Integrand size = 13, antiderivative size = 135 \[ \int \frac {\tanh ^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {a b \left (3 a^2-b^2\right ) \arctan (\sinh (x))}{\left (a^2+b^2\right )^3}+\frac {a^2 \left (a^2-3 b^2\right ) \log (\cosh (x))}{\left (a^2+b^2\right )^3}-\frac {a^2 \left (a^2-3 b^2\right ) \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}+\frac {a^3}{\left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac {\text {sech}^2(x) \left (a^2-b^2-2 a b \sinh (x)\right )}{2 \left (a^2+b^2\right )^2} \]
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Time = 0.27 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2800, 1661, 1643, 649, 209, 266} \[ \int \frac {\tanh ^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {a b \left (3 a^2-b^2\right ) \arctan (\sinh (x))}{\left (a^2+b^2\right )^3}-\frac {a^2 \left (a^2-3 b^2\right ) \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}+\frac {a^2 \left (a^2-3 b^2\right ) \log (\cosh (x))}{\left (a^2+b^2\right )^3}+\frac {\text {sech}^2(x) \left (a^2-2 a b \sinh (x)-b^2\right )}{2 \left (a^2+b^2\right )^2}+\frac {a^3}{\left (a^2+b^2\right )^2 (a+b \sinh (x))} \]
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Rule 209
Rule 266
Rule 649
Rule 1643
Rule 1661
Rule 2800
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x^3}{(a+x)^2 \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (x)\right ) \\ & = \frac {\text {sech}^2(x) \left (a^2-b^2-2 a b \sinh (x)\right )}{2 \left (a^2+b^2\right )^2}-\frac {\text {Subst}\left (\int \frac {\frac {2 a^3 b^4}{\left (a^2+b^2\right )^2}+\frac {2 a^2 b^2 x}{a^2+b^2}-\frac {2 a b^4 x^2}{\left (a^2+b^2\right )^2}}{(a+x)^2 \left (-b^2-x^2\right )} \, dx,x,b \sinh (x)\right )}{2 b^2} \\ & = \frac {\text {sech}^2(x) \left (a^2-b^2-2 a b \sinh (x)\right )}{2 \left (a^2+b^2\right )^2}-\frac {\text {Subst}\left (\int \left (\frac {2 a^3 b^2}{\left (a^2+b^2\right )^2 (a+x)^2}+\frac {2 a^2 b^2 \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3 (a+x)}+\frac {2 a b^2 \left (-b^2 \left (3 a^2-b^2\right )-a \left (a^2-3 b^2\right ) x\right )}{\left (a^2+b^2\right )^3 \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (x)\right )}{2 b^2} \\ & = -\frac {a^2 \left (a^2-3 b^2\right ) \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}+\frac {a^3}{\left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac {\text {sech}^2(x) \left (a^2-b^2-2 a b \sinh (x)\right )}{2 \left (a^2+b^2\right )^2}-\frac {a \text {Subst}\left (\int \frac {-b^2 \left (3 a^2-b^2\right )-a \left (a^2-3 b^2\right ) x}{b^2+x^2} \, dx,x,b \sinh (x)\right )}{\left (a^2+b^2\right )^3} \\ & = -\frac {a^2 \left (a^2-3 b^2\right ) \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}+\frac {a^3}{\left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac {\text {sech}^2(x) \left (a^2-b^2-2 a b \sinh (x)\right )}{2 \left (a^2+b^2\right )^2}+\frac {\left (a^2 \left (a^2-3 b^2\right )\right ) \text {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \sinh (x)\right )}{\left (a^2+b^2\right )^3}+\frac {\left (a b^2 \left (3 a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \sinh (x)\right )}{\left (a^2+b^2\right )^3} \\ & = \frac {a b \left (3 a^2-b^2\right ) \arctan (\sinh (x))}{\left (a^2+b^2\right )^3}+\frac {a^2 \left (a^2-3 b^2\right ) \log (\cosh (x))}{\left (a^2+b^2\right )^3}-\frac {a^2 \left (a^2-3 b^2\right ) \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}+\frac {a^3}{\left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac {\text {sech}^2(x) \left (a^2-b^2-2 a b \sinh (x)\right )}{2 \left (a^2+b^2\right )^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.11 \[ \int \frac {\tanh ^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {-2 a b \left (a^2+b^2\right ) \arctan (\sinh (x))+a^2 (a-i b) (a-3 i b) \log (i-\sinh (x))+a^2 (a+i b) (a+3 i b) \log (i+\sinh (x))-2 a^2 \left (a^2-3 b^2\right ) \log (a+b \sinh (x))+\left (a^4-b^4\right ) \text {sech}^2(x)+\frac {2 a^3 \left (a^2+b^2\right )}{a+b \sinh (x)}-2 a b \left (a^2+b^2\right ) \text {sech}(x) \tanh (x)}{2 \left (a^2+b^2\right )^3} \]
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Time = 1.41 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.76
method | result | size |
default | \(-\frac {2 a^{2} \left (\frac {\left (-a^{2} b -b^{3}\right ) \tanh \left (\frac {x}{2}\right )}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}+\frac {\left (a^{2}-3 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )}{2}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {\frac {2 \left (\left (a^{3} b +b^{3} a \right ) \tanh \left (\frac {x}{2}\right )^{3}+\left (-a^{4}+b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{2}+\left (-a^{3} b -b^{3} a \right ) \tanh \left (\frac {x}{2}\right )\right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{2}}+2 a \left (\frac {\left (a^{3}-3 a \,b^{2}\right ) \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )}{2}+\left (3 a^{2} b -b^{3}\right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\) | \(237\) |
risch | \(\frac {2 \,{\mathrm e}^{x} \left (a^{3} {\mathrm e}^{4 x}-{\mathrm e}^{4 x} a \,b^{2}-a^{2} b \,{\mathrm e}^{3 x}-{\mathrm e}^{3 x} b^{3}+4 a^{3} {\mathrm e}^{2 x}+{\mathrm e}^{x} a^{2} b +b^{3} {\mathrm e}^{x}+a^{3}-a \,b^{2}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+{\mathrm e}^{2 x}\right )^{2} \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}-\frac {a^{4} \ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right ) b^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {3 i a^{3} \ln \left ({\mathrm e}^{x}-i\right ) b}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {i a \ln \left ({\mathrm e}^{x}-i\right ) b^{3}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {\ln \left ({\mathrm e}^{x}-i\right ) a^{4}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {3 \ln \left ({\mathrm e}^{x}-i\right ) a^{2} b^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {3 i a^{3} \ln \left ({\mathrm e}^{x}+i\right ) b}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {i a \ln \left ({\mathrm e}^{x}+i\right ) b^{3}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {\ln \left ({\mathrm e}^{x}+i\right ) a^{4}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {3 \ln \left ({\mathrm e}^{x}+i\right ) a^{2} b^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\) | \(510\) |
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Leaf count of result is larger than twice the leaf count of optimal. 2850 vs. \(2 (133) = 266\).
