{2} + 2 i \, d^{2} e x + 4 i \, c d e - 2 i \, c^{2} f - 4 i \, f {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) + {\left (-4 i \, d e + 4 i \, c f\right )} \log \left (e^{\left (d x + c\right )} - i\right ) + {\left (-4 i \, d f x - 4 i \, c f\right )} \log \left (i \, e^{\left (d x + c\right )} + 1\right )}{2 \, a d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )}{i \, a \sinh \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 188, normalized size = 2.58 \[ \frac {i f \,x^{2}}{2 a}-\frac {i e x}{a}-\frac {2 i \ln \left ({\mathrm e}^{d x +c}-i\right ) e}{d a}+\frac {2 i \ln \left ({\mathrm e}^{d x +c}\right ) e}{d a}+\frac {2 i f c x}{d a}+\frac {i f \,c^{2}}{d^{2} a}-\frac {2 i f \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{d a}-\frac {2 i f \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{d^{2} a}-\frac {2 i f \polylog \left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {2 i f c \ln \left ({\mathrm e}^{d x +c}-i\right )}{d^{2} a}-\frac {2 i f c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, f {\left (-\frac {i \, x^{2}}{a} + 4 \, \int \frac {x}{a e^{\left (d x + c\right )} - i \, a}\,{d x}\right )} - \frac {i \, e \log \left (i \, a \sinh \left (d x + c\right ) + a\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (e+f\,x\right )}{a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {i \left (\int \frac {e \cosh {\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {f x \cosh {\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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