Optimal. Leaf size=368 \[ \frac {2 i (e+f x)^2}{a d}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {i (e+f x)^2 \coth (c+d x)}{a d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {2 i f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 f (e+f x) \text {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {4 i f^2 \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {3 f (e+f x) \text {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {i f^2 \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {3 f^2 \text {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {3 f^2 \text {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]
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Rubi [A]
time = 0.60, antiderivative size = 368, normalized size of antiderivative = 1.00, number
of steps used = 30, number of rules used = 13, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules
used = {5694, 4271, 3855, 4267, 2611, 2320, 6724, 4269, 3797, 2221, 2317, 2438, 3399}
\begin {gather*} -\frac {4 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {i f^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {3 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {3 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {3 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {3 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {2 i f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}+\frac {i (e+f x)^2 \coth (c+d x)}{a d}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {2 i (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 3855
Rule 4267
Rule 4269
Rule 4271
Rule 5694
Rule 6724
Rubi steps
\begin {align*} \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\left (i \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx\right )+\frac {\int (e+f x)^2 \text {csch}^3(c+d x) \, dx}{a}\\ &=-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {i \int (e+f x)^2 \text {csch}^2(c+d x) \, dx}{a}-\frac {\int (e+f x)^2 \text {csch}(c+d x) \, dx}{2 a}+\frac {f^2 \int \text {csch}(c+d x) \, dx}{a d^2}-\int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx\\ &=\frac {(e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {i (e+f x)^2 \coth (c+d x)}{a d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}+i \int \frac {(e+f x)^2}{a+i a \sinh (c+d x)} \, dx-\frac {\int (e+f x)^2 \text {csch}(c+d x) \, dx}{a}-\frac {(2 i f) \int (e+f x) \coth (c+d x) \, dx}{a d}+\frac {f \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a d}-\frac {f \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a d}\\ &=\frac {i (e+f x)^2}{a d}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {i (e+f x)^2 \coth (c+d x)}{a d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {i \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 {(4 i f) \int \frac {e^{2 (c+d x)} (e+f x)}{1-e^{2 (c+d x)}} \, dx}{a d}+\frac {(2 f) \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a d}-\frac {(2 f) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a d}-\frac {f^2 \int \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}+\frac {f^2 \int \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}\\ &=\frac {i (e+f x)^2}{a d}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {i (e+f x)^2 \coth (c+d x)}{a d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {2 i f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {3 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(2 i f) \int (e+f x) \coth \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{a d}-\frac {f^2 \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 {\text {Li}_2(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac {\left (2 i f^2\right ) \int \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a d^2}-\frac {\left (2 f^2\right ) \int \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (2 f^2\right ) \int \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}\\ &=\frac {2 i (e+f x)^2}{a d}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {i (e+f x)^2 \coth (c+d x)}{a d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {2 i f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {3 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(4 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 (i f^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{a d^3}-\frac {\left (2 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 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}\\ &=\frac {2 i (e+f x)^2}{a d}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {i (e+f x)^2 \coth (c+d x)}{a d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {2 i f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {3 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {i f^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {3 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {3 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {\left (4 i 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 i (e+f x)^2}{a d}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {i (e+f x)^2 \coth (c+d x)}{a d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {2 i f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {3 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {i f^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {3 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {3 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {\left (4 i 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 i (e+f x)^2}{a d}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {i (e+f x)^2 \coth (c+d x)}{a d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {2 i f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {4 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {3 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {i f^2 \text {Li}_2\left (e^{2 (c+d x)}\right )}{a d^3}-\frac {3 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {3 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {i (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. \(1370\) vs. \(2(368)=736\).
