Integrand size = 23, antiderivative size = 132 \[ \int \frac {(c+d x)^3}{a+i a \sinh (e+f x)} \, dx=\frac {(c+d x)^3}{a f}-\frac {6 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a f^2}-\frac {12 d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{e+f x}\right )}{a f^3}+\frac {12 d^3 \operatorname {PolyLog}\left (3,-i e^{e+f x}\right )}{a f^4}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f} \]
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Time = 0.22 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3399, 4269, 3797, 2221, 2611, 2320, 6724} \[ \int \frac {(c+d x)^3}{a+i a \sinh (e+f x)} \, dx=-\frac {12 d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{e+f x}\right )}{a f^3}-\frac {6 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a f^2}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{a f}+\frac {(c+d x)^3}{a f}+\frac {12 d^3 \operatorname {PolyLog}\left (3,-i e^{e+f x}\right )}{a f^4} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3399
Rule 3797
Rule 4269
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\int (c+d x)^3 \csc ^2\left (\frac {1}{2} \left (i e+\frac {\pi }{2}\right )+\frac {i f x}{2}\right ) \, dx}{2 a} \\ & = \frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f}-\frac {(3 d) \int (c+d x)^2 \coth \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \, dx}{a f} \\ & = \frac {(c+d x)^3}{a f}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f}-\frac {(6 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 f} \\ & = \frac {(c+d x)^3}{a f}-\frac {6 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a f^2}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {\left (12 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 f^2} \\ & = \frac {(c+d x)^3}{a f}-\frac {6 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a f^2}-\frac {12 d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{e+f x}\right )}{a f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {\left (12 d^3\right ) \int \operatorname {PolyLog}\left (2,-i e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a f^3} \\ & = \frac {(c+d x)^3}{a f}-\frac {6 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a f^2}-\frac {12 d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{e+f x}\right )}{a f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {\left (12 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 f^4} \\ & = \frac {(c+d x)^3}{a f}-\frac {6 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a f^2}-\frac {12 d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{e+f x}\right )}{a f^3}+\frac {12 d^3 \operatorname {PolyLog}\left (3,-i e^{e+f x}\right )}{a f^4}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a f} \\ \end{align*}
Time = 0.99 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.56 \[ \int \frac {(c+d x)^3}{a+i a \sinh (e+f x)} \, dx=\frac {2 \left (\frac {3 d e^e \left (\frac {e^{-e} (c+d x)^3}{3 d}+\frac {\left (i+e^{-e}\right ) (c+d x)^2 \log \left (1-i e^{-e-f x}\right )}{f}-\frac {2 i d e^{-e} \left (-i+e^e\right ) \left (f (c+d x) \operatorname {PolyLog}\left (2,i e^{-e-f x}\right )+d \operatorname {PolyLog}\left (3,i e^{-e-f x}\right )\right )}{f^3}\right )}{-1-i e^e}+\frac {(c+d x)^3 \sinh \left (\frac {f x}{2}\right )}{\left (\cosh \left (\frac {e}{2}\right )+i \sinh \left (\frac {e}{2}\right )\right ) \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )}\right )}{a f} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (119 ) = 238\).
Time = 1.44 (sec) , antiderivative size = 435, normalized size of antiderivative = 3.30
method | result | size |
risch | \(\frac {2 i \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 d x \,c^{2}+c^{3}\right )}{f a \left ({\mathrm e}^{f x +e}-i\right )}+\frac {6 d^{2} c \,x^{2}}{a f}+\frac {2 d^{3} x^{3}}{a f}+\frac {12 d^{2} c e x}{a \,f^{2}}+\frac {6 d^{2} c \,e^{2}}{a \,f^{3}}+\frac {6 d^{3} \ln \left (1+i {\mathrm e}^{f x +e}\right ) e^{2}}{a \,f^{4}}-\frac {4 d^{3} e^{3}}{a \,f^{4}}-\frac {12 d^{3} \operatorname {polylog}\left (2, -i {\mathrm e}^{f x +e}\right ) x}{a \,f^{3}}-\frac {6 d^{3} \ln \left (1+i {\mathrm e}^{f x +e}\right ) x^{2}}{a \,f^{2}}+\frac {12 d^{3} \operatorname {polylog}\left (3, -i {\mathrm e}^{f x +e}\right )}{a \,f^{4}}-\frac {12 d^{2} c \ln \left (1+i {\mathrm e}^{f x +e}\right ) x}{a \,f^{2}}-\frac {6 d^{3} e^{2} \ln \left ({\mathrm e}^{f x +e}-i\right )}{a \,f^{4}}+\frac {6 d^{3} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{a \,f^{4}}+\frac {12 d^{2} c e \ln \left ({\mathrm e}^{f x +e}-i\right )}{a \,f^{3}}-\frac {12 d^{2} c e \ln \left ({\mathrm e}^{f x +e}\right )}{a \,f^{3}}-\frac {6 d \ln \left ({\mathrm e}^{f x +e}-i\right ) c^{2}}{a \,f^{2}}-\frac {12 d^{2} c \ln \left (1+i {\mathrm e}^{f x +e}\right ) e}{a \,f^{3}}-\frac {12 d^{2} c \operatorname {polylog}\left (2, -i {\mathrm e}^{f x +e}\right )}{a \,f^{3}}-\frac {6 d^{3} e^{2} x}{a \,f^{3}}+\frac {6 d \ln \left ({\mathrm e}^{f x +e}\right ) c^{2}}{a \,f^{2}}\) | \(435\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (114) = 228\).
