Optimal. Leaf size=296 \[ -\frac {2 (e+f x)^2}{a d}+\frac {2 i (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^2 \coth (c+d x)}{a d}+\frac {4 f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {2 f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {2 i f (e+f x) \text {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {4 f^2 \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {2 i f (e+f x) \text {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {f^2 \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {2 i f^2 \text {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {2 i f^2 \text {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]
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Rubi [A]
time = 0.41, antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 11, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {5694, 4269,
3797, 2221, 2317, 2438, 4267, 2611, 2320, 6724, 3399} \begin {gather*} \frac {4 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {f^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {2 i f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {2 i f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {2 i f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {4 f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {2 f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {2 i (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {(e+f x)^2 \coth (c+d x)}{a d}-\frac {2 (e+f x)^2}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3399
Rule 3797
Rule 4267
Rule 4269
Rule 5694
Rule 6724
Rubi steps
\begin {align*} \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\left (i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx\right )+\frac {\int (e+f x)^2 \text {csch}^2(c+d x) \, dx}{a}\\ &=-\frac {(e+f x)^2 \coth (c+d x)}{a d}-\frac {i \int (e+f x)^2 \text {csch}(c+d x) \, dx}{a}+\frac {(2 f) \int (e+f x) \coth (c+d x) \, dx}{a d}-\int \frac {(e+f x)^2}{a+i a \sinh (c+d x)} \, dx\\ &=-\frac {(e+f x)^2}{a d}+\frac {2 i (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^2 \coth (c+d x)}{a d}-\frac {\int (e+f x)^2 \csc ^2\left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {i d x}{2}\right ) \, dx}{2 a}+\frac {(2 i f) \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a d}-\frac {(2 i f) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a d}-\frac {(4 f) \int \frac {e^{2 (c+d x)} (e+f x)}{1-e^{2 (c+d x)}} \, dx}{a d}\\ &=-\frac {(e+f x)^2}{a d}+\frac {2 i (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^2 \coth (c+d x)}{a d}+\frac {2 f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {2 i f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(2 f) \int (e+f x) \coth \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{a d}-\frac {\left (2 i f^2\right ) \int \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (2 i f^2\right ) \int \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}-\frac {\left (2 f^2\right ) \int \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a d^2}\\ &=-\frac {2 (e+f x)^2}{a d}+\frac {2 i (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^2 \coth (c+d x)}{a d}+\frac {2 f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {2 i f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(4 i f) \int \frac {e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )} (e+f x)}{1+i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}} \, dx}{a d}-\frac {\left (2 i f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac {\left (2 i f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}-\frac {f^2 \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{a d^3}\\ &=-\frac {2 (e+f x)^2}{a d}+\frac {2 i (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^2 \coth (c+d x)}{a d}+\frac {4 f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {2 f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {2 i f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {f^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {2 i f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {2 i f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (4 f^2\right ) \int \log \left (1+i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^2}\\ &=-\frac {2 (e+f x)^2}{a d}+\frac {2 i (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^2 \coth (c+d x)}{a d}+\frac {4 f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {2 f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {2 i f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {f^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {2 i f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {2 i f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (4 f^2\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{a d^3}\\ &=-\frac {2 (e+f x)^2}{a d}+\frac {2 i (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^2 \coth (c+d x)}{a d}+\frac {4 f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {2 f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {2 i f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {4 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {2 i f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {f^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {2 i f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {2 i f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^2 \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. \(659\) vs. \(2(296)=592\).
