Integrand size = 31, antiderivative size = 241 \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {12 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]
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Time = 0.35 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {5676, 3377, 2717, 32, 3399, 4269, 3797, 2221, 2611, 2320, 6724} \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {12 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {12 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f} \]
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Rule 32
Rule 2221
Rule 2320
Rule 2611
Rule 2717
Rule 3377
Rule 3399
Rule 3797
Rule 4269
Rule 5676
Rule 6724
Rubi steps \begin{align*} \text {integral}& = i \int \frac {(e+f x)^3 \sinh (c+d x)}{a+i a \sinh (c+d x)} \, dx-\frac {i \int (e+f x)^3 \sinh (c+d x) \, dx}{a} \\ & = -\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {\int (e+f x)^3 \, dx}{a}+\frac {(3 i f) \int (e+f x)^2 \cosh (c+d x) \, dx}{a d}-\int \frac {(e+f x)^3}{a+i a \sinh (c+d x)} \, dx \\ & = \frac {(e+f x)^4}{4 a f}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {\int (e+f x)^3 \csc ^2\left (\frac {1}{2} \left (i c+\frac {\pi }{2}\right )+\frac {i d x}{2}\right ) \, dx}{2 a}-\frac {\left (6 i f^2\right ) \int (e+f x) \sinh (c+d x) \, dx}{a d^2} \\ & = \frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(3 f) \int (e+f x)^2 \coth \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{a d}+\frac {\left (6 i f^3\right ) \int \cosh (c+d x) \, dx}{a d^3} \\ & = -\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(6 i f) \int \frac {e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )} (e+f x)^2}{1+i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}} \, dx}{a d} \\ & = -\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 f^2\right ) \int (e+f x) \log \left (1+i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^2} \\ & = -\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 f^3\right ) \int \operatorname {PolyLog}\left (2,-i e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^3} \\ & = -\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {\left (12 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{2 \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{a d^4} \\ & = -\frac {(e+f x)^3}{a d}+\frac {(e+f x)^4}{4 a f}-\frac {6 i f^2 (e+f x) \cosh (c+d x)}{a d^3}-\frac {i (e+f x)^3 \cosh (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {12 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^3 \sinh (c+d x)}{a d^4}+\frac {3 i f (e+f x)^2 \sinh (c+d x)}{a d^2}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2619\) vs. \(2(241)=482\).
Time = 3.58 (sec) , antiderivative size = 2619, normalized size of antiderivative = 10.87 \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Result too large to show} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 698 vs. \(2 (222 ) = 444\).
Time = 2.90 (sec) , antiderivative size = 699, normalized size of antiderivative = 2.90
method | result | size |
risch | \(\frac {f^{3} x^{4}}{4 a}+\frac {e^{4}}{4 a f}+\frac {12 f^{2} e \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}-\frac {12 f^{2} e c \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{3}}+\frac {12 f^{2} e \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}-\frac {6 f^{2} e \,c^{2}}{a \,d^{3}}-\frac {6 f \ln \left ({\mathrm e}^{d x +c}\right ) e^{2}}{a \,d^{2}}-\frac {6 f^{2} e \,x^{2}}{a d}-\frac {6 f^{3} c^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{4}}+\frac {6 f^{3} x \,c^{2}}{a \,d^{3}}+\frac {f^{2} e \,x^{3}}{a}+\frac {3 f \,e^{2} x^{2}}{2 a}+\frac {e^{3} x}{a}+\frac {4 f^{3} c^{3}}{a \,d^{4}}-\frac {2 f^{3} x^{3}}{a d}-\frac {12 f^{2} e c x}{a \,d^{2}}+\frac {12 f^{2} e c \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {12 f^{3} \operatorname {polylog}\left (3, -i {\mathrm e}^{d x +c}\right )}{a \,d^{4}}-\frac {i \left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x -3 d^{2} f^{3} x^{2}+e^{3} d^{3}-6 d^{2} e \,f^{2} x -3 d^{2} e^{2} f +6 d \,f^{3} x +6 e \,f^{2} d -6 f^{3}\right ) {\mathrm e}^{d x +c}}{2 d^{4} a}-\frac {2 i \left (f^{3} x^{3}+3 e \,f^{2} x^{2}+3 e^{2} f x +e^{3}\right )}{d a \left ({\mathrm e}^{d x +c}-i\right )}+\frac {6 f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x^{2}}{a \,d^{2}}+\frac {12 f^{2} e \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {6 f \ln \left ({\mathrm e}^{d x +c}-i\right ) e^{2}}{a \,d^{2}}+\frac {6 f^{3} c^{2} \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{4}}+\frac {12 f^{3} \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right ) x}{a \,d^{3}}-\frac {6 f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c^{2}}{a \,d^{4}}-\frac {i \left (d^{3} x^{3} f^{3}+3 d^{3} e \,f^{2} x^{2}+3 d^{3} e^{2} f x +3 d^{2} f^{3} x^{2}+e^{3} d^{3}+6 d^{2} e \,f^{2} x +3 d^{2} e^{2} f +6 d \,f^{3} x +6 e \,f^{2} d +6 f^{3}\right ) {\mathrm e}^{-d x -c}}{2 d^{4} a}\) | \(699\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 823 vs. \(2 (213) = 426\).
