Integrand size = 21, antiderivative size = 85 \[ \int \frac {\cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=-\frac {b \left (a^2+b^2\right ) \log (b+a \sinh (c+d x))}{a^4 d}+\frac {\left (a^2+b^2\right ) \sinh (c+d x)}{a^3 d}-\frac {b \sinh ^2(c+d x)}{2 a^2 d}+\frac {\sinh ^3(c+d x)}{3 a d} \]
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Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2916, 12, 786} \[ \int \frac {\cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=-\frac {b \sinh ^2(c+d x)}{2 a^2 d}-\frac {b \left (a^2+b^2\right ) \log (a \sinh (c+d x)+b)}{a^4 d}+\frac {\left (a^2+b^2\right ) \sinh (c+d x)}{a^3 d}+\frac {\sinh ^3(c+d x)}{3 a d} \]
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Rule 12
Rule 786
Rule 2916
Rule 3957
Rubi steps \begin{align*} \text {integral}& = i \int \frac {\cosh ^3(c+d x) \sinh (c+d x)}{i b+i a \sinh (c+d x)} \, dx \\ & = -\frac {i \text {Subst}\left (\int \frac {x \left (a^2-x^2\right )}{a (i b+x)} \, dx,x,i a \sinh (c+d x)\right )}{a^3 d} \\ & = -\frac {i \text {Subst}\left (\int \frac {x \left (a^2-x^2\right )}{i b+x} \, dx,x,i a \sinh (c+d x)\right )}{a^4 d} \\ & = -\frac {i \text {Subst}\left (\int \left (a^2 \left (1+\frac {b^2}{a^2}\right )-\frac {b \left (a^2+b^2\right )}{b-i x}+i b x-x^2\right ) \, dx,x,i a \sinh (c+d x)\right )}{a^4 d} \\ & = -\frac {b \left (a^2+b^2\right ) \log (b+a \sinh (c+d x))}{a^4 d}+\frac {\left (a^2+b^2\right ) \sinh (c+d x)}{a^3 d}-\frac {b \sinh ^2(c+d x)}{2 a^2 d}+\frac {\sinh ^3(c+d x)}{3 a d} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.88 \[ \int \frac {\cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {-6 b \left (a^2+b^2\right ) \log (b+a \sinh (c+d x))+6 a \left (a^2+b^2\right ) \sinh (c+d x)-3 a^2 b \sinh ^2(c+d x)+2 a^3 \sinh ^3(c+d x)}{6 a^4 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(243\) vs. \(2(81)=162\).
Time = 17.86 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.87
method | result | size |
risch | \(\frac {b x}{a^{2}}+\frac {b^{3} x}{a^{4}}+\frac {{\mathrm e}^{3 d x +3 c}}{24 d a}-\frac {b \,{\mathrm e}^{2 d x +2 c}}{8 d \,a^{2}}+\frac {3 \,{\mathrm e}^{d x +c}}{8 a d}+\frac {{\mathrm e}^{d x +c} b^{2}}{2 a^{3} d}-\frac {3 \,{\mathrm e}^{-d x -c}}{8 a d}-\frac {{\mathrm e}^{-d x -c} b^{2}}{2 a^{3} d}-\frac {b \,{\mathrm e}^{-2 d x -2 c}}{8 d \,a^{2}}-\frac {{\mathrm e}^{-3 d x -3 c}}{24 d a}+\frac {2 b c}{d \,a^{2}}+\frac {2 b^{3} c}{d \,a^{4}}-\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 b \,{\mathrm e}^{d x +c}}{a}-1\right )}{d \,a^{2}}-\frac {b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 b \,{\mathrm e}^{d x +c}}{a}-1\right )}{d \,a^{4}}\) | \(244\) |
derivativedivides | \(\frac {-\frac {1}{3 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {-a +b}{2 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {2 a^{2}-a b +2 b^{2}}{2 a^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (a^{2}+b^{2}\right ) b \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}+\frac {2 b \left (-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right ) \ln \left (-\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b \right )}{a^{4}}-\frac {1}{3 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a +b}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 a^{2}+a b +2 b^{2}}{2 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (a^{2}+b^{2}\right ) b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{4}}}{d}\) | \(245\) |
default | \(\frac {-\frac {1}{3 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {-a +b}{2 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {2 a^{2}-a b +2 b^{2}}{2 a^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (a^{2}+b^{2}\right ) b \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}+\frac {2 b \left (-\frac {a^{2}}{2}-\frac {b^{2}}{2}\right ) \ln \left (-\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a +b \right )}{a^{4}}-\frac {1}{3 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a +b}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 a^{2}+a b +2 b^{2}}{2 a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (a^{2}+b^{2}\right ) b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{4}}}{d}\) | \(245\) |
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Leaf count of result is larger than twice the leaf count of optimal. 652 vs. \(2 (81) = 162\).
