Integrand size = 13, antiderivative size = 144 \[ \int \frac {\tanh ^2(x)}{(a+b \sinh (x))^2} \, dx=-\frac {2 a^3 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}+\frac {4 a b^2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {2 a b \text {sech}(x)}{\left (a^2+b^2\right )^2}-\frac {a^2 b \cosh (x)}{\left (a^2+b^2\right )^2 (a+b \sinh (x))}-\frac {\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2} \]
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Time = 0.18 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {2810, 2743, 12, 2739, 632, 212, 2748, 3852, 8} \[ \int \frac {\tanh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {4 a b^2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2}-\frac {2 a b \text {sech}(x)}{\left (a^2+b^2\right )^2}-\frac {a^2 b \cosh (x)}{\left (a^2+b^2\right )^2 (a+b \sinh (x))}-\frac {2 a^3 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}} \]
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Rule 8
Rule 12
Rule 212
Rule 632
Rule 2739
Rule 2743
Rule 2748
Rule 2810
Rule 3852
Rubi steps \begin{align*} \text {integral}& = -\int \left (-\frac {a^2}{\left (a^2+b^2\right ) (a+b \sinh (x))^2}+\frac {2 a b^2}{\left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac {\text {sech}^2(x) \left (a^2 \left (1-\frac {b^2}{a^2}\right )-2 a b \sinh (x)\right )}{\left (a^2+b^2\right )^2}\right ) \, dx \\ & = -\frac {\int \text {sech}^2(x) \left (a^2 \left (1-\frac {b^2}{a^2}\right )-2 a b \sinh (x)\right ) \, dx}{\left (a^2+b^2\right )^2}-\frac {\left (2 a b^2\right ) \int \frac {1}{a+b \sinh (x)} \, dx}{\left (a^2+b^2\right )^2}+\frac {a^2 \int \frac {1}{(a+b \sinh (x))^2} \, dx}{a^2+b^2} \\ & = -\frac {2 a b \text {sech}(x)}{\left (a^2+b^2\right )^2}-\frac {a^2 b \cosh (x)}{\left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac {a^2 \int \frac {a}{a+b \sinh (x)} \, dx}{\left (a^2+b^2\right )^2}-\frac {\left (4 a b^2\right ) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{\left (a^2+b^2\right )^2}-\frac {\left (a^2-b^2\right ) \int \text {sech}^2(x) \, dx}{\left (a^2+b^2\right )^2} \\ & = -\frac {2 a b \text {sech}(x)}{\left (a^2+b^2\right )^2}-\frac {a^2 b \cosh (x)}{\left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac {a^3 \int \frac {1}{a+b \sinh (x)} \, dx}{\left (a^2+b^2\right )^2}+\frac {\left (8 a b^2\right ) \text {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{\left (a^2+b^2\right )^2}-\frac {\left (i \left (a^2-b^2\right )\right ) \text {Subst}(\int 1 \, dx,x,-i \tanh (x))}{\left (a^2+b^2\right )^2} \\ & = \frac {4 a b^2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {2 a b \text {sech}(x)}{\left (a^2+b^2\right )^2}-\frac {a^2 b \cosh (x)}{\left (a^2+b^2\right )^2 (a+b \sinh (x))}-\frac {\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2}+\frac {\left (2 a^3\right ) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{\left (a^2+b^2\right )^2} \\ & = \frac {4 a b^2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {2 a b \text {sech}(x)}{\left (a^2+b^2\right )^2}-\frac {a^2 b \cosh (x)}{\left (a^2+b^2\right )^2 (a+b \sinh (x))}-\frac {\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2}-\frac {\left (4 a^3\right ) \text {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{\left (a^2+b^2\right )^2} \\ & = -\frac {2 a^3 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}+\frac {4 a b^2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {2 a b \text {sech}(x)}{\left (a^2+b^2\right )^2}-\frac {a^2 b \cosh (x)}{\left (a^2+b^2\right )^2 (a+b \sinh (x))}-\frac {\left (a^2-b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.69 \[ \int \frac {\tanh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {2 a \left (a^2-2 b^2\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-2 a b \text {sech}(x)-\frac {a^2 b \cosh (x)}{a+b \sinh (x)}+\left (-a^2+b^2\right ) \tanh (x)}{\left (a^2+b^2\right )^2} \]
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Time = 0.86 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.99
method | result | size |
default | \(-\frac {2 a \left (\frac {-b^{2} \tanh \left (\frac {x}{2}\right )-a b}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {\left (a^{2}-2 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {2 \left (-a^{2}+b^{2}\right ) \tanh \left (\frac {x}{2}\right )-4 a b}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )}\) | \(142\) |
risch | \(\frac {2 a^{3} {\mathrm e}^{3 x}-4 a \,b^{2} {\mathrm e}^{3 x}-8 a^{2} b \,{\mathrm e}^{2 x}-2 b^{3} {\mathrm e}^{2 x}+6 a^{3} {\mathrm e}^{x}-4 a^{2} b +2 b^{3}}{\left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) \left (1+{\mathrm e}^{2 x}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {a^{3} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}-\frac {2 b^{2} a \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}-\frac {a^{3} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}+\frac {2 b^{2} a \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\) | \(369\) |
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Leaf count of result is larger than twice the leaf count of optimal. 900 vs. \(2 (136) = 272\).