Time = 0.26 (sec) , antiderivative size = 823, normalized size of antiderivative = 3.41 \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {2 \, d^{3} f^{3} x^{3} + 2 \, d^{3} e^{3} + 6 \, d^{2} e^{2} f + 12 \, d e f^{2} + 12 \, f^{3} + 6 \, {\left (d^{3} e f^{2} + d^{2} f^{3}\right )} x^{2} + 6 \, {\left (d^{3} e^{2} f + 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} x - 48 \, {\left ({\left (d f^{3} x + d e f^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (i \, d f^{3} x + i \, d e f^{2}\right )} e^{\left (d x + c\right )}\right )} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) + 2 \, {\left (i \, d^{3} f^{3} x^{3} + i \, d^{3} e^{3} - 3 i \, d^{2} e^{2} f + 6 i \, d e f^{2} - 6 i \, f^{3} + 3 \, {\left (i \, d^{3} e f^{2} - i \, d^{2} f^{3}\right )} x^{2} + 3 \, {\left (i \, d^{3} e^{2} f - 2 i \, d^{2} e f^{2} + 2 i \, d f^{3}\right )} x\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (d^{4} f^{3} x^{4} - 2 \, d^{3} e^{3} - 6 \, {\left (4 \, c - 1\right )} d^{2} e^{2} f + 12 \, {\left (2 \, c^{2} - 1\right )} d e f^{2} - 4 \, {\left (2 \, c^{3} - 3\right )} f^{3} + 2 \, {\left (2 \, d^{4} e f^{2} - 5 \, d^{3} f^{3}\right )} x^{3} + 6 \, {\left (d^{4} e^{2} f - 5 \, d^{3} e f^{2} + d^{2} f^{3}\right )} x^{2} + 2 \, {\left (2 \, d^{4} e^{3} - 15 \, d^{3} e^{2} f + 6 \, d^{2} e f^{2} - 6 \, d f^{3}\right )} x\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (-i \, d^{4} f^{3} x^{4} - 10 i \, d^{3} e^{3} - 6 \, {\left (-4 i \, c + i\right )} d^{2} e^{2} f - 12 \, {\left (2 i \, c^{2} + i\right )} d e f^{2} - 4 \, {\left (-2 i \, c^{3} + 3 i\right )} f^{3} - 2 \, {\left (2 i \, d^{4} e f^{2} + i \, d^{3} f^{3}\right )} x^{3} - 6 \, {\left (i \, d^{4} e^{2} f + i \, d^{3} e f^{2} + i \, d^{2} f^{3}\right )} x^{2} - 2 \, {\left (2 i \, d^{4} e^{3} + 3 i \, d^{3} e^{2} f + 6 i \, d^{2} e f^{2} + 6 i \, d f^{3}\right )} x\right )} e^{\left (d x + c\right )} - 24 \, {\left ({\left (d^{2} e^{2} f - 2 \, c d e f^{2} + c^{2} f^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (i \, d^{2} e^{2} f - 2 i \, c d e f^{2} + i \, c^{2} f^{3}\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) - 24 \, {\left ({\left (d^{2} f^{3} x^{2} + 2 \, d^{2} e f^{2} x + 2 \, c d e f^{2} - c^{2} f^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (i \, d^{2} f^{3} x^{2} + 2 i \, d^{2} e f^{2} x + 2 i \, c d e f^{2} - i \, c^{2} f^{3}\right )} e^{\left (d x + c\right )}\right )} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + 48 \, {\left (f^{3} e^{\left (2 \, d x + 2 \, c\right )} - i \, f^{3} e^{\left (d x + c\right )}\right )} {\rm polylog}\left (3, -i \, e^{\left (d x + c\right )}\right )}{4 \, {\left (a d^{4} e^{\left (2 \, d x + 2 \, c\right )} - i \, a d^{4} e^{\left (d x + c\right )}\right )}} \]
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\[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {- 2 i e^{3} - 6 i e^{2} f x - 6 i e f^{2} x^{2} - 2 i f^{3} x^{3}}{a d e^{c} e^{d x} - i a d} - \frac {i \left (\int \frac {i d e^{3}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {i d f^{3} x^{3}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {d e^{3} e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {d e^{3} e^{3 c} e^{3 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \left (- \frac {12 e^{2} f e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\right )\, dx + \int \left (- \frac {12 f^{3} x^{2} e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\right )\, dx + \int \frac {3 i d e f^{2} x^{2}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 i d e^{2} f x}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {i d e^{3} e^{2 c} e^{2 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {d f^{3} x^{3} e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {d f^{3} x^{3} e^{3 c} e^{3 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \left (- \frac {24 e f^{2} x e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\right )\, dx + \int \frac {i d f^{3} x^{3} e^{2 c} e^{2 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 d e f^{2} x^{2} e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 d e f^{2} x^{2} e^{3 c} e^{3 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 d e^{2} f x e^{c} e^{d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 d e^{2} f x e^{3 c} e^{3 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 i d e f^{2} x^{2} e^{2 c} e^{2 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx + \int \frac {3 i d e^{2} f x e^{2 c} e^{2 d x}}{e^{c} e^{2 d x} - i e^{d x}}\, dx\right ) e^{- c}}{2 a d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 671 vs. \(2 (213) = 426\).
