Integrand size = 23, antiderivative size = 305 \[ \int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx=\frac {(c+d x)^3}{3 a^2 f}-\frac {2 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{e+f x}\right )}{a^2 f^3}+\frac {4 d^3 \operatorname {PolyLog}\left (3,-i e^{e+f x}\right )}{a^2 f^4}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f} \]
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Time = 0.31 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3399, 4271, 4269, 3556, 3797, 2221, 2611, 2320, 6724} \[ \int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx=-\frac {4 d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{e+f x}\right )}{a^2 f^3}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{a^2 f^3}-\frac {2 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a^2 f^2}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{2 a^2 f^2}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 a^2 f}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{6 a^2 f}+\frac {(c+d x)^3}{3 a^2 f}+\frac {4 d^3 \operatorname {PolyLog}\left (3,-i e^{e+f x}\right )}{a^2 f^4}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{a^2 f^4} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3399
Rule 3556
Rule 3797
Rule 4269
Rule 4271
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\int (c+d x)^3 \csc ^4\left (\frac {1}{2} \left (i e+\frac {\pi }{2}\right )+\frac {i f x}{2}\right ) \, dx}{4 a^2} \\ & = \frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {\int (c+d x)^3 \text {csch}^2\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{6 a^2}+\frac {d^2 \int (c+d x) \text {csch}^2\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{a^2 f^2} \\ & = \frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\left (2 d^3\right ) \int \coth \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{a^2 f^3}-\frac {d \int (c+d x)^2 \coth \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{a^2 f} \\ & = \frac {(c+d x)^3}{3 a^2 f}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {(2 i d) \int \frac {e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)^2}{1+i e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{a^2 f} \\ & = \frac {(c+d x)^3}{3 a^2 f}-\frac {2 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\left (4 d^2\right ) \int (c+d x) \log \left (1+i e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a^2 f^2} \\ & = \frac {(c+d x)^3}{3 a^2 f}-\frac {2 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{e+f x}\right )}{a^2 f^3}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\left (4 d^3\right ) \int \operatorname {PolyLog}\left (2,-i e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a^2 f^3} \\ & = \frac {(c+d x)^3}{3 a^2 f}-\frac {2 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{e+f x}\right )}{a^2 f^3}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\left (4 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{a^2 f^4} \\ & = \frac {(c+d x)^3}{3 a^2 f}-\frac {2 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{e+f x}\right )}{a^2 f^3}+\frac {4 d^3 \operatorname {PolyLog}\left (3,-i e^{e+f x}\right )}{a^2 f^4}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f} \\ \end{align*}
Time = 2.89 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.55 \[ \int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx=\frac {\frac {2 d \left (-6 d^2 x+3 c^2 f^2 x+3 \left (1+i e^e\right ) \left (2 d^2-c^2 f^2\right ) x+3 c d f^2 x^2+d^2 f^2 x^3+6 c d \left (1+i e^e\right ) f x \log \left (1-i e^{-e-f x}\right )+3 d^2 \left (1+i e^e\right ) f x^2 \log \left (1-i e^{-e-f x}\right )+\frac {3 \left (1+i e^e\right ) \left (-2 d^2+c^2 f^2\right ) \log \left (i-e^{e+f x}\right )}{f}-6 c d \left (1+i e^e\right ) \operatorname {PolyLog}\left (2,i e^{-e-f x}\right )-6 d^2 \left (1+i e^e\right ) x \operatorname {PolyLog}\left (2,i e^{-e-f x}\right )-\frac {6 d^2 \left (1+i e^e\right ) \operatorname {PolyLog}\left (3,i e^{-e-f x}\right )}{f}\right )}{-1-i e^e}+\frac {(c+d x) \left (3 d f (c+d x) \cosh \left (\frac {f x}{2}\right )+6 i d^2 \cosh \left (e+\frac {f x}{2}\right )+i \left (c^2 f^2+2 c d f^2 x+d^2 \left (-6+f^2 x^2\right )\right ) \cosh \left (e+\frac {3 f x}{2}\right )+3 \left (c^2 f^2+2 c d f^2 x+d^2 \left (-4+f^2 x^2\right )\right ) \sinh \left (\frac {f x}{2}\right )+3 i d f (c+d x) \sinh \left (e+\frac {f x}{2}\right )\right )}{\left (\cosh \left (\frac {e}{2}\right )+i \sinh \left (\frac {e}{2}\right )\right ) \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )^3}}{3 a^2 f^3} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 722 vs. \(2 (249 ) = 498\).
