Optimal. Leaf size=15 \[ -\frac{\coth ^3(a+b x)}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0274867, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2607, 30} \[ -\frac{\coth ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2607
Rule 30
Rubi steps
\begin{align*} \int \coth ^2(a+b x) \text{csch}^2(a+b x) \, dx &=-\frac{i \operatorname{Subst}\left (\int x^2 \, dx,x,i \coth (a+b x)\right )}{b}\\ &=-\frac{\coth ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0041052, size = 15, normalized size = 1. \[ -\frac{\coth ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.017, size = 42, normalized size = 2.8 \begin{align*}{\frac{1}{b} \left ( -{\frac{\cosh \left ( bx+a \right ) }{2\, \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}}-{\frac{{\rm coth} \left (bx+a\right )}{2} \left ({\frac{2}{3}}-{\frac{ \left ({\rm csch} \left (bx+a\right ) \right ) ^{2}}{3}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.10479, size = 18, normalized size = 1.2 \begin{align*} -\frac{\coth \left (b x + a\right )^{3}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.73432, size = 378, normalized size = 25.2 \begin{align*} -\frac{8 \,{\left (\cosh \left (b x + a\right )^{2} + \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}\right )}}{3 \,{\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} - 4 \, b \cosh \left (b x + a\right )^{2} + 2 \,{\left (3 \, b \cosh \left (b x + a\right )^{2} - 2 \, b\right )} \sinh \left (b x + a\right )^{2} + 4 \,{\left (b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 3 \, b\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 \coth ^{2}{\left (a + b x \right )} \operatorname{csch}^{2}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.23816, size = 42, normalized size = 2.8 \begin{align*} -\frac{2 \,{\left (3 \, e^{\left (4 \, b x + 4 \, a\right )} + 1\right )}}{3 \, b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]