Time = 0.38 (sec) , antiderivative size = 671, normalized size of antiderivative = 2.78 \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {3}{2} \, e^{2} f {\left (\frac {2 \, x e^{\left (d x + c\right )}}{a d e^{\left (d x + c\right )} - i \, a d} + \frac {i \, d^{2} x^{2} e^{c} + i \, d x e^{c} - {\left (-i \, d x e^{\left (3 \, c\right )} + i \, e^{\left (3 \, c\right )}\right )} e^{\left (2 \, d x\right )} - {\left (d^{2} x^{2} e^{\left (2 \, c\right )} - 3 \, d x e^{\left (2 \, c\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} + {\left (d x + 1\right )} e^{\left (-d x\right )} + i \, e^{c}}{a d^{2} e^{\left (d x + 2 \, c\right )} - i \, a d^{2} e^{c}} - \frac {4 \, \log \left ({\left (e^{\left (d x + c\right )} - i\right )} e^{\left (-c\right )}\right )}{a d^{2}}\right )} + \frac {1}{2} \, e^{3} {\left (\frac {2 \, {\left (d x + c\right )}}{a d} + \frac {-5 i \, e^{\left (-d x - c\right )} + 1}{{\left (i \, a e^{\left (-d x - c\right )} + a e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} - \frac {i \, e^{\left (-d x - c\right )}}{a d}\right )} + \frac {-i \, d^{4} f^{3} x^{4} - 12 i \, d e f^{2} + 2 \, {\left (-2 i \, d^{4} e f^{2} - 5 i \, d^{3} f^{3}\right )} x^{3} - 12 i \, f^{3} + 6 \, {\left (-5 i \, d^{3} e f^{2} - i \, d^{2} f^{3}\right )} x^{2} + 12 \, {\left (-i \, d^{2} e f^{2} - i \, d f^{3}\right )} x + 2 \, {\left (-i \, d^{3} f^{3} x^{3} e^{\left (2 \, c\right )} + 3 \, {\left (-i \, d^{3} e f^{2} + i \, d^{2} f^{3}\right )} x^{2} e^{\left (2 \, c\right )} + 6 \, {\left (i \, d^{2} e f^{2} - i \, d f^{3}\right )} x e^{\left (2 \, c\right )} + 6 \, {\left (-i \, d e f^{2} + i \, f^{3}\right )} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} + {\left (d^{4} f^{3} x^{4} e^{c} + 2 \, {\left (2 \, d^{4} e f^{2} - d^{3} f^{3}\right )} x^{3} e^{c} - 6 \, {\left (d^{3} e f^{2} - d^{2} f^{3}\right )} x^{2} e^{c} + 12 \, {\left (d^{2} e f^{2} - d f^{3}\right )} x e^{c} - 12 \, {\left (d e f^{2} - f^{3}\right )} e^{c}\right )} e^{\left (d x\right )}}{4 \, {\left (a d^{4} e^{\left (d x + c\right )} - i \, a d^{4}\right )}} + \frac {12 \, {\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 )} e f^{2}}{a d^{3}} + \frac {6 \, {\left (d^{2} x^{2} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(-i \, e^{\left (d x + c\right )})\right )} f^{3}}{a d^{4}} - \frac {2 \, {\left (d^{3} f^{3} x^{3} + 3 \, d^{3} e f^{2} x^{2}\right )}}{a d^{4}} \]
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\[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \sinh \left (d x + c\right )^{2}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^3 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]
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