Optimal. Leaf size=57 \[ \frac{\left (a^2-b^2\right ) \log (\cosh (x))}{a^3}-\frac{\left (a^2-b^2\right ) \log (a+b \cosh (x))}{a^3}-\frac{b \text{sech}(x)}{a^2}+\frac{\text{sech}^2(x)}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0988762, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2721, 894} \[ \frac{\left (a^2-b^2\right ) \log (\cosh (x))}{a^3}-\frac{\left (a^2-b^2\right ) \log (a+b \cosh (x))}{a^3}-\frac{b \text{sech}(x)}{a^2}+\frac{\text{sech}^2(x)}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2721
Rule 894
Rubi steps
\begin{align*} \int \frac{\tanh ^3(x)}{a+b \cosh (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{b^2-x^2}{x^3 (a+x)} \, dx,x,b \cosh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{b^2}{a x^3}-\frac{b^2}{a^2 x^2}+\frac{-a^2+b^2}{a^3 x}+\frac{a^2-b^2}{a^3 (a+x)}\right ) \, dx,x,b \cosh (x)\right )\\ &=\frac{\left (a^2-b^2\right ) \log (\cosh (x))}{a^3}-\frac{\left (a^2-b^2\right ) \log (a+b \cosh (x))}{a^3}-\frac{b \text{sech}(x)}{a^2}+\frac{\text{sech}^2(x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0925007, size = 46, normalized size = 0.81 \[ \frac{2 \left (a^2-b^2\right ) (\log (\cosh (x))-\log (a+b \cosh (x)))+a^2 \text{sech}^2(x)-2 a b \text{sech}(x)}{2 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.03, size = 140, normalized size = 2.5 \begin{align*} -{\frac{1}{a}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b-a-b \right ) }+{\frac{{b}^{2}}{{a}^{3}}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b-a-b \right ) }+{\frac{1}{a}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) }-{\frac{{b}^{2}}{{a}^{3}}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) }-2\,{\frac{1}{a \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) }}-2\,{\frac{b}{{a}^{2} \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) }}+2\,{\frac{1}{a \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.56926, size = 130, normalized size = 2.28 \begin{align*} -\frac{2 \,{\left (b e^{\left (-x\right )} - a e^{\left (-2 \, x\right )} + b e^{\left (-3 \, x\right )}\right )}}{2 \, a^{2} e^{\left (-2 \, x\right )} + a^{2} e^{\left (-4 \, x\right )} + a^{2}} - \frac{{\left (a^{2} - b^{2}\right )} \log \left (2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} + b\right )}{a^{3}} + \frac{{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.00553, size = 1166, normalized size = 20.46 \begin{align*} -\frac{2 \, a b \cosh \left (x\right )^{3} + 2 \, a b \sinh \left (x\right )^{3} - 2 \, a^{2} \cosh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \,{\left (3 \, a b \cosh \left (x\right ) - a^{2}\right )} \sinh \left (x\right )^{2} +{\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{4} + 4 \,{\left (a^{2} - b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} +{\left (a^{2} - b^{2}\right )} \sinh \left (x\right )^{4} + 2 \,{\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \,{\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} - b^{2} + 4 \,{\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{3} +{\left (a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac{2 \,{\left (b \cosh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) -{\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{4} + 4 \,{\left (a^{2} - b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} +{\left (a^{2} - b^{2}\right )} \sinh \left (x\right )^{4} + 2 \,{\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \,{\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{2} + a^{2} - b^{2}\right )} \sinh \left (x\right )^{2} + a^{2} - b^{2} + 4 \,{\left ({\left (a^{2} - b^{2}\right )} \cosh \left (x\right )^{3} +{\left (a^{2} - b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \,{\left (3 \, a b \cosh \left (x\right )^{2} - 2 \, a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )}{a^{3} \cosh \left (x\right )^{4} + 4 \, a^{3} \cosh \left (x\right ) \sinh \left (x\right )^{3} + a^{3} \sinh \left (x\right )^{4} + 2 \, a^{3} \cosh \left (x\right )^{2} + a^{3} + 2 \,{\left (3 \, a^{3} \cosh \left (x\right )^{2} + a^{3}\right )} \sinh \left (x\right )^{2} + 4 \,{\left (a^{3} \cosh \left (x\right )^{3} + a^{3} \cosh \left (x\right )\right )} \sinh \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{3}{\left (x \right )}}{a + b \cosh{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.15605, size = 155, normalized size = 2.72 \begin{align*} \frac{{\left (a^{2} - b^{2}\right )} \log \left (e^{\left (-x\right )} + e^{x}\right )}{a^{3}} - \frac{{\left (a^{2} b - b^{3}\right )} \log \left ({\left | b{\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, a \right |}\right )}{a^{3} b} - \frac{3 \, a^{2}{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 3 \, b^{2}{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 4 \, a b{\left (e^{\left (-x\right )} + e^{x}\right )} - 4 \, a^{2}}{2 \, a^{3}{\left (e^{\left (-x\right )} + e^{x}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]