Optimal. Leaf size=90 \[ -\frac {2 i f \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{a d^2}+\frac {i (e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {i e x}{a}-\frac {i f x^2}{2 a} \]
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Rubi [A] time = 0.11, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5557, 3318, 4184, 3475} \[ -\frac {2 i f \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{a d^2}+\frac {i (e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {i e x}{a}-\frac {i f x^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 3318
Rule 3475
Rule 4184
Rule 5557
Rubi steps
\begin {align*} \int \frac {(e+f x) \sinh (c+d x)}{a+i a \sinh (c+d x)} \, dx &=i \int \frac {e+f x}{a+i a \sinh (c+d x)} \, dx-\frac {i \int (e+f x) \, dx}{a}\\ &=-\frac {i e x}{a}-\frac {i f x^2}{2 a}+\frac {i \int (e+f x) \csc ^2\left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {i d x}{2}\right ) \, dx}{2 a}\\ &=-\frac {i e x}{a}-\frac {i f x^2}{2 a}+\frac {i (e+f x) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(i f) \int \coth \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{a d}\\ &=-\frac {i e x}{a}-\frac {i f x^2}{2 a}-\frac {2 i f \log \left (\cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}+\frac {i (e+f x) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}\\ \end {align*}
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Mathematica [B] time = 0.63, size = 239, normalized size = 2.66 \[ \frac {-i \cosh \left (\frac {d x}{2}\right ) \left (2 f \log (\cosh (c+d x))+4 i f \tan ^{-1}\left (\sinh \left (\frac {d x}{2}\right ) \text {sech}\left (c+\frac {d x}{2}\right )\right )+d^2 x (2 e+f x)\right )+2 d^2 e x \sinh \left (c+\frac {d x}{2}\right )+d^2 f x^2 \sinh \left (c+\frac {d x}{2}\right )-2 d f x \cosh \left (c+\frac {d x}{2}\right )+2 f \sinh \left (c+\frac {d x}{2}\right ) \log (\cosh (c+d x))+4 i f \sinh \left (c+\frac {d x}{2}\right ) \tan ^{-1}\left (\sinh \left (\frac {d x}{2}\right ) \text {sech}\left (c+\frac {d x}{2}\right )\right )+4 i d e \sinh \left (\frac {d x}{2}\right )+2 i d f x \sinh \left (\frac {d x}{2}\right )}{2 a d^2 \left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 95, normalized size = 1.06 \[ -\frac {d^{2} f x^{2} + 2 \, d^{2} e x + 4 \, d e - {\left (-i \, d^{2} f x^{2} + {\left (-2 i \, d^{2} e + 4 i \, d f\right )} x\right )} e^{\left (d x + c\right )} + 4 \, {\left (i \, f e^{\left (d x + c\right )} + f\right )} \log \left (e^{\left (d x + c\right )} - i\right )}{2 \, {\left (a d^{2} e^{\left (d x + c\right )} - i \, a d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.61, size = 133, normalized size = 1.48 \[ -\frac {i \, d^{2} f x^{2} e^{\left (d x + 2 \, c\right )} + d^{2} f x^{2} e^{c} + 2 i \, d^{2} x e^{\left (d x + 2 \, c + 1\right )} - 4 i \, d f x e^{\left (d x + 2 \, c\right )} + 2 \, d^{2} x e^{\left (c + 1\right )} + 4 i \, f e^{\left (d x + 2 \, c\right )} \log \left (e^{\left (d x + c\right )} - i\right ) + 4 \, f e^{c} \log \left (e^{\left (d x + c\right )} - i\right ) + 4 \, d e^{\left (c + 1\right )}}{2 \, {\left (a d^{2} e^{\left (d x + 2 \, c\right )} - i \, a d^{2} e^{c}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 86, normalized size = 0.96 \[ -\frac {i f \,x^{2}}{2 a}-\frac {i e x}{a}+\frac {2 i f x}{a d}+\frac {2 i f c}{a \,d^{2}}-\frac {2 \left (f x +e \right )}{d a \left ({\mathrm e}^{d x +c}-i\right )}-\frac {2 i f \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 108, normalized size = 1.20 \[ \frac {1}{2} \, f {\left (\frac {-i \, d x^{2} + {\left (d x^{2} e^{c} - 4 \, x e^{c}\right )} e^{\left (d x\right )}}{i \, a d e^{\left (d x + c\right )} + a d} - \frac {4 i \, \log \left ({\left (e^{\left (d x + c\right )} - i\right )} e^{\left (-c\right )}\right )}{a d^{2}}\right )} + \frac {1}{2} \, e {\left (-\frac {2 i \, {\left (d x + c\right )}}{a d} - \frac {4}{{\left (a e^{\left (-d x - c\right )} + i \, a\right )} d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.55, size = 74, normalized size = 0.82 \[ -\frac {f\,x^2\,1{}\mathrm {i}}{2\,a}-\frac {2\,\left (e+f\,x\right )}{a\,d\,\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )}+\frac {x\,\left (2\,f-d\,e\right )\,1{}\mathrm {i}}{a\,d}-\frac {f\,\ln \left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-\mathrm {i}\right )\,2{}\mathrm {i}}{a\,d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 83, normalized size = 0.92 \[ \frac {2 e e^{c} + 2 f x e^{c}}{- i a d e^{c} - a d e^{- d x}} - \frac {i f x^{2}}{2 a} + \frac {x \left (- i d e - 2 i f\right )}{a d} - \frac {2 i f \log {\left (i e^{c} + e^{- d x} \right )}}{a d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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