Time = 0.29 (sec) , antiderivative size = 900, normalized size of antiderivative = 6.25 \[ \int \frac {\tanh ^2(x)}{(a+b \sinh (x))^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {\tanh ^2(x)}{(a+b \sinh (x))^2} \, dx=\int \frac {\tanh ^{2}{\left (x \right )}}{\left (a + b \sinh {\left (x \right )}\right )^{2}}\, dx \]
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Time = 0.33 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.55 \[ \int \frac {\tanh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {{\left (a^{2} - 2 \, b^{2}\right )} a \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (3 \, a^{3} e^{\left (-x\right )} + 2 \, a^{2} b - b^{3} + {\left (4 \, a^{2} b + b^{3}\right )} e^{\left (-2 \, x\right )} + {\left (a^{3} - 2 \, a b^{2}\right )} e^{\left (-3 \, x\right )}\right )}}{a^{4} b + 2 \, a^{2} b^{3} + b^{5} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-x\right )} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-3 \, x\right )} - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-4 \, x\right )}} \]
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Time = 0.32 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.26 \[ \int \frac {\tanh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {{\left (a^{3} - 2 \, a b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (a^{3} e^{\left (3 \, x\right )} - 2 \, a b^{2} e^{\left (3 \, x\right )} - 4 \, a^{2} b e^{\left (2 \, x\right )} - b^{3} e^{\left (2 \, x\right )} + 3 \, a^{3} e^{x} - 2 \, a^{2} b + b^{3}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b e^{\left (4 \, x\right )} + 2 \, a e^{\left (3 \, x\right )} + 2 \, a e^{x} - b\right )}} \]
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Time = 1.89 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.62 \[ \int \frac {\tanh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {2\,\left (a^2\,b^9-2\,a^4\,b^7\right )}{b^3\,\left (a^3+a\,b^2\right )\,\left (a^3\,b^3+a\,b^5\right )}-\frac {2\,{\mathrm {e}}^{2\,x}\,\left (4\,a^4\,b^7+a^2\,b^9\right )}{b^3\,\left (a^3+a\,b^2\right )\,\left (a^3\,b^3+a\,b^5\right )}+\frac {6\,a^5\,b^3\,{\mathrm {e}}^x}{\left (a^3+a\,b^2\right )\,\left (a^3\,b^3+a\,b^5\right )}-\frac {2\,a\,{\mathrm {e}}^{3\,x}\,\left (2\,a^2\,b^9-a^4\,b^7\right )}{b^4\,\left (a^3+a\,b^2\right )\,\left (a^3\,b^3+a\,b^5\right )}}{2\,a\,{\mathrm {e}}^x-b+2\,a\,{\mathrm {e}}^{3\,x}+b\,{\mathrm {e}}^{4\,x}}-\frac {a\,\ln \left (\frac {2\,{\mathrm {e}}^x\,\left (2\,a\,b^2-a^3\right )}{b\,{\left (a^2+b^2\right )}^2}-\frac {2\,a\,\left (a^2-2\,b^2\right )\,\left (b-a\,{\mathrm {e}}^x\right )}{b\,{\left (a^2+b^2\right )}^{5/2}}\right )\,\left (a^2-2\,b^2\right )}{{\left (a^2+b^2\right )}^{5/2}}+\frac {a\,\ln \left (\frac {2\,{\mathrm {e}}^x\,\left (2\,a\,b^2-a^3\right )}{b\,{\left (a^2+b^2\right )}^2}+\frac {2\,a\,\left (a^2-2\,b^2\right )\,\left (b-a\,{\mathrm {e}}^x\right )}{b\,{\left (a^2+b^2\right )}^{5/2}}\right )\,\left (a^2-2\,b^2\right )}{{\left (a^2+b^2\right )}^{5/2}} \]
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