Integrand size = 13, antiderivative size = 93 \[ \int \frac {\text {sech}^2(x)}{(a+b \sinh (x))^2} \, dx=-\frac {6 a b^2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {b \text {sech}(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\text {sech}(x) \left (3 a b+\left (a^2-2 b^2\right ) \sinh (x)\right )}{\left (a^2+b^2\right )^2} \]
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Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2773, 2945, 12, 2739, 632, 212} \[ \int \frac {\text {sech}^2(x)}{(a+b \sinh (x))^2} \, dx=-\frac {6 a b^2 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {b \text {sech}(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\text {sech}(x) \left (\left (a^2-2 b^2\right ) \sinh (x)+3 a b\right )}{\left (a^2+b^2\right )^2} \]
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Rule 12
Rule 212
Rule 632
Rule 2739
Rule 2773
Rule 2945
Rubi steps \begin{align*} \text {integral}& = -\frac {b \text {sech}(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {\int \frac {\text {sech}^2(x) (-a+2 b \sinh (x))}{a+b \sinh (x)} \, dx}{a^2+b^2} \\ & = -\frac {b \text {sech}(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\text {sech}(x) \left (3 a b+\left (a^2-2 b^2\right ) \sinh (x)\right )}{\left (a^2+b^2\right )^2}+\frac {\int \frac {3 a b^2}{a+b \sinh (x)} \, dx}{\left (a^2+b^2\right )^2} \\ & = -\frac {b \text {sech}(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\text {sech}(x) \left (3 a b+\left (a^2-2 b^2\right ) \sinh (x)\right )}{\left (a^2+b^2\right )^2}+\frac {\left (3 a b^2\right ) \int \frac {1}{a+b \sinh (x)} \, dx}{\left (a^2+b^2\right )^2} \\ & = -\frac {b \text {sech}(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\text {sech}(x) \left (3 a b+\left (a^2-2 b^2\right ) \sinh (x)\right )}{\left (a^2+b^2\right )^2}+\frac {\left (6 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 {b \text {sech}(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\text {sech}(x) \left (3 a b+\left (a^2-2 b^2\right ) \sinh (x)\right )}{\left (a^2+b^2\right )^2}-\frac {\left (12 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 {6 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 {b \text {sech}(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\text {sech}(x) \left (3 a b+\left (a^2-2 b^2\right ) \sinh (x)\right )}{\left (a^2+b^2\right )^2} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.01 \[ \int \frac {\text {sech}^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {6 a b^2 \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+2 a b \text {sech}(x)-\frac {b^3 \cosh (x)}{a+b \sinh (x)}+a^2 \tanh (x)-b^2 \tanh (x)}{\left (a^2+b^2\right )^2} \]
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Time = 50.27 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.48
method | result | size |
default | \(-\frac {2 b^{2} \left (\frac {-\frac {b^{2} \tanh \left (\frac {x}{2}\right )}{a}-b}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {3 a \,\operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {2 \left (\left (-a^{2}+b^{2}\right ) \tanh \left (\frac {x}{2}\right )-2 a b \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )}\) | \(138\) |
risch | \(-\frac {2 \left (-3 a \,b^{2} {\mathrm e}^{3 x}-3 a^{2} b \,{\mathrm e}^{2 x}+2 a^{3} {\mathrm e}^{x}-a \,b^{2} {\mathrm e}^{x}-a^{2} b +2 b^{3}\right )}{\left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) \left (1+{\mathrm e}^{2 x}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {3 b^{2} a \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}-\frac {3 b^{2} a \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\) | \(229\) |
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Leaf count of result is larger than twice the leaf count of optimal. 802 vs. \(2 (89) = 178\).
