Integrand size = 29, antiderivative size = 139 \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {i (e+f x)^4}{4 a f}-\frac {2 i (e+f x)^3 \log \left (1+i e^{c+d x}\right )}{a d}-\frac {6 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^3}-\frac {12 i f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{a d^4} \]
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Time = 0.15 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {5678, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {12 i f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{a d^4}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^3}-\frac {6 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^2}-\frac {2 i (e+f x)^3 \log \left (1+i e^{c+d x}\right )}{a d}+\frac {i (e+f x)^4}{4 a f} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 5678
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \frac {i (e+f x)^4}{4 a f}+2 \int \frac {e^{c+d x} (e+f x)^3}{a+i a e^{c+d x}} \, dx \\ & = \frac {i (e+f x)^4}{4 a f}-\frac {2 i (e+f x)^3 \log \left (1+i e^{c+d x}\right )}{a d}+\frac {(6 i f) \int (e+f x)^2 \log \left (1+i e^{c+d x}\right ) \, dx}{a d} \\ & = \frac {i (e+f x)^4}{4 a f}-\frac {2 i (e+f x)^3 \log \left (1+i e^{c+d x}\right )}{a d}-\frac {6 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^2}+\frac {\left (12 i f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx}{a d^2} \\ & = \frac {i (e+f x)^4}{4 a f}-\frac {2 i (e+f x)^3 \log \left (1+i e^{c+d x}\right )}{a d}-\frac {6 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^3}-\frac {\left (12 i f^3\right ) \int \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) \, dx}{a d^3} \\ & = \frac {i (e+f x)^4}{4 a f}-\frac {2 i (e+f x)^3 \log \left (1+i e^{c+d x}\right )}{a d}-\frac {6 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^3}-\frac {\left (12 i f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4} \\ & = \frac {i (e+f x)^4}{4 a f}-\frac {2 i (e+f x)^3 \log \left (1+i e^{c+d x}\right )}{a d}-\frac {6 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^3}-\frac {12 i f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{a d^4} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.85 \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {i \left (\frac {(e+f x)^4}{f}-\frac {8 (e+f x)^3 \log \left (1+i e^{c+d x}\right )}{d}-\frac {24 f \left (d^2 (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )-2 d f (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )+2 f^2 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )\right )}{d^4}\right )}{4 a} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 646 vs. \(2 (124 ) = 248\).
Time = 6.17 (sec) , antiderivative size = 647, normalized size of antiderivative = 4.65
method | result | size |
risch | \(\frac {i f^{3} x^{4}}{4 a}-\frac {i e^{4}}{4 a f}-\frac {i e^{3} x}{a}+\frac {i f^{2} e \,x^{3}}{a}+\frac {3 i f \,e^{2} x^{2}}{2 a}+\frac {6 i e^{2} f c \ln \left ({\mathrm e}^{d x +c}-i\right )}{d^{2} a}-\frac {6 i e^{2} f c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} a}-\frac {6 i f^{2} e x \,c^{2}}{d^{2} a}-\frac {6 i f^{2} e \ln \left (1+i {\mathrm e}^{d x +c}\right ) x^{2}}{d a}+\frac {6 i f^{2} e \ln \left (1+i {\mathrm e}^{d x +c}\right ) c^{2}}{d^{3} a}-\frac {12 i f^{2} e \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right ) x}{d^{2} a}+\frac {6 i f \,e^{2} c x}{d a}-\frac {6 i f \,e^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{d a}-\frac {6 i f \,e^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{d^{2} a}-\frac {6 i e \,f^{2} c^{2} \ln \left ({\mathrm e}^{d x +c}-i\right )}{d^{3} a}+\frac {6 i e \,f^{2} c^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{d^{3} a}+\frac {3 i f^{3} c^{4}}{2 d^{4} a}+\frac {2 i \ln \left ({\mathrm e}^{d x +c}\right ) e^{3}}{d a}-\frac {2 i \ln \left ({\mathrm e}^{d x +c}-i\right ) e^{3}}{d a}-\frac {12 i f^{3} \operatorname {polylog}\left (4, -i {\mathrm e}^{d x +c}\right )}{a \,d^{4}}-\frac {6 i f^{3} \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right ) x^{2}}{d^{2} a}+\frac {3 i f \,e^{2} c^{2}}{d^{2} a}-\frac {6 i f \,e^{2} \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right )}{d^{2} a}+\frac {2 i f^{3} c^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{d^{4} a}-\frac {2 i f^{3} c^{3} \ln \left ({\mathrm e}^{d x +c}\right )}{d^{4} a}-\frac {4 i f^{2} e \,c^{3}}{d^{3} a}+\frac {12 i f^{3} \operatorname {polylog}\left (3, -i {\mathrm e}^{d x +c}\right ) x}{d^{3} a}+\frac {2 i f^{3} c^{3} x}{d^{3} a}-\frac {2 i f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x^{3}}{d a}+\frac {12 i f^{2} e \operatorname {polylog}\left (3, -i {\mathrm e}^{d x +c}\right )}{d^{3} a}-\frac {2 i f^{3} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c^{3}}{d^{4} a}\) | \(647\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (114) = 228\).