time = 21.34, size = 1370, normalized size = 3.72 \begin {gather*} \frac {2 f \left (d \left (d e^c x (2 e+f x)-2 \left (-i+e^c\right ) (e+f x) \log \left (1+i e^{c+d x}\right )\right )-2 \left (-i+e^c\right ) f \text {PolyLog}\left (2,-i e^{c+d x}\right )\right )}{a d^3 \left (-1-i e^c\right )}+\frac {8 i d e e^{2 c} f x-8 i d e \left (-1+e^{2 c}\right ) f x+4 i d e^{2 c} f^2 x^2-4 i d \left (-1+e^{2 c}\right ) f^2 x^2+6 d e^2 \left (-1+e^{2 c}\right ) \tanh ^{-1}\left (e^{c+d x}\right )-\frac {4 \left (-1+e^{2 c}\right ) f^2 \tanh ^{-1}\left (e^{c+d x}\right )}{d}+4 i e \left (-1+e^{2 c}\right ) f \left (2 d x-\log \left (1-e^{2 (c+d x)}\right )\right )+6 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 )+\frac {2 i \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}+\frac {3 \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^2 \left (-1+e^{2 c}\right )}+\frac {\text {csch}(c) \text {csch}^2(c+d x) \left (2 e f \cosh \left (\frac {d x}{2}\right )+2 f^2 x \cosh \left (\frac {d x}{2}\right )+2 e f \cosh \left (\frac {3 d x}{2}\right )+2 f^2 x \cosh \left (\frac {3 d x}{2}\right )+5 i d e^2 \cosh \left (c-\frac {d x}{2}\right )+10 i d e f x \cosh \left (c-\frac {d x}{2}\right )+5 i d f^2 x^2 \cosh \left (c-\frac {d x}{2}\right )-i d e^2 \cosh \left (c+\frac {d x}{2}\right )-2 i d e f x \cosh \left (c+\frac {d x}{2}\right )-i d f^2 x^2 \cosh \left (c+\frac {d x}{2}\right )-2 e f \cosh \left (2 c+\frac {d x}{2}\right )-2 f^2 x \cosh \left (2 c+\frac {d x}{2}\right )+i d e^2 \cosh \left (c+\frac {3 d x}{2}\right )+2 i d e f x \cosh \left (c+\frac {3 d x}{2}\right )+i d f^2 x^2 \cosh \left (c+\frac {3 d x}{2}\right )-2 e f \cosh \left (2 c+\frac {3 d x}{2}\right )-2 f^2 x \cosh \left (2 c+\frac {3 d x}{2}\right )-3 i d e^2 \cosh \left (3 c+\frac {3 d x}{2}\right )-6 i d e f x \cosh \left (3 c+\frac {3 d x}{2}\right )-3 i d f^2 x^2 \cosh \left (3 c+\frac {3 d x}{2}\right )-4 i d e^2 \cosh \left (c+\frac {5 d x}{2}\right )-8 i d e f x \cosh \left (c+\frac {5 d x}{2}\right )-4 i d f^2 x^2 \cosh \left (c+\frac {5 d x}{2}\right )+2 i d e^2 \cosh \left (3 c+\frac {5 d x}{2}\right )+4 i d e f x \cosh \left (3 c+\frac {5 d x}{2}\right )+2 i d f^2 x^2 \cosh \left (3 c+\frac {5 d x}{2}\right )-d e^2 \sinh \left (\frac {d x}{2}\right )-2 d e f x \sinh \left (\frac {d x}{2}\right )-d f^2 x^2 \sinh \left (\frac {d x}{2}\right )-d e^2 \sinh \left (\frac {3 d x}{2}\right )-2 d e f x \sinh \left (\frac {3 d x}{2}\right )-d f^2 x^2 \sinh \left (\frac {3 d x}{2}\right )+2 i e f \sinh \left (c-\frac {d x}{2}\right )+2 i f^2 x \sinh \left (c-\frac {d x}{2}\right )+2 i e f \sinh \left (c+\frac {d x}{2}\right )+2 i f^2 x \sinh \left (c+\frac {d x}{2}\right )-3 d e^2 \sinh \left (2 c+\frac {d x}{2}\right )-6 d e f x \sinh \left (2 c+\frac {d x}{2}\right )-3 d f^2 x^2 \sinh \left (2 c+\frac {d x}{2}\right )+2 i e f \sinh \left (c+\frac {3 d x}{2}\right )+2 i f^2 x \sinh \left (c+\frac {3 d x}{2}\right )-d e^2 \sinh \left (2 c+\frac {3 d x}{2}\right )-2 d e f x \sinh \left (2 c+\frac {3 d x}{2}\right )-d f^2 x^2 \sinh \left (2 c+\frac {3 d x}{2}\right )-2 i e f \sinh \left (3 c+\frac {3 d x}{2}\right )-2 i f^2 x \sinh \left (3 c+\frac {3 d x}{2}\right )+2 d e^2 \sinh \left (2 c+\frac {5 d x}{2}\right )+4 d e f x \sinh \left (2 c+\frac {5 d x}{2}\right )+2 d f^2 x^2 \sinh \left (2 c+\frac {5 d x}{2}\right )\right )}{8 a d^2 \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*}
Warning: Unable to verify antiderivative.