Time = 0.25 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.76 \[ \int \frac {(c+d x)^3}{a+i a \sinh (e+f x)} \, dx=-\frac {2 \, {\left (i \, d^{3} e^{3} - 3 i \, c d^{2} e^{2} f + 3 i \, c^{2} d e f^{2} - i \, c^{3} f^{3} + 6 \, {\left (-i \, d^{3} f x - i \, c d^{2} f + {\left (d^{3} f x + c d^{2} f\right )} e^{\left (f x + e\right )}\right )} {\rm Li}_2\left (-i \, e^{\left (f x + e\right )}\right ) - {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + 3 \, c^{2} d f^{3} x + d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2}\right )} e^{\left (f x + e\right )} + 3 \, {\left (-i \, d^{3} e^{2} + 2 i \, c d^{2} e f - i \, c^{2} d f^{2} + {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} e^{\left (f x + e\right )}\right )} \log \left (e^{\left (f x + e\right )} - i\right ) + 3 \, {\left (-i \, d^{3} f^{2} x^{2} - 2 i \, c d^{2} f^{2} x + i \, d^{3} e^{2} - 2 i \, c d^{2} e f + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f\right )} e^{\left (f x + e\right )}\right )} \log \left (i \, e^{\left (f x + e\right )} + 1\right ) - 6 \, {\left (d^{3} e^{\left (f x + e\right )} - i \, d^{3}\right )} {\rm polylog}\left (3, -i \, e^{\left (f x + e\right )}\right )\right )}}{a f^{4} e^{\left (f x + e\right )} - i \, a f^{4}} \]
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\[ \int \frac {(c+d x)^3}{a+i a \sinh (e+f x)} \, dx=\frac {2 i c^{3} + 6 i c^{2} d x + 6 i c d^{2} x^{2} + 2 i d^{3} x^{3}}{a f e^{e} e^{f x} - i a f} - \frac {6 i d \left (\int \frac {c^{2}}{e^{e} e^{f x} - i}\, dx + \int \frac {d^{2} x^{2}}{e^{e} e^{f x} - i}\, dx + \int \frac {2 c d x}{e^{e} e^{f x} - i}\, dx\right )}{a f} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (114) = 228\).
Time = 0.31 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.80 \[ \int \frac {(c+d x)^3}{a+i a \sinh (e+f x)} \, dx=6 \, c^{2} d {\left (\frac {x e^{\left (f x + e\right )}}{a f e^{\left (f x + e\right )} - i \, a f} - \frac {\log \left ({\left (e^{\left (f x + e\right )} - i\right )} e^{\left (-e\right )}\right )}{a f^{2}}\right )} - \frac {2 \, c^{3}}{{\left (i \, a e^{\left (-f x - e\right )} - a\right )} f} - \frac {2 \, {\left (-i \, d^{3} x^{3} - 3 i \, c d^{2} x^{2}\right )}}{a f e^{\left (f x + e\right )} - i \, a f} - \frac {12 \, {\left (f x \log \left (i \, e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (-i \, e^{\left (f x + e\right )}\right )\right )} c d^{2}}{a f^{3}} - \frac {6 \, {\left (f^{2} x^{2} \log \left (i \, e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-i \, e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(-i \, e^{\left (f x + e\right )})\right )} d^{3}}{a f^{4}} + \frac {2 \, {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2}\right )}}{a f^{4}} \]
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\[ \int \frac {(c+d x)^3}{a+i a \sinh (e+f x)} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{i \, a \sinh \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(c+d x)^3}{a+i a \sinh (e+f x)} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}} \,d x \]
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