Optimal. Leaf size=287 \[ \frac {4 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {2 f^2 \cosh (c+d x)}{a d^3}-\frac {i f^2 \sinh (c+d x) \cosh (c+d x)}{4 a d^3}+\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}-\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {i (e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 a d}+\frac {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f} \]
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Rubi [A] time = 0.55, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {5557, 3311, 32, 2635, 8, 3296, 2638, 3318, 4184, 3716, 2190, 2279, 2391} \[ \frac {4 i f^2 \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}+\frac {2 f^2 \cosh (c+d x)}{a d^3}-\frac {i f^2 \sinh (c+d x) \cosh (c+d x)}{4 a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}-\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{a d}-\frac {i (e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{2 a d}+\frac {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 32
Rule 2190
Rule 2279
Rule 2391
Rule 2635
Rule 2638
Rule 3296
Rule 3311
Rule 3318
Rule 3716
Rule 4184
Rule 5557
Rubi steps
\begin {align*} \int \frac {(e+f x)^2 \sinh ^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=i \int \frac {(e+f x)^2 \sinh ^2(c+d x)}{a+i a \sinh (c+d x)} \, dx-\frac {i \int (e+f x)^2 \sinh ^2(c+d x) \, dx}{a}\\ &=-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}+\frac {i \int (e+f x)^2 \, dx}{2 a}+\frac {\int (e+f x)^2 \sinh (c+d x) \, dx}{a}-\frac {\left (i f^2\right ) \int \sinh ^2(c+d x) \, dx}{2 a d^2}-\int \frac {(e+f x)^2 \sinh (c+d x)}{a+i a \sinh (c+d x)} \, dx\\ &=\frac {i (e+f x)^3}{6 a f}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 a d^2}-i \int \frac {(e+f x)^2}{a+i a \sinh (c+d x)} \, dx+\frac {i \int (e+f x)^2 \, dx}{a}-\frac {(2 f) \int (e+f x) \cosh (c+d x) \, dx}{a d}+\frac {\left (i f^2\right ) \int 1 \, dx}{4 a d^2}\\ &=\frac {i f^2 x}{4 a d^2}+\frac {i (e+f x)^3}{2 a f}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 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 {\left (2 f^2\right ) \int \sinh (c+d x) \, dx}{a d^2}\\ &=\frac {i f^2 x}{4 a d^2}+\frac {i (e+f x)^3}{2 a f}+\frac {2 f^2 \cosh (c+d x)}{a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 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 {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f}+\frac {2 f^2 \cosh (c+d x)}{a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 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 {(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 {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f}+\frac {2 f^2 \cosh (c+d x)}{a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}+\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 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 {\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 {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f}+\frac {2 f^2 \cosh (c+d x)}{a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}+\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 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 {\left (4 i f^2\right ) \operatorname {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 {i f^2 x}{4 a d^2}-\frac {i (e+f x)^2}{a d}+\frac {i (e+f x)^3}{2 a f}+\frac {2 f^2 \cosh (c+d x)}{a d^3}+\frac {(e+f x)^2 \cosh (c+d x)}{a d}+\frac {4 i f (e+f x) \log \left (1+i 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 {2 f (e+f x) \sinh (c+d x)}{a d^2}-\frac {i f^2 \cosh (c+d x) \sinh (c+d x)}{4 a d^3}-\frac {i (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {i f (e+f x) \sinh ^2(c+d x)}{2 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}\\ \end {align*}
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Mathematica [B] time = 4.98, size = 1661, normalized size = 5.79 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 587, normalized size = 2.05 \[ \frac {2 \, d^{2} f^{2} x^{2} + 2 \, d^{2} e^{2} + 2 \, d e f + f^{2} + 2 \, {\left (2 \, d^{2} e f + d f^{2}\right )} x + {\left (64 i \, f^{2} e^{\left (3 \, d x + 3 \, c\right )} + 64 \, f^{2} e^{\left (2 \, d x + 2 \, c\right )}\right )} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) + {\left (-2 i \, d^{2} f^{2} x^{2} - 2 i \, d^{2} e^{2} + 2 i \, d e f - i \, f^{2} + {\left (-4 i \, d^{2} e f + 2 i \, d f^{2}\right )} x\right )} e^{\left (5 \, d x + 5 \, c\right )} + {\left (6 \, d^{2} f^{2} x^{2} + 6 \, d^{2} e^{2} - 14 \, d e f + 15 \, f^{2} + 2 \, {\left (6 \, d^{2} e f - 7 \, d f^{2}\right )} x\right )} e^{\left (4 \, d x + 4 \, c\right )} + {\left (8 i \, d^{3} f^{2} x^{3} - 8 i \, d^{2} e^{2} + {\left (-64 i \, c + 16 i\right )} d e f + {\left (32 