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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. 1106 vs. \(2 (343 ) = 686\).
time = 3.25, size = 1107, normalized size = 3.01
method | result | size |
risch | \(-\frac {2 i \ln \left (1-{\mathrm e}^{d x +c}\right ) c \,f^{2}}{a \,d^{3}}-\frac {3 f^{2} \polylog \left (3, -{\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {2 i f^{2} c \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{3}}-\frac {4 i f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}-\frac {8 i f^{2} c \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {4 i f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}+\frac {4 i f^{2} x^{2}}{a d}+\frac {4 i c^{2} f^{2}}{a \,d^{3}}-\frac {6 d e f x \,{\mathrm e}^{4 d x +4 c}-2 i e f \,{\mathrm e}^{3 d x +3 c}+i d \,f^{2} x^{2} {\mathrm e}^{d x +c}+3 d \,f^{2} x^{2} {\mathrm e}^{4 d x +4 c}-6 i d e f x \,{\mathrm e}^{3 d x +3 c}+4 d \,e^{2}+2 i e f \,{\mathrm e}^{d x +c}+2 i d e f x \,{\mathrm e}^{d x +c}-3 i d \,f^{2} x^{2} {\mathrm e}^{3 d x +3 c}-2 i f^{2} x \,{\mathrm e}^{3 d x +3 c}-3 i d \,e^{2} {\mathrm e}^{3 d x +3 c}-5 d \,e^{2} {\mathrm e}^{2 d x +2 c}+8 d e f x +3 d \,e^{2} {\mathrm e}^{4 d x +4 c}+2 f^{2} x \,{\mathrm e}^{4 d x +4 c}+2 e f \,{\mathrm e}^{4 d x +4 c}-2 f^{2} x \,{\mathrm e}^{2 d x +2 c}-2 e f \,{\mathrm e}^{2 d x +2 c}+4 d \,f^{2} x^{2}-10 d e f x \,{\mathrm e}^{2 d x +2 c}-5 d \,f^{2} x^{2} {\mathrm e}^{2 d x +2 c}+2 i f^{2} x \,{\mathrm e}^{d x +c}+i d \,e^{2} {\mathrm e}^{d x +c}}{\left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} d^{2} \left ({\mathrm e}^{d x +c}-i\right ) a}-\frac {2 i f^{2} \polylog \left (2, {\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {2 i f^{2} \polylog \left (2, -{\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {3 e f \polylog \left (2, {\mathrm e}^{d x +c}\right )}{a \,d^{2}}-\frac {3 e^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a d}+\frac {3 e^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 a d}+\frac {f^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{3}}-\frac {f^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{a \,d^{3}}+\frac {3 f^{2} \polylog \left (3, {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {4 i f^{2} c \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{3}}+\frac {8 i f^{2} c x}{a \,d^{2}}-\frac {2 i e f \ln \left ({\mathrm e}^{d x +c}+1\right )}{a \,d^{2}}-\frac {4 i e f \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{2}}-\frac {2 i e f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}-\frac {2 i \ln \left ({\mathrm e}^{d x +c}+1\right ) f^{2} x}{a \,d^{2}}-\frac {2 i \ln \left (1-{\mathrm e}^{d x +c}\right ) f^{2} x}{a \,d^{2}}-\frac {4 i f^{2} \polylog \left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {8 i e f \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}-\frac {3 f^{2} \polylog \left (2, {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}+\frac {3 \ln \left ({\mathrm e}^{d x +c}+1\right ) f^{2} x^{2}}{2 a d}-\frac {3 \ln \left (1-{\mathrm e}^{d x +c}\right ) f^{2} x^{2}}{2 a d}+\frac {3 \ln \left ({\mathrm e}^{d x +c}+1\right ) e f x}{a d}-\frac {3 \ln \left (1-{\mathrm e}^{d x +c}\right ) e f x}{a d}-\frac {3 \ln \left (1-{\mathrm e}^{d x +c}\right ) c e f}{a \,d^{2}}-\frac {3 f^{2} c^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a \,d^{3}}+\frac {3 \ln \left (1-{\mathrm e}^{d x +c}\right ) c^{2} f^{2}}{2 a \,d^{3}}+\frac {3 e f c \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}+\frac {3 \polylog \left (2, -{\mathrm e}^{d x +c}\right ) f^{2} x}{a \,d^{2}}+\frac {3 e f \polylog \left (2, -{\mathrm e}^{d x +c}\right )}{a \,d^{2}}\) | \(1107\) |
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. 897 vs. \(2 (339) = 678\).
