Integrand size = 27, antiderivative size = 130 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b \arctan (\sinh (c+d x))}{\left (a^2+b^2\right ) d}+\frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}+\frac {a \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d}-\frac {\left (a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {b^4 \log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right ) d} \]
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Time = 0.17 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2916, 12, 908, 649, 209, 266} \[ \int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b \arctan (\sinh (c+d x))}{d \left (a^2+b^2\right )}+\frac {a \log (\cosh (c+d x))}{d \left (a^2+b^2\right )}+\frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\left (a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {b^4 \log (a+b \sinh (c+d x))}{a^3 d \left (a^2+b^2\right )}-\frac {\text {csch}^2(c+d x)}{2 a d} \]
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Rule 12
Rule 209
Rule 266
Rule 649
Rule 908
Rule 2916
Rubi steps \begin{align*} \text {integral}& = -\frac {b \text {Subst}\left (\int \frac {b^3}{x^3 (a+x) \left (-b^2-x^2\right )} \, dx,x,b \sinh (c+d x)\right )}{d} \\ & = -\frac {b^4 \text {Subst}\left (\int \frac {1}{x^3 (a+x) \left (-b^2-x^2\right )} \, dx,x,b \sinh (c+d x)\right )}{d} \\ & = -\frac {b^4 \text {Subst}\left (\int \left (-\frac {1}{a b^2 x^3}+\frac {1}{a^2 b^2 x^2}+\frac {a^2-b^2}{a^3 b^4 x}+\frac {1}{a^3 \left (a^2+b^2\right ) (a+x)}+\frac {-b^2-a x}{b^4 \left (a^2+b^2\right ) \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (c+d x)\right )}{d} \\ & = \frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}-\frac {\left (a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {b^4 \log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right ) d}-\frac {\text {Subst}\left (\int \frac {-b^2-a x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right ) d} \\ & = \frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}-\frac {\left (a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {b^4 \log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right ) d}+\frac {a \text {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right ) d}+\frac {b^2 \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right ) d} \\ & = \frac {b \arctan (\sinh (c+d x))}{\left (a^2+b^2\right ) d}+\frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}+\frac {a \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d}-\frac {\left (a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {b^4 \log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right ) d} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.26 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {2 b \text {csch}(c+d x)}{a^2}-\frac {\text {csch}^2(c+d x)}{a}-\frac {2 (a-b) (a+b) \log (\sinh (c+d x))}{a^3}+\frac {\left (a-\sqrt {-b^2}\right ) \log \left (\sqrt {-b^2}-b \sinh (c+d x)\right )}{a^2+b^2}-\frac {2 b^4 \log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right )}+\frac {\left (a+\sqrt {-b^2}\right ) \log \left (\sqrt {-b^2}+b \sinh (c+d x)\right )}{a^2+b^2}}{2 d} \]
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Time = 7.07 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.43
method | result | size |
derivativedivides | \(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}{2}+2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {b^{4} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{a^{3} \left (a^{2}+b^{2}\right )}+\frac {4 a \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+8 b \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2}+4 b^{2}}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-4 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(186\) |
default | \(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}{2}+2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {b^{4} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{a^{3} \left (a^{2}+b^{2}\right )}+\frac {4 a \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+8 b \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2}+4 b^{2}}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-4 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(186\) |
risch | \(-\frac {2 a \,d^{2} x}{a^{2} d^{2}+b^{2} d^{2}}-\frac {2 a d c}{a^{2} d^{2}+b^{2} d^{2}}+\frac {2 x}{a}+\frac {2 c}{d a}-\frac {2 b^{2} x}{a^{3}}-\frac {2 b^{2} c}{d \,a^{3}}+\frac {2 b^{4} x}{a^{3} \left (a^{2}+b^{2}\right )}+\frac {2 b^{4} c}{d \,a^{3} \left (a^{2}+b^{2}\right )}-\frac {2 \,{\mathrm e}^{d x +c} \left (-b \,{\mathrm e}^{2 d x +2 c}+a \,{\mathrm e}^{d x +c}+b \right )}{a^{2} d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) b}{\left (a^{2}+b^{2}\right ) d}+\frac {\ln \left ({\mathrm e}^{d x +c}+i\right ) a}{\left (a^{2}+b^{2}\right ) d}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) b}{\left (a^{2}+b^{2}\right ) d}+\frac {\ln \left ({\mathrm e}^{d x +c}-i\right ) a}{\left (a^{2}+b^{2}\right ) d}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d a}+\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \,a^{3}}-\frac {b^{4} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{d \,a^{3} \left (a^{2}+b^{2}\right )}\) | \(358\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1035 vs. \(2 (128) = 256\).
Time = 0.35 (sec) , antiderivative size = 1035, normalized size of antiderivative = 7.96 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.82 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b^{4} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{5} + a^{3} b^{2}\right )} d} - \frac {2 \, b \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac {a \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d} - \frac {2 \, {\left (b e^{\left (-d x - c\right )} - a e^{\left (-2 \, d x - 2 \, c\right )} - b e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{{\left (2 \, a^{2} e^{\left (-2 \, d x - 2 \, c\right )} - a^{2} e^{\left (-4 \, d x - 4 \, c\right )} - a^{2}\right )} d} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{3} d} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{3} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (128) = 256\).
Time = 0.29 (sec) , antiderivative size = 263, normalized size of antiderivative = 2.02 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {2 \, b^{5} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{5} b + a^{3} b^{3}} - \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} b}{a^{2} + b^{2}} - \frac {a \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{a^{2} + b^{2}} + \frac {2 \, {\left (a^{2} - b^{2}\right )} \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a^{3}} - \frac {3 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 3 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 4 \, a^{2}}{a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2}}}{2 \, d} \]
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Time = 4.63 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.51 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\ln \left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )}{a\,d-b\,d\,1{}\mathrm {i}}-\frac {\frac {2}{a\,d}-\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{a^2\,d}}{{\mathrm {e}}^{2\,c+2\,d\,x}-1}-\frac {2}{a\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {b^4\,\ln \left (2\,a\,{\mathrm {e}}^{c+d\,x}-b+b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{d\,a^5+d\,a^3\,b^2}-\frac {\ln \left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )\,\left (a^2-b^2\right )}{a^3\,d}+\frac {\ln \left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{-b\,d+a\,d\,1{}\mathrm {i}} \]
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