Time = 0.27 (sec) , antiderivative size = 652, normalized size of antiderivative = 7.67 \[ \int \frac {\cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {a^{3} \cosh \left (d x + c\right )^{6} + a^{3} \sinh \left (d x + c\right )^{6} - 3 \, a^{2} b \cosh \left (d x + c\right )^{5} + 24 \, {\left (a^{2} b + b^{3}\right )} d x \cosh \left (d x + c\right )^{3} + 3 \, {\left (2 \, a^{3} \cosh \left (d x + c\right ) - a^{2} b\right )} \sinh \left (d x + c\right )^{5} + 3 \, {\left (3 \, a^{3} + 4 \, a b^{2}\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, a^{3} \cosh \left (d x + c\right )^{2} - 5 \, a^{2} b \cosh \left (d x + c\right ) + 3 \, a^{3} + 4 \, a b^{2}\right )} \sinh \left (d x + c\right )^{4} - 3 \, a^{2} b \cosh \left (d x + c\right ) + 2 \, {\left (10 \, a^{3} \cosh \left (d x + c\right )^{3} - 15 \, a^{2} b \cosh \left (d x + c\right )^{2} + 12 \, {\left (a^{2} b + b^{3}\right )} d x + 6 \, {\left (3 \, a^{3} + 4 \, a b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - a^{3} - 3 \, {\left (3 \, a^{3} + 4 \, a b^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, a^{3} \cosh \left (d x + c\right )^{4} - 10 \, a^{2} b \cosh \left (d x + c\right )^{3} + 24 \, {\left (a^{2} b + b^{3}\right )} d x \cosh \left (d x + c\right ) - 3 \, a^{3} - 4 \, a b^{2} + 6 \, {\left (3 \, a^{3} + 4 \, a b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} - 24 \, {\left ({\left (a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, {\left (a^{2} b + b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a^{2} b + b^{3}\right )} \sinh \left (d x + c\right )^{3}\right )} \log \left (\frac {2 \, {\left (a \sinh \left (d x + c\right ) + b\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 3 \, {\left (2 \, a^{3} \cosh \left (d x + c\right )^{5} - 5 \, a^{2} b \cosh \left (d x + c\right )^{4} + 24 \, {\left (a^{2} b + b^{3}\right )} d x \cosh \left (d x + c\right )^{2} + 4 \, {\left (3 \, a^{3} + 4 \, a b^{2}\right )} \cosh \left (d x + c\right )^{3} - a^{2} b - 2 \, {\left (3 \, a^{3} + 4 \, a b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \, {\left (a^{4} d \cosh \left (d x + c\right )^{3} + 3 \, a^{4} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{4} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{4} d \sinh \left (d x + c\right )^{3}\right )}} \]
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\[ \int \frac {\cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {\cosh ^{3}{\left (c + d x \right )}}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (81) = 162\).
Time = 0.22 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.15 \[ \int \frac {\cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=-\frac {{\left (3 \, a b e^{\left (-d x - c\right )} - a^{2} - 3 \, {\left (3 \, a^{2} + 4 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{24 \, a^{3} d} - \frac {{\left (a^{2} b + b^{3}\right )} {\left (d x + c\right )}}{a^{4} d} - \frac {3 \, a b e^{\left (-2 \, d x - 2 \, c\right )} + a^{2} e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, {\left (3 \, a^{2} + 4 \, b^{2}\right )} e^{\left (-d x - c\right )}}{24 \, a^{3} d} - \frac {{\left (a^{2} b + b^{3}\right )} \log \left (-2 \, b e^{\left (-d x - c\right )} + a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )}{a^{4} d} \]
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Time = 0.32 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.71 \[ \int \frac {\cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {\frac {a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 3 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 12 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 12 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{a^{3}} - \frac {24 \, {\left (a^{2} b + b^{3}\right )} \log \left ({\left | a {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, b \right |}\right )}{a^{4}}}{24 \, d} \]
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Time = 2.47 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.12 \[ \int \frac {\cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {x\,\left (a^2\,b+b^3\right )}{a^4}-\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,a\,d}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,a\,d}-\frac {b\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,a^2\,d}-\frac {b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,a^2\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (3\,a^2+4\,b^2\right )}{8\,a^3\,d}-\frac {\ln \left (2\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-a+a\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )\,\left (a^2\,b+b^3\right )}{a^4\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a^2+4\,b^2\right )}{8\,a^3\,d} \]
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