Optimal. Leaf size=67 \[ -\frac {2 a \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{b^2 \sqrt {a-b} \sqrt {a+b}}-\frac {\sinh (x)}{b (a+b \cosh (x))}+\frac {x}{b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2693, 2735, 2659, 208} \[ -\frac {2 a \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{b^2 \sqrt {a-b} \sqrt {a+b}}-\frac {\sinh (x)}{b (a+b \cosh (x))}+\frac {x}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 2659
Rule 2693
Rule 2735
Rubi steps
\begin {align*} \int \frac {\sinh ^2(x)}{(a+b \cosh (x))^2} \, dx &=-\frac {\sinh (x)}{b (a+b \cosh (x))}+\frac {\int \frac {\cosh (x)}{a+b \cosh (x)} \, dx}{b}\\ &=\frac {x}{b^2}-\frac {\sinh (x)}{b (a+b \cosh (x))}-\frac {a \int \frac {1}{a+b \cosh (x)} \, dx}{b^2}\\ &=\frac {x}{b^2}-\frac {\sinh (x)}{b (a+b \cosh (x))}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b^2}\\ &=\frac {x}{b^2}-\frac {2 a \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^2 \sqrt {a+b}}-\frac {\sinh (x)}{b (a+b \cosh (x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 61, normalized size = 0.91 \[ \frac {\frac {2 a \tan ^{-1}\left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}}-\frac {b \sinh (x)}{a+b \cosh (x)}+x}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.59, size = 700, normalized size = 10.45 \[ \left [\frac {{\left (a^{2} b - b^{3}\right )} x \cosh \relax (x)^{2} + {\left (a^{2} b - b^{3}\right )} x \sinh \relax (x)^{2} + 2 \, a^{2} b - 2 \, b^{3} + {\left (a b \cosh \relax (x)^{2} + a b \sinh \relax (x)^{2} + 2 \, a^{2} \cosh \relax (x) + a b + 2 \, {\left (a b \cosh \relax (x) + a^{2}\right )} \sinh \relax (x)\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {b^{2} \cosh \relax (x)^{2} + b^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x) + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) + b}\right ) + {\left (a^{2} b - b^{3}\right )} x + 2 \, {\left (a^{3} - a b^{2} + {\left (a^{3} - a b^{2}\right )} x\right )} \cosh \relax (x) + 2 \, {\left (a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} x \cosh \relax (x) + {\left (a^{3} - a b^{2}\right )} x\right )} \sinh \relax (x)}{a^{2} b^{3} - b^{5} + {\left (a^{2} b^{3} - b^{5}\right )} \cosh \relax (x)^{2} + {\left (a^{2} b^{3} - b^{5}\right )} \sinh \relax (x)^{2} + 2 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cosh \relax (x) + 2 \, {\left (a^{3} b^{2} - a b^{4} + {\left (a^{2} b^{3} - b^{5}\right )} \cosh \relax (x)\right )} \sinh \relax (x)}, \frac {{\left (a^{2} b - b^{3}\right )} x \cosh \relax (x)^{2} + {\left (a^{2} b - b^{3}\right )} x \sinh \relax (x)^{2} + 2 \, a^{2} b - 2 \, b^{3} + 2 \, {\left (a b \cosh \relax (x)^{2} + a b \sinh \relax (x)^{2} + 2 \, a^{2} \cosh \relax (x) + a b + 2 \, {\left (a b \cosh \relax (x) + a^{2}\right )} \sinh \relax (x)\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \relax (x) + b \sinh \relax (x) + a\right )}}{a^{2} - b^{2}}\right ) + {\left (a^{2} b - b^{3}\right )} x + 2 \, {\left (a^{3} - a b^{2} + {\left (a^{3} - a b^{2}\right )} x\right )} \cosh \relax (x) + 2 \, {\left (a^{3} - a b^{2} + {\left (a^{2} b - b^{3}\right )} x \cosh \relax (x) + {\left (a^{3} - a b^{2}\right )} x\right )} \sinh \relax (x)}{a^{2} b^{3} - b^{5} + {\left (a^{2} b^{3} - b^{5}\right )} \cosh \relax (x)^{2} + {\left (a^{2} b^{3} - b^{5}\right )} \sinh \relax (x)^{2} + 2 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cosh \relax (x) + 2 \, {\left (a^{3} b^{2} - a b^{4} + {\left (a^{2} b^{3} - b^{5}\right )} \cosh \relax (x)\right )} \sinh \relax (x)}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 68, normalized size = 1.01 \[ -\frac {2 \, a \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}} b^{2}} + \frac {x}{b^{2}} + \frac {2 \, {\left (a e^{x} + b\right )}}{{\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} + b\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 99, normalized size = 1.48 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b^{2}}+\frac {2 \tanh \left (\frac {x}{2}\right )}{b \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b -a -b \right )}-\frac {2 a \arctanh \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{2} \sqrt {\left (a +b \right ) \left (a -b \right )}} \]
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 = 1.12, size = 139, normalized size = 2.07 \[ \frac {x}{b^2}+\frac {\frac {2}{b}+\frac {2\,a\,{\mathrm {e}}^x}{b^2}}{b+2\,a\,{\mathrm {e}}^x+b\,{\mathrm {e}}^{2\,x}}+\frac {a\,\ln \left (\frac {2\,a\,{\mathrm {e}}^x}{b^3}-\frac {2\,a\,\left (b+a\,{\mathrm {e}}^x\right )}{b^3\,\sqrt {a+b}\,\sqrt {a-b}}\right )}{b^2\,\sqrt {a+b}\,\sqrt {a-b}}-\frac {a\,\ln \left (\frac {2\,a\,{\mathrm {e}}^x}{b^3}+\frac {2\,a\,\left (b+a\,{\mathrm {e}}^x\right )}{b^3\,\sqrt {a+b}\,\sqrt {a-b}}\right )}{b^2\,\sqrt {a+b}\,\sqrt {a-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________