Integrand size = 14, antiderivative size = 31 \[ \int \frac {1}{a^2+b^2 \cosh ^2(x)} \, dx=\frac {\text {arctanh}\left (\frac {a \tanh (x)}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3260, 214} \[ \int \frac {1}{a^2+b^2 \cosh ^2(x)} \, dx=\frac {\text {arctanh}\left (\frac {a \tanh (x)}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}} \]
[In]
[Out]
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 = 0.45 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a^2+b^2 \cosh ^2(x)} \, dx=\frac {\text {arctanh}\left (\frac {a \tanh (x)}{\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(97\) vs. \(2(27)=54\).
Time = 0.32 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.16
method | result | size |
default | \(\frac {\ln \left (\sqrt {a^{2}+b^{2}}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )+2 a \tanh \left (\frac {x}{2}\right )+\sqrt {a^{2}+b^{2}}\right )}{2 a \sqrt {a^{2}+b^{2}}}-\frac {\ln \left (\sqrt {a^{2}+b^{2}}\, \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 a \tanh \left (\frac {x}{2}\right )+\sqrt {a^{2}+b^{2}}\right )}{2 a \sqrt {a^{2}+b^{2}}}\) | \(98\) |
risch | \(\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a^{2} \sqrt {a^{2}+b^{2}}+b^{2} \sqrt {a^{2}+b^{2}}-2 a^{3}-2 b^{2} a}{b^{2} \sqrt {a^{2}+b^{2}}}\right )}{2 a \sqrt {a^{2}+b^{2}}}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a^{2} \sqrt {a^{2}+b^{2}}+b^{2} \sqrt {a^{2}+b^{2}}+2 a^{3}+2 b^{2} a}{b^{2} \sqrt {a^{2}+b^{2}}}\right )}{2 a \sqrt {a^{2}+b^{2}}}\) | \(146\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (27) = 54\).
Time = 0.25 (sec) , antiderivative size = 288, normalized size of antiderivative = 9.29 \[ \int \frac {1}{a^2+b^2 \cosh ^2(x)} \, dx=\frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {b^{4} \cosh \left (x\right )^{4} + 4 \, b^{4} \cosh \left (x\right ) \sinh \left (x\right )^{3} + b^{4} \sinh \left (x\right )^{4} + 8 \, a^{4} + 8 \, a^{2} b^{2} + b^{4} + 2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b^{4} \cosh \left (x\right )^{2} + 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (b^{4} \cosh \left (x\right )^{3} + {\left (2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) - 4 \, {\left (a b^{2} \cosh \left (x\right )^{2} + 2 \, a b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a b^{2} \sinh \left (x\right )^{2} + 2 \, a^{3} + a b^{2}\right )} \sqrt {a^{2} + b^{2}}}{b^{2} \cosh \left (x\right )^{4} + 4 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + b^{2} \sinh \left (x\right )^{4} + 2 \, {\left (2 \, a^{2} + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} + b^{2}\right )} \sinh \left (x\right )^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \left (x\right )^{3} + {\left (2 \, a^{2} + b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}\right )}{2 \, {\left (a^{3} + a b^{2}\right )}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 15.58 (sec) , antiderivative size = 1129, normalized size of antiderivative = 36.42 \[ \int \frac {1}{a^2+b^2 \cosh ^2(x)} \, dx=\text {Too large to display} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (27) = 54\).
Time = 0.32 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.45 \[ \int \frac {1}{a^2+b^2 \cosh ^2(x)} \, dx=-\frac {\log \left (\frac {b^{2} e^{\left (-2 \, x\right )} + 2 \, a^{2} + b^{2} - 2 \, \sqrt {a^{2} + b^{2}} a}{b^{2} e^{\left (-2 \, x\right )} + 2 \, a^{2} + b^{2} + 2 \, \sqrt {a^{2} + b^{2}} a}\right )}{2 \, \sqrt {a^{2} + b^{2}} a} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (27) = 54\).
Time = 0.27 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.55 \[ \int \frac {1}{a^2+b^2 \cosh ^2(x)} \, dx=\frac {\log \left (\frac {b^{2} e^{\left (2 \, x\right )} + 2 \, a^{2} + b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left | a \right |}}{b^{2} e^{\left (2 \, x\right )} + 2 \, a^{2} + b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left | a \right |}}\right )}{2 \, \sqrt {a^{2} + b^{2}} {\left | a \right |}} \]
[In]
[Out]
Time = 0.77 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.52 \[ \int \frac {1}{a^2+b^2 \cosh ^2(x)} \, dx=\frac {\mathrm {atan}\left (\frac {2\,a^2\,{\left (-a^4-a^2\,b^2\right )}^{3/2}+b^2\,{\left (-a^4-a^2\,b^2\right )}^{3/2}+b^2\,{\mathrm {e}}^{2\,x}\,{\left (-a^4-a^2\,b^2\right )}^{3/2}}{2\,a^8+4\,a^6\,b^2+2\,a^4\,b^4}\right )}{\sqrt {-a^4-a^2\,b^2}} \]
[In]
[Out]