Integrand size = 13, antiderivative size = 102 \[ \int \frac {\cosh ^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {b \left (a^2+b^2\right )^2 \log (b+a \sinh (x))}{a^6}+\frac {\left (a^2+b^2\right )^2 \sinh (x)}{a^5}-\frac {b \left (2 a^2+b^2\right ) \sinh ^2(x)}{2 a^4}+\frac {\left (2 a^2+b^2\right ) \sinh ^3(x)}{3 a^3}-\frac {b \sinh ^4(x)}{4 a^2}+\frac {\sinh ^5(x)}{5 a} \]
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Time = 0.14 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3957, 2916, 12, 786} \[ \int \frac {\cosh ^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {b \sinh ^4(x)}{4 a^2}-\frac {b \left (a^2+b^2\right )^2 \log (a \sinh (x)+b)}{a^6}+\frac {\left (a^2+b^2\right )^2 \sinh (x)}{a^5}-\frac {b \left (2 a^2+b^2\right ) \sinh ^2(x)}{2 a^4}+\frac {\left (2 a^2+b^2\right ) \sinh ^3(x)}{3 a^3}+\frac {\sinh ^5(x)}{5 a} \]
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Rule 12
Rule 786
Rule 2916
Rule 3957
Rubi steps \begin{align*} \text {integral}& = i \int \frac {\cosh ^5(x) \sinh (x)}{i b+i a \sinh (x)} \, dx \\ & = -\frac {i \text {Subst}\left (\int \frac {x \left (a^2-x^2\right )^2}{a (i b+x)} \, dx,x,i a \sinh (x)\right )}{a^5} \\ & = -\frac {i \text {Subst}\left (\int \frac {x \left (a^2-x^2\right )^2}{i b+x} \, dx,x,i a \sinh (x)\right )}{a^6} \\ & = -\frac {i \text {Subst}\left (\int \left (\left (a^2+b^2\right )^2-\frac {b \left (a^2+b^2\right )^2}{b-i x}+i b \left (2 a^2+b^2\right ) x-\left (2 a^2+b^2\right ) x^2-i b x^3+x^4\right ) \, dx,x,i a \sinh (x)\right )}{a^6} \\ & = -\frac {b \left (a^2+b^2\right )^2 \log (b+a \sinh (x))}{a^6}+\frac {\left (a^2+b^2\right )^2 \sinh (x)}{a^5}-\frac {b \left (2 a^2+b^2\right ) \sinh ^2(x)}{2 a^4}+\frac {\left (2 a^2+b^2\right ) \sinh ^3(x)}{3 a^3}-\frac {b \sinh ^4(x)}{4 a^2}+\frac {\sinh ^5(x)}{5 a} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.95 \[ \int \frac {\cosh ^5(x)}{a+b \text {csch}(x)} \, dx=\frac {-60 b \left (a^2+b^2\right )^2 \log (b+a \sinh (x))+60 a \left (a^2+b^2\right )^2 \sinh (x)-30 a^2 b \left (2 a^2+b^2\right ) \sinh ^2(x)+20 a^3 \left (2 a^2+b^2\right ) \sinh ^3(x)-15 a^4 b \sinh ^4(x)+12 a^5 \sinh ^5(x)}{60 a^6} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(377\) vs. \(2(94)=188\).
Time = 0.13 (sec) , antiderivative size = 378, normalized size of antiderivative = 3.71
\[-\frac {1}{5 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{5}}-\frac {2 a +b}{4 a^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {11 a^{2}+6 a b +4 b^{2}}{12 a^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {7 a^{3}+9 a^{2} b +4 a \,b^{2}+4 b^{3}}{8 a^{4} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {8 a^{4}+7 a^{3} b +16 a^{2} b^{2}+4 a \,b^{3}+8 b^{4}}{8 a^{5} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {b \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a^{6}}-\frac {2 b \left (\frac {1}{2} a^{4}+a^{2} b^{2}+\frac {1}{2} b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} b -2 a \tanh \left (\frac {x}{2}\right )-b \right )}{a^{6}}-\frac {1}{5 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {-2 a +b}{4 a^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {11 a^{2}-6 a b +4 b^{2}}{12 a^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {-7 a^{3}+9 a^{2} b -4 a \,b^{2}+4 b^{3}}{8 a^{4} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {8 a^{4}-7 a^{3} b +16 a^{2} b^{2}-4 a \,b^{3}+8 b^{4}}{8 a^{5} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {b \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a^{6}}\]
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Leaf count of result is larger than twice the leaf count of optimal. 1398 vs. \(2 (94) = 188\).
