Optimal. Leaf size=35 \[ \frac {\tanh ^{-1}\left (\frac {a \tanh (x)}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3181, 208} \[ \frac {\tanh ^{-1}\left (\frac {a \tanh (x)}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 3181
Rubi steps
\begin {align*} \int \frac {1}{a^2-b^2 \cosh ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{a^2-\left (a^2-b^2\right ) x^2} \, dx,x,\coth (x)\right )\\ &=\frac {\tanh ^{-1}\left (\frac {a \tanh (x)}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 35, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {a \tanh (x)}{\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a^2-b^2 \cosh ^2(x)} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.37, size = 388, normalized size = 11.09 \[ \left [\frac {\sqrt {a^{2} - b^{2}} \log \left (\frac {b^{4} \cosh \relax (x)^{4} + 4 \, b^{4} \cosh \relax (x) \sinh \relax (x)^{3} + b^{4} \sinh \relax (x)^{4} + 8 \, a^{4} - 8 \, a^{2} b^{2} + b^{4} - 2 \, {\left (2 \, a^{2} b^{2} - b^{4}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b^{4} \cosh \relax (x)^{2} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \relax (x)^{2} + 4 \, {\left (b^{4} \cosh \relax (x)^{3} - {\left (2 \, a^{2} b^{2} - b^{4}\right )} \cosh \relax (x)\right )} \sinh \relax (x) + 4 \, {\left (a b^{2} \cosh \relax (x)^{2} + 2 \, a b^{2} \cosh \relax (x) \sinh \relax (x) + a b^{2} \sinh \relax (x)^{2} - 2 \, a^{3} + a b^{2}\right )} \sqrt {a^{2} - b^{2}}}{b^{2} \cosh \relax (x)^{4} + 4 \, b^{2} \cosh \relax (x) \sinh \relax (x)^{3} + b^{2} \sinh \relax (x)^{4} - 2 \, {\left (2 \, a^{2} - b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b^{2} \cosh \relax (x)^{2} - 2 \, a^{2} + b^{2}\right )} \sinh \relax (x)^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \relax (x)^{3} - {\left (2 \, a^{2} - b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x)}\right )}{2 \, {\left (a^{3} - a b^{2}\right )}}, \frac {\sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {{\left (b^{2} \cosh \relax (x)^{2} + 2 \, b^{2} \cosh \relax (x) \sinh \relax (x) + b^{2} \sinh \relax (x)^{2} - 2 \, a^{2} + b^{2}\right )} \sqrt {-a^{2} + b^{2}}}{2 \, {\left (a^{3} - a b^{2}\right )}}\right )}{a^{3} - a b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.59, size = 50, normalized size = 1.43 \[ -\frac {\arctan \left (\frac {b^{2} e^{\left (2 \, x\right )} - 2 \, a^{2} + b^{2}}{2 \, \sqrt {-a^{2} + b^{2}} a}\right )}{\sqrt {-a^{2} + b^{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.13, size = 74, normalized size = 2.11
method | result | size |
default | \(\frac {\arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a \sqrt {\left (a +b \right ) \left (a -b \right )}}+\frac {\arctanh \left (\frac {\left (a +b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{a \sqrt {\left (a +b \right ) \left (a -b \right )}}\) | \(74\) |
risch | \(\frac {\ln \left ({\mathrm e}^{2 x}-\frac {2 a^{2} \sqrt {a^{2}-b^{2}}-b^{2} \sqrt {a^{2}-b^{2}}-2 a^{3}+2 b^{2} a}{b^{2} \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, a}-\frac {\ln \left ({\mathrm e}^{2 x}-\frac {2 a^{2} \sqrt {a^{2}-b^{2}}-b^{2} \sqrt {a^{2}-b^{2}}+2 a^{3}-2 b^{2} a}{b^{2} \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, a}\) | \(166\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.38, size = 106, normalized size = 3.03 \[ -\frac {\mathrm {atan}\left (\frac {b^2\,{\left (a^2\,b^2-a^4\right )}^{3/2}-2\,a^2\,{\left (a^2\,b^2-a^4\right )}^{3/2}+b^2\,{\mathrm {e}}^{2\,x}\,{\left (a^2\,b^2-a^4\right )}^{3/2}}{2\,a^8-4\,a^6\,b^2+2\,a^4\,b^4}\right )}{\sqrt {a^2\,b^2-a^4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 40.60, size = 892, normalized size = 25.49 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________