time = 0.52, size = 897, normalized size = 2.44 \begin {gather*} \frac {2 i \, f^{2} x^{2}}{a d} - \frac {1}{2} \, {\left (\frac {2 \, {\left (-i \, e^{\left (-d x - c\right )} - 5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 i \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4\right )}}{{\left (a e^{\left (-d x - c\right )} - 2 i \, a e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, a e^{\left (-3 \, d x - 3 \, c\right )} + i \, a e^{\left (-4 \, d x - 4 \, c\right )} + a e^{\left (-5 \, d x - 5 \, c\right )} + i \, a\right )} d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{a d}\right )} e^{2} + \frac {4 i \, f x e}{a d} - \frac {4 \, d f^{2} x^{2} + 8 \, d f x e + {\left (3 \, d f^{2} x^{2} e^{\left (4 \, c\right )} + 2 \, {\left (f^{2} e^{\left (4 \, c\right )} + 3 \, d f e^{\left (4 \, c + 1\right )}\right )} x + 2 \, f e^{\left (4 \, c + 1\right )}\right )} e^{\left (4 \, d x\right )} + {\left (-3 i \, d f^{2} x^{2} e^{\left (3 \, c\right )} - 2 \, {\left (i \, f^{2} e^{\left (3 \, c\right )} + 3 i \, d f e^{\left (3 \, c + 1\right )}\right )} x - 2 i \, f e^{\left (3 \, c + 1\right )}\right )} e^{\left (3 \, d x\right )} - {\left (5 \, d f^{2} x^{2} e^{\left (2 \, c\right )} + 2 \, {\left (f^{2} e^{\left (2 \, c\right )} + 5 \, d f e^{\left (2 \, c + 1\right )}\right )} x + 2 \, f e^{\left (2 \, c + 1\right )}\right )} e^{\left (2 \, d x\right )} + {\left (i \, d f^{2} x^{2} e^{c} - 2 \, {\left (-i \, d f e^{\left (c + 1\right )} - i \, f^{2} e^{c}\right )} x + 2 i \, f e^{\left (c + 1\right )}\right )} e^{\left (d x\right )}}{a d^{2} e^{\left (5 \, d x + 5 \, c\right )} - i \, a d^{2} e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a d^{2} e^{\left (3 \, d x + 3 \, c\right )} + 2 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}} - \frac {4 i \, f e \log \left (i \, e^{\left (d x + c\right )} + 1\right )}{a d^{2}} + \frac {3 \, {\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}}{2 \, a d^{3}} - \frac {3 \, {\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}}{2 \, a d^{3}} - \frac {4 i \, {\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 {{\left (2 i \, d f e + f^{2}\right )} x}{a d^{2}} + \frac {{\left (2 i \, d f e - f^{2}\right )} x}{a d^{2}} + \frac {{\left (3 \, d f e - 2 i \, 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 {{\left (3 \, d f e + 2 i \, 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 {{\left (2 i \, d f e + f^{2}\right )} \log \left (e^{\left (d x + c\right )} + 1\right )}{a d^{3}} - \frac {{\left (2 i \, d f e - f^{2}\right )} \log \left (e^{\left (d x + c\right )} - 1\right )}{a d^{3}} + \frac {d^{3} f^{2} x^{3} + {\left (3 \, d f e + 2 i \, f^{2}\right )} d^{2} x^{2}}{2 \, a d^{3}} - \frac {d^{3} f^{2} x^{3} + {\left (3 \, d f e - 2 i \, f^{2}\right )} d^{2} x^{2}}{2 \, 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. 2238 vs. \(2 (339) = 678\).
time = 0.38, size = 2238, normalized size = 6.08 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 )}^3\,\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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