Integrand size = 13, antiderivative size = 69 \[ \int \frac {\tanh ^2(x)}{a+b \sinh (x)} \, dx=-\frac {2 a^2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {b \text {sech}(x)}{a^2+b^2}-\frac {a \tanh (x)}{a^2+b^2} \]
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Time = 0.06 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2806, 3852, 8, 2686, 2739, 632, 212} \[ \int \frac {\tanh ^2(x)}{a+b \sinh (x)} \, dx=-\frac {2 a^2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {a \tanh (x)}{a^2+b^2}-\frac {b \text {sech}(x)}{a^2+b^2} \]
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Rule 8
Rule 212
Rule 632
Rule 2686
Rule 2739
Rule 2806
Rule 3852
Rubi steps \begin{align*} \text {integral}& = -\frac {a \int \text {sech}^2(x) \, dx}{a^2+b^2}+\frac {a^2 \int \frac {1}{a+b \sinh (x)} \, dx}{a^2+b^2}+\frac {b \int \text {sech}(x) \tanh (x) \, dx}{a^2+b^2} \\ & = -\frac {(i a) \text {Subst}(\int 1 \, dx,x,-i \tanh (x))}{a^2+b^2}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^2+b^2}-\frac {b \text {Subst}(\int 1 \, dx,x,\text {sech}(x))}{a^2+b^2} \\ & = -\frac {b \text {sech}(x)}{a^2+b^2}-\frac {a \tanh (x)}{a^2+b^2}-\frac {\left (4 a^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 )}{a^2+b^2} \\ & = -\frac {2 a^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 )^{3/2}}-\frac {b \text {sech}(x)}{a^2+b^2}-\frac {a \tanh (x)}{a^2+b^2} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \frac {\tanh ^2(x)}{a+b \sinh (x)} \, dx=\frac {-b \text {sech}(x)+a \left (\frac {2 a \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-\tanh (x)\right )}{a^2+b^2} \]
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Time = 0.75 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.22
method | result | size |
default | \(\frac {8 a^{2} \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (4 a^{2}+4 b^{2}\right ) \sqrt {a^{2}+b^{2}}}+\frac {-2 a \tanh \left (\frac {x}{2}\right )-2 b}{\left (a^{2}+b^{2}\right ) \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )}\) | \(84\) |
risch | \(\frac {-2 \,{\mathrm e}^{x} b +2 a}{\left (1+{\mathrm e}^{2 x}\right ) \left (a^{2}+b^{2}\right )}+\frac {a^{2} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {a^{2} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\) | \(145\) |
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Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (65) = 130\).
Time = 0.28 (sec) , antiderivative size = 257, normalized size of antiderivative = 3.72 \[ \int \frac {\tanh ^2(x)}{a+b \sinh (x)} \, dx=\frac {2 \, a^{3} + 2 \, a b^{2} + {\left (a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} + a^{2}\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - b}\right ) - 2 \, {\left (a^{2} b + b^{3}\right )} \cosh \left (x\right ) - 2 \, {\left (a^{2} b + b^{3}\right )} \sinh \left (x\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{2}} \]
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\[ \int \frac {\tanh ^2(x)}{a+b \sinh (x)} \, dx=\int \frac {\tanh ^{2}{\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.29 \[ \int \frac {\tanh ^2(x)}{a+b \sinh (x)} \, dx=\frac {a^{2} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (b e^{\left (-x\right )} + a\right )}}{a^{2} + b^{2} + {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, x\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.26 \[ \int \frac {\tanh ^2(x)}{a+b \sinh (x)} \, dx=\frac {a^{2} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (b e^{x} - a\right )}}{{\left (a^{2} + b^{2}\right )} {\left (e^{\left (2 \, x\right )} + 1\right )}} \]
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Time = 1.65 (sec) , antiderivative size = 330, normalized size of antiderivative = 4.78 \[ \int \frac {\tanh ^2(x)}{a+b \sinh (x)} \, dx=\frac {\frac {2\,a}{a^2+b^2}-\frac {2\,b\,{\mathrm {e}}^x}{a^2+b^2}}{{\mathrm {e}}^{2\,x}+1}-\frac {2\,\mathrm {atan}\left (\left (\frac {b^3\,\sqrt {-a^6-3\,a^4\,b^2-3\,a^2\,b^4-b^6}}{2}+\frac {a^2\,b\,\sqrt {-a^6-3\,a^4\,b^2-3\,a^2\,b^4-b^6}}{2}\right )\,\left ({\mathrm {e}}^x\,\left (\frac {2\,a^2}{b^2\,\sqrt {a^4}\,{\left (a^2+b^2\right )}^2}+\frac {2\,\left (a^3\,\sqrt {a^4}+a\,b^2\,\sqrt {a^4}\right )}{a\,b^2\,\sqrt {-{\left (a^2+b^2\right )}^3}\,\left (a^2+b^2\right )\,\sqrt {-a^6-3\,a^4\,b^2-3\,a^2\,b^4-b^6}}\right )-\frac {2\,\left (b^3\,\sqrt {a^4}+a^2\,b\,\sqrt {a^4}\right )}{a\,b^2\,\sqrt {-{\left (a^2+b^2\right )}^3}\,\left (a^2+b^2\right )\,\sqrt {-a^6-3\,a^4\,b^2-3\,a^2\,b^4-b^6}}\right )\right )\,\sqrt {a^4}}{\sqrt {-a^6-3\,a^4\,b^2-3\,a^2\,b^4-b^6}} \]
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