time = 11.07, size = 659, normalized size = 2.23 \begin {gather*} \frac {2 f \left (d \left (-\frac {d e^c x (2 e+f x)}{-i+e^c}+2 (e+f x) \log \left (1+i e^{c+d x}\right )\right )+2 f \text {PolyLog}\left (2,-i e^{c+d x}\right )\right )}{a d^3}+\frac {-4 e e^{2 c} f x+4 e \left (-1+e^{2 c}\right ) f x-2 e^{2 c} f^2 x^2+2 \left (-1+e^{2 c}\right ) f^2 x^2+2 i e^2 \left (-1+e^{2 c}\right ) \tanh ^{-1}\left (e^{c+d x}\right )-\frac {2 e \left (-1+e^{2 c}\right ) f \left (2 d x-\log \left (1-e^{2 (c+d x)}\right )\right )}{d}+\frac {2 i e \left (-1+e^{2 c}\right ) f \left (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 )}{d}-\frac {\left (-1+e^{2 c}\right ) f^2 \left (2 d x \left (d x-\log \left (1-e^{2 (c+d x)}\right )\right )-\text {PolyLog}\left (2,e^{2 (c+d x)}\right )\right )}{d^2}+\frac {i \left (-1+e^{2 c}\right ) f^2 \left (-d^2 x^2 \log \left (1-e^{c+d x}\right )+d^2 x^2 \log \left (1+e^{c+d x}\right )+2 d x \text {PolyLog}\left (2,-e^{c+d x}\right )-2 d x \text {PolyLog}\left (2,e^{c+d x}\right )-2 \text {PolyLog}\left (3,-e^{c+d x}\right )+2 \text {PolyLog}\left (3,e^{c+d x}\right )\right )}{d^2}}{a d \left (-1+e^{2 c}\right )}+\frac {\text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-e^2 \sinh \left (\frac {d x}{2}\right )-2 e f x \sinh \left (\frac {d x}{2}\right )-f^2 x^2 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {\text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^2 \sinh \left (\frac {d x}{2}\right )+2 e f x \sinh \left (\frac {d x}{2}\right )+f^2 x^2 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d}-\frac {2 \left (e^2 \sinh \left (\frac {d x}{2}\right )+2 e f x \sinh \left (\frac {d x}{2}\right )+f^2 x^2 \sinh \left (\frac {d x}{2}\right )\right )}{a d \left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )+i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \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. 846 vs. \(2 (275 ) = 550\).
time = 3.01, size = 847, normalized size = 2.86
method | result | size |
risch | \(\frac {2 e f \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a}-\frac {8 f^{2} c x}{d^{2} a}+\frac {2 f^{2} \polylog \left (2, -{\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {2 f^{2} \polylog \left (2, {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {4 f^{2} \polylog \left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {2 i e f c \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a}-\frac {2 i \ln \left (1-{\mathrm e}^{d x +c}\right ) c e f}{d^{2} a}-\frac {2 i \ln \left (1-{\mathrm e}^{d x +c}\right ) e f x}{d a}+\frac {2 i \ln \left ({\mathrm e}^{d x +c}+1\right ) e f x}{d a}-\frac {8 e f \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} a}+\frac {2 e f \ln \left ({\mathrm e}^{d x +c}+1\right )}{d^{2} a}-\frac {2 f^{2} c \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{3} a}+\frac {8 f^{2} c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{3} a}+\frac {2 f^{2} \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{d^{2} a}+\frac {2 f^{2} \ln \left (1-{\mathrm e}^{d x +c}\right ) x}{d^{2} a}+\frac {2 f^{2} \ln \left (1-{\mathrm e}^{d x +c}\right ) c}{d^{3} a}+\frac {2 i f^{2} \polylog \left (2, -{\mathrm e}^{d x +c}\right ) x}{d^{2} a}-\frac {2 i f^{2} \polylog \left (3, -{\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {i e^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{d a}-\frac {4 f^{2} c \ln \left ({\mathrm e}^{d x +c}-i\right )}{d^{3} a}+\frac {i e^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{d a}+\frac {4 f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{d^{2} a}+\frac {4 e f \ln \left ({\mathrm e}^{d x +c}-i\right )}{d^{2} a}+\frac {2 i f^{2} \polylog \left (3, {\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {2 i f^{2} \polylog \left (2, {\mathrm e}^{d x +c}\right ) x}{d^{2} a}-\frac {4 f^{2} x^{2}}{a d}-\frac {4 f^{2} c^{2}}{a \,d^{3}}+\frac {i f^{2} \ln \left ({\mathrm e}^{d x +c}+1\right ) x^{2}}{d a}+\frac {i f^{2} \ln \left (1-{\mathrm e}^{d x +c}\right ) c^{2}}{d^{3} a}-\frac {i f^{2} c^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{3} a}-\frac {i f^{2} \ln \left (1-{\mathrm e}^{d x +c}\right ) x^{2}}{d a}-\frac {2 i \left (f^{2} x^{2} {\mathrm e}^{2 d x +2 c}+2 e f x \,{\mathrm e}^{2 d x +2 c}+e^{2} {\mathrm e}^{2 d x +2 c}-2 x^{2} f^{2}-i {\mathrm e}^{d x +c} f^{2} x^{2}-4 e f x -2 i {\mathrm e}^{d x +c} e f x -2 e^{2}-i {\mathrm e}^{d x +c} e^{2}\right )}{\left ({\mathrm e}^{2 d x +2 c}-1\right ) \left ({\mathrm e}^{d x +c}-i\right ) a d}+\frac {2 i e f \polylog \left (2, -{\mathrm e}^{d x +c}\right )}{d^{2} a}-\frac {2 i e f \polylog \left (2, {\mathrm e}^{d x +c}\right )}{d^{2} a}+\frac {4 f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}\) | \(847\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 611 vs. \(2 (271) = 542\).
