Integrand size = 15, antiderivative size = 35 \[ \int \frac {1}{a^2-b^2 \cosh ^2(x)} \, dx=\frac {\text {arctanh}\left (\frac {a \tanh (x)}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}} \]
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Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3260, 214} \[ \int \frac {1}{a^2-b^2 \cosh ^2(x)} \, dx=\frac {\text {arctanh}\left (\frac {a \tanh (x)}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}} \]
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Rule 214
Rule 3260
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{a^2-\left (a^2-b^2\right ) x^2} \, dx,x,\coth (x)\right ) \\ & = \frac {\text {arctanh}\left (\frac {a \tanh (x)}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^2-b^2 \cosh ^2(x)} \, dx=\frac {\text {arctanh}\left (\frac {a \tanh (x)}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(31)=62\).
Time = 0.22 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.11
method | result | size |
default | \(\frac {\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a \sqrt {\left (a +b \right ) \left (a -b \right )}}+\frac {\operatorname {arctanh}\left (\frac {\left (a +b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a \sqrt {\left (a +b \right ) \left (a -b \right )}}\) | \(74\) |
risch | \(\frac {\ln \left ({\mathrm e}^{2 x}-\frac {2 a^{2} \sqrt {a^{2}-b^{2}}-b^{2} \sqrt {a^{2}-b^{2}}-2 a^{3}+2 b^{2} a}{b^{2} \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, a}-\frac {\ln \left ({\mathrm e}^{2 x}-\frac {2 a^{2} \sqrt {a^{2}-b^{2}}-b^{2} \sqrt {a^{2}-b^{2}}+2 a^{3}-2 b^{2} a}{b^{2} \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, a}\) | \(166\) |
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (31) = 62\).
Time = 0.27 (sec) , antiderivative size = 388, normalized size of antiderivative = 11.09 \[ \int \frac {1}{a^2-b^2 \cosh ^2(x)} \, dx=\left [\frac {\sqrt {a^{2} - b^{2}} \log \left (\frac {b^{4} \cosh \left (x\right )^{4} + 4 \, b^{4} \cosh \left (x\right ) \sinh \left (x\right )^{3} + b^{4} \sinh \left (x\right )^{4} + 8 \, a^{4} - 8 \, a^{2} b^{2} + b^{4} - 2 \, {\left (2 \, a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b^{4} \cosh \left (x\right )^{2} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (b^{4} \cosh \left (x\right )^{3} - {\left (2 \, a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 4 \, {\left (a b^{2} \cosh \left (x\right )^{2} + 2 \, a b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a b^{2} \sinh \left (x\right )^{2} - 2 \, a^{3} + a b^{2}\right )} \sqrt {a^{2} - b^{2}}}{b^{2} \cosh \left (x\right )^{4} + 4 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + b^{2} \sinh \left (x\right )^{4} - 2 \, {\left (2 \, a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (x\right )^{2} - 2 \, a^{2} + b^{2}\right )} \sinh \left (x\right )^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \left (x\right )^{3} - {\left (2 \, a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}\right )}{2 \, {\left (a^{3} - a b^{2}\right )}}, \frac {\sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {{\left (b^{2} \cosh \left (x\right )^{2} + 2 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + b^{2} \sinh \left (x\right )^{2} - 2 \, a^{2} + b^{2}\right )} \sqrt {-a^{2} + b^{2}}}{2 \, {\left (a^{3} - a b^{2}\right )}}\right )}{a^{3} - a b^{2}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 874 vs. \(2 (27) = 54\).
Time = 14.55 (sec) , antiderivative size = 874, normalized size of antiderivative = 24.97 \[ \int \frac {1}{a^2-b^2 \cosh ^2(x)} \, dx=\text {Too large to display} \]
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Exception generated. \[ \int \frac {1}{a^2-b^2 \cosh ^2(x)} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.43 \[ \int \frac {1}{a^2-b^2 \cosh ^2(x)} \, dx=-\frac {\arctan \left (\frac {b^{2} e^{\left (2 \, x\right )} - 2 \, a^{2} + b^{2}}{2 \, \sqrt {-a^{2} + b^{2}} a}\right )}{\sqrt {-a^{2} + b^{2}} a} \]
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Time = 0.40 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.03 \[ \int \frac {1}{a^2-b^2 \cosh ^2(x)} \, dx=-\frac {\mathrm {atan}\left (\frac {b^2\,{\left (a^2\,b^2-a^4\right )}^{3/2}-2\,a^2\,{\left (a^2\,b^2-a^4\right )}^{3/2}+b^2\,{\mathrm {e}}^{2\,x}\,{\left (a^2\,b^2-a^4\right )}^{3/2}}{2\,a^8-4\,a^6\,b^2+2\,a^4\,b^4}\right )}{\sqrt {a^2\,b^2-a^4}} \]
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