Time = 1.94 (sec) , antiderivative size = 723, normalized size of antiderivative = 2.37
method | result | size |
risch | \(\frac {2 f^{2} c^{3} {\mathrm e}^{f x +e}-4 i d^{3} x \,{\mathrm e}^{2 f x +2 e}-4 i c \,d^{2} {\mathrm e}^{2 f x +2 e}-2 i f \,c^{2} d \,{\mathrm e}^{2 f x +2 e}-8 d^{3} x \,{\mathrm e}^{f x +e}-8 c \,d^{2} {\mathrm e}^{f x +e}-2 i f \,d^{3} x^{2} {\mathrm e}^{2 f x +2 e}-\frac {2 i c^{3} f^{2}}{3}-2 i f^{2} c \,d^{2} x^{2}+4 i d^{3} x +6 f^{2} c \,d^{2} x^{2} {\mathrm e}^{f x +e}+6 f^{2} c^{2} d x \,{\mathrm e}^{f x +e}-4 f c \,d^{2} x \,{\mathrm e}^{f x +e}-2 i f^{2} c^{2} d x -4 i f c \,d^{2} x \,{\mathrm e}^{2 f x +2 e}-\frac {2 i f^{2} d^{3} x^{3}}{3}+4 i c \,d^{2}-2 f \,d^{3} x^{2} {\mathrm e}^{f x +e}-2 f \,c^{2} d \,{\mathrm e}^{f x +e}+2 f^{2} d^{3} x^{3} {\mathrm e}^{f x +e}}{\left ({\mathrm e}^{f x +e}-i\right )^{3} f^{3} a^{2}}-\frac {2 d^{3} e^{2} x}{a^{2} f^{3}}-\frac {2 d^{3} \ln \left (1+i {\mathrm e}^{f x +e}\right ) x^{2}}{a^{2} f^{2}}-\frac {4 d^{3} \operatorname {polylog}\left (2, -i {\mathrm e}^{f x +e}\right ) x}{a^{2} f^{3}}+\frac {2 d^{2} c \,x^{2}}{a^{2} f}+\frac {2 d^{2} c \,e^{2}}{a^{2} f^{3}}-\frac {4 d^{2} c \operatorname {polylog}\left (2, -i {\mathrm e}^{f x +e}\right )}{a^{2} f^{3}}+\frac {4 d^{3} \ln \left ({\mathrm e}^{f x +e}-i\right )}{a^{2} f^{4}}-\frac {4 d^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{a^{2} f^{4}}-\frac {4 d^{3} e^{3}}{3 a^{2} f^{4}}+\frac {4 d^{3} \operatorname {polylog}\left (3, -i {\mathrm e}^{f x +e}\right )}{a^{2} f^{4}}+\frac {4 d^{2} c e x}{a^{2} f^{2}}-\frac {4 d^{2} c \ln \left (1+i {\mathrm e}^{f x +e}\right ) x}{a^{2} f^{2}}-\frac {4 d^{2} c \ln \left (1+i {\mathrm e}^{f x +e}\right ) e}{a^{2} f^{3}}-\frac {4 d^{2} \ln \left ({\mathrm e}^{f x +e}\right ) c e}{a^{2} f^{3}}+\frac {4 d^{2} \ln \left ({\mathrm e}^{f x +e}-i\right ) c e}{a^{2} f^{3}}+\frac {2 d^{3} x^{3}}{3 a^{2} f}+\frac {2 d^{3} \ln \left (1+i {\mathrm e}^{f x +e}\right ) e^{2}}{a^{2} f^{4}}+\frac {2 d \ln \left ({\mathrm e}^{f x +e}\right ) c^{2}}{a^{2} f^{2}}-\frac {2 d \ln \left ({\mathrm e}^{f x +e}-i\right ) c^{2}}{a^{2} f^{2}}+\frac {2 d^{3} \ln \left ({\mathrm e}^{f x +e}\right ) e^{2}}{a^{2} f^{4}}-\frac {2 d^{3} \ln \left ({\mathrm e}^{f x +e}-i\right ) e^{2}}{a^{2} f^{4}}\) | \(723\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 917 vs. \(2 (239) = 478\).