i \, c^{2} - 16 i\right )} f^{2} + {\left (24 i \, d^{3} e f - 40 i \, d^{2} f^{2}\right )} x^{2} + {\left (24 i \, d^{3} e^{2} - 80 i \, d^{2} e f + 16 i \, d f^{2}\right )} x\right )} e^{\left (3 \, d x + 3 \, c\right )} + 8 \, {\left (d^{3} f^{2} x^{3} + 5 \, d^{2} e^{2} - 2 \, {\left (4 \, c - 1\right )} d e f + 2 \, {\left (2 \, c^{2} + 1\right )} f^{2} + {\left (3 \, d^{3} e f + d^{2} f^{2}\right )} x^{2} + {\left (3 \, d^{3} e^{2} + 2 \, d^{2} e f + 2 \, d f^{2}\right )} x\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (-6 i \, d^{2} f^{2} x^{2} - 6 i \, d^{2} e^{2} - 14 i \, d e f - 15 i \, f^{2} + {\left (-12 i \, d^{2} e f - 14 i \, d f^{2}\right )} x\right )} e^{\left (d x + c\right )} + {\left ({\left (64 i \, d e f - 64 i \, c f^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} + 64 \, {\left (d e f - c f^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) + {\left ({\left (64 i \, d f^{2} x + 64 i \, c f^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} + 64 \, {\left (d f^{2} x + c f^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )}\right )} \log \left (i \, e^{\left (d x + c\right )} + 1\right )}{16 \, a d^{3} e^{\left (3 \, d x + 3 \, c\right )} - 16 i \, a d^{3} e^{\left (2 \, d x + 2 \, c\right )}} \]
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 \[ \int \frac {{\left (f x + e\right )}^{2} \sinh \left (d x + c\right )^{3}}{i \, a \sinh \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 508, normalized size = 1.77 \[ \frac {4 i f^{2} c \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {4 i f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}+\frac {4 i \ln \left ({\mathrm e}^{d x +c}-i\right ) e f}{a \,d^{2}}+\frac {3 i e^{2} x}{2 a}+\frac {\left (d^{2} f^{2} x^{2}+2 d^{2} e f x +d^{2} e^{2}-2 d \,f^{2} x -2 d e f +2 f^{2}\right ) {\mathrm e}^{d x +c}}{2 a \,d^{3}}+\frac {\left (d^{2} f^{2} x^{2}+2 d^{2} e f x +d^{2} e^{2}+2 d \,f^{2} x +2 d e f +2 f^{2}\right ) {\mathrm e}^{-d x -c}}{2 a \,d^{3}}+\frac {i x^{3} f^{2}}{2 a}+\frac {2 x^{2} f^{2}+4 e f x +2 e^{2}}{d a \left ({\mathrm e}^{d x +c}-i\right )}-\frac {4 i f^{2} c x}{a \,d^{2}}+\frac {4 i f^{2} \polylog \left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {4 i \ln \left ({\mathrm e}^{d x +c}\right ) e f}{a \,d^{2}}-\frac {4 i f^{2} c \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{3}}+\frac {3 i e f \,x^{2}}{2 a}-\frac {2 i f^{2} c^{2}}{a \,d^{3}}+\frac {i \left (2 d^{2} f^{2} x^{2}+4 d^{2} e f x +2 d^{2} e^{2}+2 d \,f^{2} x +2 d e f +f^{2}\right ) {\mathrm e}^{-2 d x -2 c}}{16 a \,d^{3}}-\frac {2 i f^{2} x^{2}}{a d}+\frac {4 i f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}-\frac {i \left (2 d^{2} f^{2} x^{2}+4 d^{2} e f x +2 d^{2} e^{2}-2 d \,f^{2} x -2 d e f +f^{2}\right ) {\mathrm e}^{2 d x +2 c}}{16 a \,d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2}{a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]
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 \[ \frac {2 i e^{2} e^{c} + 4 i e f x e^{c} + 2 i f^{2} x^{2} e^{c}}{- a d e^{c} + i a d e^{- d x}} - \frac {i \left (\int \left (- \frac {i d e^{2}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {i d f^{2} x^{2}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {d e^{2} e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {4 d e^{2} e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {d e^{2} e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {16 e f e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {16 f^{2} x e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {2 i d e f x}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {4 i d e^{2} e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {i d e^{2} e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {d f^{2} x^{2} e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {4 d f^{2} x^{2} e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {d f^{2} x^{2} e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {4 i d f^{2} x^{2} e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {i d f^{2} x^{2} e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \left (- \frac {2 d e f x e^{c} e^{d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \left (- \frac {8 d e f x e^{3 c} e^{3 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\right )\, dx + \int \frac {2 d e f x e^{5 c} e^{5 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {8 i d e f x e^{2 c} e^{2 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx + \int \frac {2 i d e f x e^{4 c} e^{4 d x}}{e^{c} e^{3 d x} - i e^{2 d x}}\, dx\right ) e^{- 2 c}}{4 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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