time = 0.45, size = 611, normalized size = 2.06 \begin {gather*} -\frac {2 \, f^{2} x^{2}}{a d} - {\left (\frac {2 \, {\left (e^{\left (-d x - c\right )} - i \, e^{\left (-2 \, d x - 2 \, c\right )} + 2 i\right )}}{{\left (a e^{\left (-d x - c\right )} - i \, a e^{\left (-2 \, d x - 2 \, c\right )} - a e^{\left (-3 \, d x - 3 \, c\right )} + i \, a\right )} d} - \frac {i \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac {i \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{a d}\right )} e^{2} - \frac {8 \, f x e}{a d} - \frac {2 \, {\left (-2 i \, f^{2} x^{2} - 4 i \, f x e - {\left (-i \, f^{2} x^{2} e^{\left (2 \, c\right )} - 2 i \, f x e^{\left (2 \, c + 1\right )}\right )} e^{\left (2 \, d x\right )} + {\left (f^{2} x^{2} e^{c} + 2 \, f x e^{\left (c + 1\right )}\right )} e^{\left (d x\right )}\right )}}{a d e^{\left (3 \, d x + 3 \, c\right )} - i \, a d e^{\left (2 \, d x + 2 \, c\right )} - a d e^{\left (d x + c\right )} + i \, a d} + \frac {2 \, f e \log \left (e^{\left (d x + c\right )} + 1\right )}{a d^{2}} + \frac {4 \, f e \log \left (e^{\left (d x + c\right )} - i\right )}{a d^{2}} + \frac {2 \, f e \log \left (e^{\left (d x + c\right )} - 1\right )}{a d^{2}} + \frac {i \, {\left (d^{2} x^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (d x + c\right )})\right )} f^{2}}{a d^{3}} - \frac {i \, {\left (d^{2} x^{2} \log \left (-e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (d x + c\right )})\right )} f^{2}}{a d^{3}} + \frac {4 \, {\left (d x \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right )\right )} f^{2}}{a d^{3}} - \frac {2 \, {\left (-i \, d f e - f^{2}\right )} {\left (d x \log \left (e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (d x + c\right )}\right )\right )}}{a d^{3}} + \frac {2 \, {\left (-i \, d f e + f^{2}\right )} {\left (d x \log \left (-e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (d x + c\right )}\right )\right )}}{a d^{3}} + \frac {i \, d^{3} f^{2} x^{3} - 3 \, {\left (-i \, d f e + f^{2}\right )} d^{2} x^{2}}{3 \, a d^{3}} - \frac {i \, d^{3} f^{2} x^{3} - 3 \, {\left (-i \, d f e - f^{2}\right )} d^{2} x^{2}}{3 \, a d^{3}} \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. 1381 vs. \(2 (271) = 542\).
time = 0.37, size = 1381, normalized size = 4.67 \begin {gather*} \frac {4 i \, c^{2} f^{2} - 8 i \, c d f e + 4 i \, d^{2} e^{2} + 4 \, {\left (f^{2} e^{\left (3 \, d x + 3 \, c\right )} - i \, f^{2} e^{\left (2 \, d x + 2 \, c\right )} - f^{2} e^{\left (d x + c\right )} + i \, f^{2}\right )} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - 2 \, {\left (d f^{2} x + d f e - i \, f^{2} + {\left (-i \, d f^{2} x - i \, d f e - f^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d f^{2} x + d f e - i \, f^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (i \, d f^{2} x + i \, d f e + f^{2}\right )} e^{\left (d x + c\right )}\right )} {\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) + 2 \, {\left (d f^{2} x + d f e + i \, f^{2} - {\left (i \, d f^{2} x + i \, d f e - f^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d f^{2} x + d f e + i \, f^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (-i \, d f^{2} x - i \, d f e + f^{2}\right )} e^{\left (d x + c\right )}\right )} {\rm Li}_2\left (e^{\left (d x + c\right )}\right ) - 4 \, {\left (d^{2} f^{2} x^{2} - c^{2} f^{2} + 2 \, {\left (d^{2} f x + c d f\right )} e\right )} e^{\left (3 \, d x + 3 \, c\right )} - 2 \, {\left (-i \, d^{2} f^{2} x^{2} + 2 i \, c^{2} f^{2} + i \, d^{2} e^{2} + 2 \, {\left (-i \, d^{2} f x - 2 i \, c d f\right )} e\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (d^{2} f^{2} x^{2} - 2 \, c^{2} f^{2} - d^{2} e^{2} + 