Time = 0.26 (sec) , antiderivative size = 917, normalized size of antiderivative = 3.01 \[ \int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx=-\frac {2 \, {\left (-i \, d^{3} e^{3} - 3 i \, c^{2} d e f^{2} + i \, c^{3} f^{3} + 6 i \, d^{3} e + 3 \, {\left (i \, c d^{2} e^{2} - 2 i \, c d^{2}\right )} f + 6 \, {\left (i \, d^{3} f x + i \, c d^{2} f + {\left (d^{3} f x + c d^{2} f\right )} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, {\left (-i \, d^{3} f x - i \, c d^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, {\left (d^{3} f x + c d^{2} f\right )} e^{\left (f x + e\right )}\right )} {\rm Li}_2\left (-i \, e^{\left (f x + e\right )}\right ) - {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2} - 6 \, d^{3} e + 3 \, {\left (c^{2} d f^{3} - 2 \, d^{3} f\right )} x\right )} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, {\left (i \, d^{3} f^{3} x^{3} + i \, d^{3} e^{3} - 6 i \, d^{3} e + {\left (3 i \, c^{2} d e + i \, c^{2} d\right )} f^{2} + {\left (3 i \, c d^{2} f^{3} + i \, d^{3} f^{2}\right )} x^{2} + {\left (-3 i \, c d^{2} e^{2} + 2 i \, c d^{2}\right )} f + {\left (3 i \, c^{2} d f^{3} + 2 i \, c d^{2} f^{2} - 4 i \, d^{3} f\right )} x\right )} e^{\left (2 \, f x + 2 \, e\right )} + 3 \, {\left (d^{3} f^{2} x^{2} + d^{3} e^{3} - c^{3} f^{3} - 6 \, d^{3} e + {\left (3 \, c^{2} d e + c^{2} d\right )} f^{2} - {\left (3 \, c d^{2} e^{2} - 4 \, c d^{2}\right )} f + 2 \, {\left (c d^{2} f^{2} - d^{3} f\right )} x\right )} e^{\left (f x + e\right )} + 3 \, {\left (i \, d^{3} e^{2} - 2 i \, c d^{2} e f + i \, c^{2} d f^{2} - 2 i \, d^{3} + {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2} - 2 \, d^{3}\right )} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, {\left (-i \, d^{3} e^{2} + 2 i \, c d^{2} e f - i \, c^{2} d f^{2} + 2 i \, d^{3}\right )} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2} - 2 \, d^{3}\right )} e^{\left (f x + e\right )}\right )} \log \left (e^{\left (f x + e\right )} - i\right ) + 3 \, {\left (i \, d^{3} f^{2} x^{2} + 2 i \, c d^{2} f^{2} x - i \, d^{3} e^{2} + 2 i \, c d^{2} e f + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f\right )} e^{\left (3 \, f x + 3 \, e\right )} + 3 \, {\left (-i \, d^{3} f^{2} x^{2} - 2 i \, c d^{2} f^{2} x + i \, d^{3} e^{2} - 2 i \, c d^{2} e f\right )} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f\right )} e^{\left (f x + e\right )}\right )} \log \left (i \, e^{\left (f x + e\right )} + 1\right ) - 6 \, {\left (d^{3} e^{\left (3 \, f x + 3 \, e\right )} - 3 i \, d^{3} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, d^{3} e^{\left (f x + e\right )} + i \, d^{3}\right )} {\rm polylog}\left (3, -i \, e^{\left (f x + e\right )}\right )\right )}}{3 \, {\left (a^{2} f^{4} e^{\left (3 \, f x + 3 \, e\right )} - 3 i \, a^{2} f^{4} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, a^{2} f^{4} e^{\left (f x + e\right )} + i \, a^{2} f^{4}\right )}} \]
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\[ \int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx=\frac {- 2 i c^{3} f^{2} - 6 i c^{2} d f^{2} x - 6 i c d^{2} f^{2} x^{2} + 12 i c d^{2} - 2 i d^{3} f^{2} x^{3} + 12 i d^{3} x + \left (- 6 i c^{2} d f e^{2 e} - 12 i c d^{2} f x e^{2 e} - 12 i c d^{2} e^{2 e} - 6 i d^{3} f x^{2} e^{2 e} - 12 i d^{3} x e^{2 e}\right ) e^{2 f x} + \left (6 c^{3} f^{2} e^{e} + 18 c^{2} d f^{2} x e^{e} - 6 c^{2} d f e^{e} + 18 c d^{2} f^{2} x^{2} e^{e} - 12 c d^{2} f x e^{e} - 24 c d^{2} e^{e} + 6 d^{3} f^{2} x^{3} e^{e} - 6 d^{3} f x^{2} e^{e} - 24 d^{3} x e^{e}\right ) e^{f x}}{3 a^{2} f^{3} e^{3 e} e^{3 f x} - 9 i a^{2} f^{3} e^{2 e} e^{2 f x} - 9 a^{2} f^{3} e^{e} e^{f x} + 3 i a^{2} f^{3}} - \frac {2 i d \left (\int \left (- \frac {2 d^{2}}{e^{e} e^{f x} - i}\right )\, dx + \int \frac {c^{2} f^{2}}{e^{e} e^{f x} - i}\, dx + \int \frac {d^{2} f^{2} x^{2}}{e^{e} e^{f x} - i}\, dx + \int \frac {2 c d f^{2} x}{e^{e} e^{f x} - i}\, dx\right )}{a^{2} f^{3}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 635 vs. \(2 (239) = 478\).
