Optimal. Leaf size=38 \[ \frac{x}{a}-\frac{\coth ^3(x) (1-\text{sech}(x))}{3 a}-\frac{\coth (x) (3-2 \text{sech}(x))}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0893435, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3888, 3882, 8} \[ \frac{x}{a}-\frac{\coth ^3(x) (1-\text{sech}(x))}{3 a}-\frac{\coth (x) (3-2 \text{sech}(x))}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3888
Rule 3882
Rule 8
Rubi steps
\begin{align*} \int \frac{\coth ^2(x)}{a+a \text{sech}(x)} \, dx &=-\frac{\int \coth ^4(x) (-a+a \text{sech}(x)) \, dx}{a^2}\\ &=-\frac{\coth ^3(x) (1-\text{sech}(x))}{3 a}+\frac{\int \coth ^2(x) (3 a-2 a \text{sech}(x)) \, dx}{3 a^2}\\ &=-\frac{\coth (x) (3-2 \text{sech}(x))}{3 a}-\frac{\coth ^3(x) (1-\text{sech}(x))}{3 a}-\frac{\int -3 a \, dx}{3 a^2}\\ &=\frac{x}{a}-\frac{\coth (x) (3-2 \text{sech}(x))}{3 a}-\frac{\coth ^3(x) (1-\text{sech}(x))}{3 a}\\ \end{align*}
Mathematica [A] time = 0.0718144, size = 33, normalized size = 0.87 \[ \frac{6 x-4 \tanh (x)-4 \coth (x)-2 \text{csch}(x)+6 x \text{sech}(x)}{6 a \text{sech}(x)+6 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.03, size = 56, normalized size = 1.5 \begin{align*} -{\frac{1}{12\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}-{\frac{1}{a}\tanh \left ({\frac{x}{2}} \right ) }+{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{4\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.09619, size = 63, normalized size = 1.66 \begin{align*} \frac{x}{a} - \frac{2 \,{\left (5 \, e^{\left (-x\right )} - 3 \, e^{\left (-3 \, x\right )} + 4\right )}}{3 \,{\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.23765, size = 149, normalized size = 3.92 \begin{align*} -\frac{2 \, \cosh \left (x\right )^{2} -{\left ({\left (3 \, x + 4\right )} \cosh \left (x\right ) + 3 \, x + 4\right )} \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} + \cosh \left (x\right )}{3 \,{\left (a \cosh \left (x\right ) + a\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*} \frac{\int \frac{\coth ^{2}{\left (x \right )}}{\operatorname{sech}{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17326, size = 54, normalized size = 1.42 \begin{align*} \frac{x}{a} - \frac{1}{2 \, a{\left (e^{x} - 1\right )}} + \frac{15 \, e^{\left (2 \, x\right )} + 24 \, e^{x} + 13}{6 \, a{\left (e^{x} + 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]