Optimal. Leaf size=163 \[ \frac {2 i (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {2 f \log \left (\cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {i f \text {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {i f \text {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {(e+f x) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]
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Rubi [A]
time = 0.18, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {5694, 4269,
3556, 4267, 2317, 2438, 3399} \begin {gather*} \frac {i f \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {i f \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {2 f \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )\right )}{a d^2}+\frac {2 i (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {(e+f x) \coth (c+d x)}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2438
Rule 3399
Rule 3556
Rule 4267
Rule 4269
Rule 5694
Rubi steps
\begin {align*} \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\left (i \int \frac {(e+f x) \text {csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx\right )+\frac {\int (e+f x) \text {csch}^2(c+d x) \, dx}{a}\\ &=-\frac {(e+f x) \coth (c+d x)}{a d}-\frac {i \int (e+f x) \text {csch}(c+d x) \, dx}{a}+\frac {f \int \coth (c+d x) \, dx}{a d}-\int \frac {e+f x}{a+i a \sinh (c+d x)} \, dx\\ &=\frac {2 i (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {f \log (\sinh (c+d x))}{a d^2}-\frac {\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 f) \int \log \left (1-e^{c+d x}\right ) \, dx}{a d}-\frac {(i f) \int \log \left (1+e^{c+d x}\right ) \, dx}{a d}\\ &=\frac {2 i (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {f \log (\sinh (c+d x))}{a d^2}-\frac {(e+f x) \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(i f) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^2}-\frac {(i f) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{a d^2}+\frac {f \int \coth \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{a d}\\ &=\frac {2 i (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {2 f \log \left (\cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {i f \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {i f \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {(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] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(454\) vs. \(2(163)=326\).
time = 3.46, size = 454, normalized size = 2.79 \begin {gather*} \frac {\left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right ) \left (-d (e+f x) \cosh \left (\frac {1}{2} (c+d x)\right ) \left (i+\coth \left (\frac {1}{2} (c+d x)\right )\right )+4 i f \text {ArcTan}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )+2 f \log (\cosh (c+d x)) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )+2 f \log (\sinh (c+d x)) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )+2 i c f \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )-2 i f \left ((c+d x) \left (\log \left (1-e^{-c-d x}\right )-\log \left (1+e^{-c-d x}\right )\right )+\text {PolyLog}\left (2,-e^{-c-d x}\right )-\text {PolyLog}\left (2,e^{-c-d x}\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )-4 d (e+f x) \sinh \left (\frac {1}{2} (c+d x)\right )+2 f (c+d x) \left (-i \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )+2 d e \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right ) \left (-i \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )-i d (e+f x) \sinh \left (\frac {1}{2} (c+d x)\right ) \left (-i+\tanh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{2 d^2 (a+i a \sinh (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 315 vs. \(2 (143 ) = 286\).
time = 3.07, size = 316, normalized size = 1.94
method | result | size |
risch | \(-\frac {2 i \left (f x \,{\mathrm e}^{2 d x +2 c}+e \,{\mathrm e}^{2 d x +2 c}-2 f x -i {\mathrm e}^{d x +c} f x -2 e -i {\mathrm e}^{d x +c} e \right )}{\left ({\mathrm e}^{2 d x +2 c}-1\right ) \left ({\mathrm e}^{d x +c}-i\right ) a d}+\frac {i \ln \left ({\mathrm e}^{d x +c}+1\right ) f x}{a d}-\frac {i \ln \left (1-{\mathrm e}^{d x +c}\right ) f x}{a d}-\frac {i \ln \left (1-{\mathrm e}^{d x +c}\right ) c f}{a \,d^{2}}+\frac {i e \ln \left ({\mathrm e}^{d x +c}+1\right )}{a d}-\frac {i e \ln \left ({\mathrm e}^{d x +c}-1\right )}{a d}+\frac {2 f \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{2}}+\frac {f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}-\frac {4 f \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {i f \polylog \left (2, -{\mathrm e}^{d x +c}\right )}{a \,d^{2}}-\frac {i f \polylog \left (2, {\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {f \ln \left ({\mathrm e}^{d x +c}+1\right )}{a \,d^{2}}+\frac {i f c \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}\) | \(316\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 520 vs. \(2 (142) = 284\).
time = 0.47, size = 520, normalized size = 3.19 \begin {gather*} \frac {-2 i \, c f + {\left (i \, f e^{\left (3 \, d x + 3 \, c\right )} + f e^{\left (2 \, d x + 2 \, c\right )} - i \, f e^{\left (d x + c\right )} - f\right )} {\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) + {\left (-i \, f e^{\left (3 \, d x + 3 \, c\right )} - f e^{\left (2 \, d x + 2 \, c\right )} + i \, f e^{\left (d x + c\right )} + f\right )} {\rm Li}_2\left (e^{\left (d x + c\right )}\right ) + 4 i \, d e - 2 \, {\left (2 \, d f x + c f\right )} e^{\left (3 \, d x + 3 \, c\right )} - 2 \, {\left (-i \, d f x - i \, c f + i \, d e\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (d f x + c f - d e\right )} e^{\left (d x + c\right )} - {\left (d f x + d e - {\left (i \, d f x + i \, d e + f\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d f x + d e - i \, f\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (-i \, d f x - i \, d e - f\right )} e^{\left (d x + c\right )} - i \, f\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) + 2 \, {\left (f e^{\left (3 \, d x + 3 \, c\right )} - i \, f e^{\left (2 \, d x + 2 \, c\right )} - f e^{\left (d x + c\right )} + i \, f\right )} \log \left (e^{\left (d x + c\right )} - i\right ) - {\left ({\left (c - i\right )} f - d e - {\left ({\left (i \, c + 1\right )} f - i \, d e\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left ({\left (c - i\right )} f - d e\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left ({\left (-i \, c - 1\right )} f + i \, d e\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} - 1\right ) + {\left (d f x + c f + {\left (-i \, d f x - i \, c f\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d f x + c f\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (i \, d f x + i \, c f\right )} e^{\left (d x + c\right )}\right )} \log \left (-e^{\left (d x + c\right )} + 1\right )}{a d^{2} e^{\left (3 \, d x + 3 \, c\right )} - i \, a d^{2} e^{\left (2 \, d x + 2 \, c\right )} - a d^{2} e^{\left (d x + c\right )} + i \, a d^{2}} \end {gather*}
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 \begin {gather*} - \frac {i \left (\int \frac {e \operatorname {csch}^{2}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {f x \operatorname {csch}^{2}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx\right )}{a} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {e+f\,x}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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