Time = 0.38 (sec) , antiderivative size = 635, normalized size of antiderivative = 2.08 \[ \int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx=2 \, c^{2} d {\left (\frac {f x e^{\left (3 \, f x + 3 \, e\right )} - {\left (3 i \, f x e^{\left (2 \, e\right )} + i \, e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} - e^{\left (f x + e\right )}}{a^{2} f^{2} e^{\left (3 \, f x + 3 \, e\right )} - 3 i \, a^{2} f^{2} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, a^{2} f^{2} e^{\left (f x + e\right )} + i \, a^{2} f^{2}} - \frac {\log \left (-i \, {\left (i \, e^{\left (f x + e\right )} + 1\right )} e^{\left (-e\right )}\right )}{a^{2} f^{2}}\right )} + \frac {2}{3} \, c^{3} {\left (\frac {3 \, e^{\left (-f x - e\right )}}{{\left (3 \, a^{2} e^{\left (-f x - e\right )} - 3 i \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + i \, a^{2}\right )} f} + \frac {i}{{\left (3 \, a^{2} e^{\left (-f x - e\right )} - 3 i \, a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - a^{2} e^{\left (-3 \, f x - 3 \, e\right )} + i \, a^{2}\right )} f}\right )} - \frac {2 \, {\left (i \, d^{3} f^{2} x^{3} + 3 i \, c d^{2} f^{2} x^{2} - 6 i \, d^{3} x - 6 i \, c d^{2} - 3 \, {\left (-i \, d^{3} f x^{2} e^{\left (2 \, e\right )} - 2 i \, c d^{2} e^{\left (2 \, e\right )} + 2 \, {\left (-i \, c d^{2} f e^{\left (2 \, e\right )} - i \, d^{3} e^{\left (2 \, e\right )}\right )} x\right )} e^{\left (2 \, f x\right )} - 3 \, {\left (d^{3} f^{2} x^{3} e^{e} - 4 \, c d^{2} e^{e} + {\left (3 \, c d^{2} f^{2} e^{e} - d^{3} f e^{e}\right )} x^{2} - 2 \, {\left (c d^{2} f e^{e} + 2 \, d^{3} e^{e}\right )} x\right )} e^{\left (f x\right )}\right )}}{3 \, {\left (a^{2} f^{3} e^{\left (3 \, f x + 3 \, e\right )} - 3 i \, a^{2} f^{3} e^{\left (2 \, f x + 2 \, e\right )} - 3 \, a^{2} f^{3} e^{\left (f x + e\right )} + i \, a^{2} f^{3}\right )}} - \frac {4 \, {\left (f x \log \left (i \, e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (-i \, e^{\left (f x + e\right )}\right )\right )} c d^{2}}{a^{2} f^{3}} - \frac {4 \, d^{3} x}{a^{2} f^{3}} - \frac {2 \, {\left (f^{2} x^{2} \log \left (i \, e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-i \, e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(-i \, e^{\left (f x + e\right )})\right )} d^{3}}{a^{2} f^{4}} + \frac {4 \, d^{3} \log \left (e^{\left (f x + e\right )} - i\right )}{a^{2} f^{4}} + \frac {2 \, {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2}\right )}}{3 \, a^{2} f^{4}} \]
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\[ \int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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