Time = 0.25 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.15 \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {i \, d^{4} f^{3} x^{4} + 4 i \, d^{4} e f^{2} x^{3} + 6 i \, d^{4} e^{2} f x^{2} + 4 i \, d^{4} e^{3} x + 8 i \, c d^{3} e^{3} - 12 i \, c^{2} d^{2} e^{2} f + 8 i \, c^{3} d e f^{2} - 2 i \, c^{4} f^{3} - 48 i \, f^{3} {\rm polylog}\left (4, -i \, e^{\left (d x + c\right )}\right ) - 24 \, {\left (i \, d^{2} f^{3} x^{2} + 2 i \, d^{2} e f^{2} x + i \, d^{2} e^{2} f\right )} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - 8 \, {\left (i \, d^{3} e^{3} - 3 i \, c d^{2} e^{2} f + 3 i \, c^{2} d e f^{2} - i \, c^{3} f^{3}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) - 8 \, {\left (i \, d^{3} f^{3} x^{3} + 3 i \, d^{3} e f^{2} x^{2} + 3 i \, d^{3} e^{2} f x + 3 i \, c d^{2} e^{2} f - 3 i \, c^{2} d e f^{2} + i \, c^{3} f^{3}\right )} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) - 48 \, {\left (-i \, d f^{3} x - i \, d e f^{2}\right )} {\rm polylog}\left (3, -i \, e^{\left (d x + c\right )}\right )}{4 \, a d^{4}} \]
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\[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+i a \sinh (c+d x)} \, dx=- \frac {i \left (\int \frac {e^{3} \cosh {\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {f^{3} x^{3} \cosh {\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {3 e f^{2} x^{2} \cosh {\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {3 e^{2} f x \cosh {\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx\right )}{a} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (114) = 228\).
Time = 0.29 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.90 \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {i \, e^{3} \log \left (i \, a \sinh \left (d x + c\right ) + a\right )}{a d} - \frac {6 i \, {\left (d x \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right )\right )} e^{2} f}{a d^{2}} - \frac {i \, {\left (f^{3} x^{4} + 4 \, e f^{2} x^{3} + 6 \, e^{2} f x^{2}\right )}}{4 \, a} - \frac {6 i \, {\left (d^{2} x^{2} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(-i \, e^{\left (d x + c\right )})\right )} e f^{2}}{a d^{3}} - \frac {2 i \, {\left (d^{3} x^{3} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + 3 \, d^{2} x^{2} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - 6 \, d x {\rm Li}_{3}(-i \, e^{\left (d x + c\right )}) + 6 \, {\rm Li}_{4}(-i \, e^{\left (d x + c\right )})\right )} f^{3}}{a d^{4}} + \frac {i \, d^{4} f^{3} x^{4} + 4 i \, d^{4} e f^{2} x^{3} + 6 i \, d^{4} e^{2} f x^{2}}{2 \, a d^{4}} \]
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\[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^3 \cosh (c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^3}{a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]
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