Time = 0.35 (sec) , antiderivative size = 2850, normalized size of antiderivative = 21.11 \[ \int \frac {\tanh ^3(x)}{(a+b \sinh (x))^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {\tanh ^3(x)}{(a+b \sinh (x))^2} \, dx=\int \frac {\tanh ^{3}{\left (x \right )}}{\left (a + b \sinh {\left (x \right )}\right )^{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (133) = 266\).
Time = 0.34 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.78 \[ \int \frac {\tanh ^3(x)}{(a+b \sinh (x))^2} \, dx=-\frac {2 \, {\left (3 \, a^{3} b - a b^{3}\right )} \arctan \left (e^{\left (-x\right )}\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (4 \, a^{3} e^{\left (-3 \, x\right )} + {\left (a^{3} - a b^{2}\right )} e^{\left (-x\right )} - {\left (a^{2} b + b^{3}\right )} e^{\left (-2 \, x\right )} + {\left (a^{2} b + b^{3}\right )} e^{\left (-4 \, x\right )} + {\left (a^{3} - a b^{2}\right )} e^{\left (-5 \, x\right )}\right )}}{a^{4} b + 2 \, a^{2} b^{3} + b^{5} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-x\right )} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-2 \, x\right )} + 4 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-3 \, x\right )} - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-4 \, x\right )} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-5 \, x\right )} - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-6 \, x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (133) = 266\).
Time = 0.30 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.27 \[ \int \frac {\tanh ^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} {\left (3 \, a^{3} b - a b^{3}\right )}}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac {{\left (a^{4} b - 3 \, a^{2} b^{3}\right )} \log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac {2 \, {\left (a^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} - a b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + a^{2} b {\left (e^{\left (-x\right )} - e^{x}\right )} + b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )} + 6 \, a^{3} - 2 \, a b^{2}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} - 2 \, a {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4 \, b {\left (e^{\left (-x\right )} - e^{x}\right )} - 8 \, a\right )}} \]
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Time = 4.84 (sec) , antiderivative size = 501, normalized size of antiderivative = 3.71 \[ \int \frac {\tanh ^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {2\,\left (a^8+2\,a^6\,b^2-2\,a^2\,b^6-b^8\right )}{\left (a^2+b^2\right )\,{\left (a^4+2\,a^2\,b^2+b^4\right )}^2}-\frac {2\,{\mathrm {e}}^x\,\left (a^7\,b+3\,a^5\,b^3+3\,a^3\,b^5+a\,b^7\right )}{\left (a^2+b^2\right )\,{\left (a^4+2\,a^2\,b^2+b^4\right )}^2}}{{\mathrm {e}}^{2\,x}+1}-\frac {\frac {2\,\left (a^2-b^2\right )}{a^4+2\,a^2\,b^2+b^4}-\frac {4\,a\,b\,{\mathrm {e}}^x}{a^4+2\,a^2\,b^2+b^4}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}-\frac {a\,\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}{-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}}-\frac {\ln \left (15\,a^6\,b^3-a^2\,b^7-30\,a^4\,b^5-4\,a^8\,b+8\,a^9\,{\mathrm {e}}^x+a^2\,b^7\,{\mathrm {e}}^{2\,x}+30\,a^4\,b^5\,{\mathrm {e}}^{2\,x}-15\,a^6\,b^3\,{\mathrm {e}}^{2\,x}+4\,a^8\,b\,{\mathrm {e}}^{2\,x}+2\,a^3\,b^6\,{\mathrm {e}}^x+60\,a^5\,b^4\,{\mathrm {e}}^x-30\,a^7\,b^2\,{\mathrm {e}}^x\right )\,\left (a^4-3\,a^2\,b^2\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {2\,{\mathrm {e}}^x\,\left (a^7\,b^2+2\,a^5\,b^4+a^3\,b^6\right )}{b\,\left (a^2\,b+b^3\right )\,\left (a^2+b^2\right )\,\left (2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {a\,\ln \left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3} \]
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