Optimal. Leaf size=79 \[ \frac{\left (a^2-b^2\right ) \tanh (x)}{a^3}-\frac{b \left (a^2-b^2\right ) \log (\tanh (x))}{a^4}-\frac{b \left (a^2-b^2\right ) \log (a+b \coth (x))}{a^4}+\frac{b \tanh ^2(x)}{2 a^2}-\frac{\tanh ^3(x)}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.099762, antiderivative size = 79, 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 = {3516, 894} \[ \frac{\left (a^2-b^2\right ) \tanh (x)}{a^3}-\frac{b \left (a^2-b^2\right ) \log (\tanh (x))}{a^4}-\frac{b \left (a^2-b^2\right ) \log (a+b \coth (x))}{a^4}+\frac{b \tanh ^2(x)}{2 a^2}-\frac{\tanh ^3(x)}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3516
Rule 894
Rubi steps
\begin{align*} \int \frac{\text{sech}^4(x)}{a+b \coth (x)} \, dx &=-\left (b \operatorname{Subst}\left (\int \frac{-b^2+x^2}{x^4 (a+x)} \, dx,x,b \coth (x)\right )\right )\\ &=-\left (b \operatorname{Subst}\left (\int \left (-\frac{b^2}{a x^4}+\frac{b^2}{a^2 x^3}+\frac{a^2-b^2}{a^3 x^2}+\frac{-a^2+b^2}{a^4 x}+\frac{a^2-b^2}{a^4 (a+x)}\right ) \, dx,x,b \coth (x)\right )\right )\\ &=-\frac{b \left (a^2-b^2\right ) \log (a+b \coth (x))}{a^4}-\frac{b \left (a^2-b^2\right ) \log (\tanh (x))}{a^4}+\frac{\left (a^2-b^2\right ) \tanh (x)}{a^3}+\frac{b \tanh ^2(x)}{2 a^2}-\frac{\tanh ^3(x)}{3 a}\\ \end{align*}
Mathematica [A] time = 0.258318, size = 68, normalized size = 0.86 \[ \frac{\left (4 a^3-6 a b^2\right ) \tanh (x)-6 b \left (b^2-a^2\right ) (\log (\cosh (x))-\log (a \sinh (x)+b \cosh (x)))+a^2 \text{sech}^2(x) (2 a \tanh (x)-3 b)}{6 a^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.053, size = 257, normalized size = 3.3 \begin{align*} 2\,{\frac{ \left ( \tanh \left ( x/2 \right ) \right ) ^{5}}{a \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}-2\,{\frac{ \left ( \tanh \left ( x/2 \right ) \right ) ^{5}{b}^{2}}{{a}^{3} \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+2\,{\frac{b \left ( \tanh \left ( x/2 \right ) \right ) ^{4}}{{a}^{2} \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+{\frac{4}{3\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3} \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-3}}-4\,{\frac{ \left ( \tanh \left ( x/2 \right ) \right ) ^{3}{b}^{2}}{{a}^{3} \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+2\,{\frac{ \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b}{{a}^{2} \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+2\,{\frac{\tanh \left ( x/2 \right ) }{a \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}-2\,{\frac{\tanh \left ( x/2 \right ){b}^{2}}{{a}^{3} \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+{\frac{b}{{a}^{2}}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) }-{\frac{{b}^{3}}{{a}^{4}}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) }-{\frac{b}{{a}^{2}}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b+2\,a\tanh \left ( x/2 \right ) +b \right ) }+{\frac{{b}^{3}}{{a}^{4}}\ln \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b+2\,a\tanh \left ( x/2 \right ) +b \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.7053, size = 180, normalized size = 2.28 \begin{align*} \frac{2 \,{\left (2 \, a^{2} - 3 \, b^{2} + 3 \,{\left (2 \, a^{2} - a b - 2 \, b^{2}\right )} e^{\left (-2 \, x\right )} - 3 \,{\left (a b + b^{2}\right )} e^{\left (-4 \, x\right )}\right )}}{3 \,{\left (3 \, a^{3} e^{\left (-2 \, x\right )} + 3 \, a^{3} e^{\left (-4 \, x\right )} + a^{3} e^{\left (-6 \, x\right )} + a^{3}\right )}} - \frac{{\left (a^{2} b - b^{3}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} + a + b\right )}{a^{4}} + \frac{{\left (a^{2} b - b^{3}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.76842, size = 2187, normalized size = 27.68 \begin{align*} \text{result too large to display} \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{\operatorname{sech}^{4}{\left (x \right )}}{a + b \coth{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.14789, size = 271, normalized size = 3.43 \begin{align*} -\frac{{\left (a^{3} b + a^{2} b^{2} - a b^{3} - b^{4}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - a + b \right |}\right )}{a^{5} + a^{4} b} + \frac{{\left (a^{2} b - b^{3}\right )} \log \left (e^{\left (2 \, x\right )} + 1\right )}{a^{4}} - \frac{11 \, a^{2} b e^{\left (6 \, x\right )} - 11 \, b^{3} e^{\left (6 \, x\right )} + 45 \, a^{2} b e^{\left (4 \, x\right )} - 12 \, a b^{2} e^{\left (4 \, x\right )} - 33 \, b^{3} e^{\left (4 \, x\right )} + 24 \, a^{3} e^{\left (2 \, x\right )} + 45 \, a^{2} b e^{\left (2 \, x\right )} - 24 \, a b^{2} e^{\left (2 \, x\right )} - 33 \, b^{3} e^{\left (2 \, x\right )} + 8 \, a^{3} + 11 \, a^{2} b - 12 \, a b^{2} - 11 \, b^{3}}{6 \, a^{4}{\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]