2 \, {\left (d^{2} f x + 2 \, c d f\right )} e\right )} e^{\left (d x + c\right )} - {\left (d^{2} f^{2} x^{2} - 2 i \, d f^{2} x + d^{2} e^{2} + 2 \, {\left (d^{2} f x - i \, d f\right )} e - {\left (i \, d^{2} f^{2} x^{2} + 2 \, d f^{2} x + i \, d^{2} e^{2} - 2 \, {\left (-i \, d^{2} f x - d f\right )} e\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d^{2} f^{2} x^{2} - 2 i \, d f^{2} x + d^{2} e^{2} + 2 \, {\left (d^{2} f x - i \, d f\right )} e\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (-i \, d^{2} f^{2} x^{2} - 2 \, d f^{2} x - i \, d^{2} e^{2} - 2 \, {\left (i \, d^{2} f x + d f\right )} e\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) - 4 \, {\left (i \, c f^{2} - i \, d f e + {\left (c f^{2} - d f e\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (-i \, c f^{2} + i \, d f e\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (c f^{2} - d f e\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) - {\left (2 \, {\left (c - i\right )} d f e - {\left (c^{2} - 2 i \, c\right )} f^{2} - d^{2} e^{2} + {\left (2 \, {\left (-i \, c - 1\right )} d f e - {\left (-i \, c^{2} - 2 \, c\right )} f^{2} + i \, d^{2} e^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (2 \, {\left (c - i\right )} d f e - {\left (c^{2} - 2 i \, c\right )} f^{2} - d^{2} e^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (2 \, {\left (i \, c + 1\right )} d f e - {\left (i \, c^{2} + 2 \, c\right )} f^{2} - i \, d^{2} e^{2}\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} - 1\right ) - 4 \, {\left (-i \, d f^{2} x - i \, c f^{2} - {\left (d f^{2} x + c f^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (i \, d f^{2} x + i \, c f^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (d f^{2} x + c f^{2}\right )} e^{\left (d x + c\right )}\right )} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + {\left (d^{2} f^{2} x^{2} + 2 i \, d f^{2} x - {\left (c^{2} - 2 i \, c\right )} f^{2} + 2 \, {\left (d^{2} f x + c d f\right )} e + {\left (-i \, d^{2} f^{2} x^{2} + 2 \, d f^{2} x + {\left (i \, c^{2} + 2 \, c\right )} f^{2} - 2 \, {\left (i \, d^{2} f x + i \, c d f\right )} e\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d^{2} f^{2} x^{2} + 2 i \, d f^{2} x - {\left (c^{2} - 2 i \, c\right )} f^{2} + 2 \, {\left (d^{2} f x + c d f\right )} e\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (i \, d^{2} f^{2} x^{2} - 2 \, d f^{2} x + {\left (-i \, c^{2} - 2 \, c\right )} f^{2} - 2 \, {\left (-i \, d^{2} f x - i \, c d f\right )} e\right )} e^{\left (d x + c\right )}\right )} \log \left (-e^{\left (d x + c\right )} + 1\right ) - 2 \, {\left (i \, f^{2} e^{\left (3 \, d x + 3 \, c\right )} + f^{2} e^{\left (2 \, d x + 2 \, c\right )} - i \, f^{2} e^{\left (d x + c\right )} - f^{2}\right )} {\rm polylog}\left (3, -e^{\left (d x + c\right )}\right ) - 2 \, {\left (-i \, f^{2} e^{\left (3 \, d x + 3 \, c\right )} - f^{2} e^{\left (2 \, d x + 2 \, c\right )} + i \, f^{2} e^{\left (d x + c\right )} + f^{2}\right )} {\rm polylog}\left (3, e^{\left (d x + c\right )}\right )}{a d^{3} e^{\left (3 \, d x + 3 \, c\right )} - i \, a d^{3} e^{\left (2 \, d x + 2 \, c\right )} - a d^{3} e^{\left (d x + c\right )} + i \, a d^{3}} \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^{2} \operatorname {csch}^{2}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {f^{2} x^{2} \operatorname {csch}^{2}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {2 e 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 = -0.00 \begin {gather*} \int \frac {{\left (e+f\,x\right )}^2}{{\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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