Time = 0.26 (sec) , antiderivative size = 1398, normalized size of antiderivative = 13.71 \[ \int \frac {\cosh ^5(x)}{a+b \text {csch}(x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\cosh ^5(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\cosh ^{5}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (94) = 188\).
Time = 0.18 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.37 \[ \int \frac {\cosh ^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {{\left (15 \, a^{3} b e^{\left (-x\right )} - 6 \, a^{4} - 10 \, {\left (5 \, a^{4} + 4 \, a^{2} b^{2}\right )} e^{\left (-2 \, x\right )} + 60 \, {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} e^{\left (-3 \, x\right )} - 60 \, {\left (5 \, a^{4} + 14 \, a^{2} b^{2} + 8 \, b^{4}\right )} e^{\left (-4 \, x\right )}\right )} e^{\left (5 \, x\right )}}{960 \, a^{5}} - \frac {15 \, a^{3} b e^{\left (-4 \, x\right )} + 6 \, a^{4} e^{\left (-5 \, x\right )} + 60 \, {\left (5 \, a^{4} + 14 \, a^{2} b^{2} + 8 \, b^{4}\right )} e^{\left (-x\right )} + 60 \, {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} e^{\left (-2 \, x\right )} + 10 \, {\left (5 \, a^{4} + 4 \, a^{2} b^{2}\right )} e^{\left (-3 \, x\right )}}{960 \, a^{5}} - \frac {{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} x}{a^{6}} - \frac {{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (94) = 188\).
Time = 0.29 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.90 \[ \int \frac {\cosh ^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {6 \, a^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{5} + 15 \, a^{3} b {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 80 \, a^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 40 \, a^{2} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 240 \, a^{3} b {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 120 \, a b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 480 \, a^{4} {\left (e^{\left (-x\right )} - e^{x}\right )} + 960 \, a^{2} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )} + 480 \, b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}}{960 \, a^{5}} - \frac {{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | -a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a^{6}} \]
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Time = 2.97 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.24 \[ \int \frac {\cosh ^5(x)}{a+b \text {csch}(x)} \, dx=\frac {{\mathrm {e}}^{5\,x}}{160\,a}-\frac {{\mathrm {e}}^{-5\,x}}{160\,a}-\frac {{\mathrm {e}}^{-2\,x}\,\left (3\,a^2\,b+2\,b^3\right )}{16\,a^4}-\frac {{\mathrm {e}}^{2\,x}\,\left (3\,a^2\,b+2\,b^3\right )}{16\,a^4}+\frac {{\mathrm {e}}^x\,\left (5\,a^4+14\,a^2\,b^2+8\,b^4\right )}{16\,a^5}-\frac {b\,{\mathrm {e}}^{-4\,x}}{64\,a^2}-\frac {b\,{\mathrm {e}}^{4\,x}}{64\,a^2}-\frac {\ln \left (2\,b\,{\mathrm {e}}^x-a+a\,{\mathrm {e}}^{2\,x}\right )\,\left (a^4\,b+2\,a^2\,b^3+b^5\right )}{a^6}-\frac {{\mathrm {e}}^{-x}\,\left (5\,a^4+14\,a^2\,b^2+8\,b^4\right )}{16\,a^5}-\frac {{\mathrm {e}}^{-3\,x}\,\left (5\,a^2+4\,b^2\right )}{96\,a^3}+\frac {{\mathrm {e}}^{3\,x}\,\left (5\,a^2+4\,b^2\right )}{96\,a^3}+\frac {b\,x\,{\left (a^2+b^2\right )}^2}{a^6} \]
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