Time = 0.30 (sec) , antiderivative size = 802, normalized size of antiderivative = 8.62 \[ \int \frac {\text {sech}^2(x)}{(a+b \sinh (x))^2} \, dx=-\frac {2 \, a^{4} b - 2 \, a^{2} b^{3} - 4 \, b^{5} + 6 \, {\left (a^{3} b^{2} + a b^{4}\right )} \cosh \left (x\right )^{3} + 6 \, {\left (a^{3} b^{2} + a b^{4}\right )} \sinh \left (x\right )^{3} + 6 \, {\left (a^{4} b + a^{2} b^{3}\right )} \cosh \left (x\right )^{2} + 6 \, {\left (a^{4} b + a^{2} b^{3} + 3 \, {\left (a^{3} b^{2} + a b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 3 \, {\left (a b^{3} \cosh \left (x\right )^{4} + a b^{3} \sinh \left (x\right )^{4} + 2 \, a^{2} b^{2} \cosh \left (x\right )^{3} + 2 \, a^{2} b^{2} \cosh \left (x\right ) - a b^{3} + 2 \, {\left (2 \, a b^{3} \cosh \left (x\right ) + a^{2} b^{2}\right )} \sinh \left (x\right )^{3} + 6 \, {\left (a b^{3} \cosh \left (x\right )^{2} + a^{2} b^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 2 \, {\left (2 \, a b^{3} \cosh \left (x\right )^{3} + 3 \, a^{2} b^{2} \cosh \left (x\right )^{2} + a^{2} b^{2}\right )} \sinh \left (x\right )\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 (2 \, a^{5} + a^{3} b^{2} - a b^{4}\right )} \cosh \left (x\right ) - 2 \, {\left (2 \, a^{5} + a^{3} b^{2} - a b^{4} - 9 \, {\left (a^{3} b^{2} + a b^{4}\right )} \cosh \left (x\right )^{2} - 6 \, {\left (a^{4} b + a^{2} b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7} - {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cosh \left (x\right )^{4} - {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \sinh \left (x\right )^{4} - 2 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cosh \left (x\right )^{3} - 2 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6} + 2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 6 \, {\left ({\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cosh \left (x\right )^{2} + {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} - 2 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cosh \left (x\right ) - 2 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6} + 2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cosh \left (x\right )^{3} + 3 \, {\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )} \]
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\[ \int \frac {\text {sech}^2(x)}{(a+b \sinh (x))^2} \, dx=\int \frac {\operatorname {sech}^{2}{\left (x \right )}}{\left (a + b \sinh {\left (x \right )}\right )^{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (89) = 178\).
Time = 0.29 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.31 \[ \int \frac {\text {sech}^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {3 \, a b^{2} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (3 \, a^{2} b e^{\left (-2 \, x\right )} - 3 \, a b^{2} e^{\left (-3 \, x\right )} + a^{2} b - 2 \, b^{3} + {\left (2 \, a^{3} - a b^{2}\right )} e^{\left (-x\right )}\right )}}{a^{4} b + 2 \, a^{2} b^{3} + b^{5} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-x\right )} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-3 \, x\right )} - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-4 \, x\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.80 \[ \int \frac {\text {sech}^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {3 \, a b^{2} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (3 \, a b^{2} e^{\left (3 \, x\right )} + 3 \, a^{2} b e^{\left (2 \, x\right )} - 2 \, a^{3} e^{x} + a b^{2} e^{x} + a^{2} b - 2 \, b^{3}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b e^{\left (4 \, x\right )} + 2 \, a e^{\left (3 \, x\right )} + 2 \, a e^{x} - b\right )}} \]
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Time = 1.74 (sec) , antiderivative size = 302, normalized size of antiderivative = 3.25 \[ \int \frac {\text {sech}^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {6\,a^4\,b^4\,{\mathrm {e}}^{2\,x}}{\left (a^3+a\,b^2\right )\,\left (a^3\,b^3+a\,b^5\right )}-\frac {2\,\left (2\,a^2\,b^6-a^4\,b^4\right )}{\left (a^3+a\,b^2\right )\,\left (a^3\,b^3+a\,b^5\right )}+\frac {6\,a^3\,b^5\,{\mathrm {e}}^{3\,x}}{\left (a^3+a\,b^2\right )\,\left (a^3\,b^3+a\,b^5\right )}+\frac {2\,a\,{\mathrm {e}}^x\,\left (a^2\,b^6-2\,a^4\,b^4\right )}{b\,\left (a^3+a\,b^2\right )\,\left (a^3\,b^3+a\,b^5\right )}}{2\,a\,{\mathrm {e}}^x-b+2\,a\,{\mathrm {e}}^{3\,x}+b\,{\mathrm {e}}^{4\,x}}-\frac {3\,a\,b^2\,\ln \left (-\frac {6\,a\,b\,{\mathrm {e}}^x}{{\left (a^2+b^2\right )}^2}-\frac {6\,a\,b\,\left (b-a\,{\mathrm {e}}^x\right )}{{\left (a^2+b^2\right )}^{5/2}}\right )}{{\left (a^2+b^2\right )}^{5/2}}+\frac {3\,a\,b^2\,\ln \left (\frac {6\,a\,b\,\left (b-a\,{\mathrm {e}}^x\right )}{{\left (a^2+b^2\right )}^{5/2}}-\frac {6\,a\,b\,{\mathrm {e}}^x}{{\left (a^2+b^2\right )}^2}\right )}{{\left (a^2+b^2\right )}^{5/2}} \]
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