Optimal. Leaf size=81 \[ \frac{x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{2 b^2}+\frac{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}{b^3}+\frac{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4}+\frac{x^3}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0588508, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2159, 2158, 2157, 29} \[ \frac{x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{2 b^2}+\frac{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}{b^3}+\frac{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4}+\frac{x^3}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2159
Rule 2158
Rule 2157
Rule 29
Rubi steps
\begin{align*} \int \frac{x^3}{\coth ^{-1}(\tanh (a+b x))} \, dx &=\frac{x^3}{3 b}-\frac{\left (-b x+\coth ^{-1}(\tanh (a+b x))\right ) \int \frac{x^2}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b}\\ &=\frac{x^3}{3 b}+\frac{x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{2 b^2}+\frac{\left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2 \int \frac{x}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b^2}\\ &=\frac{x^3}{3 b}+\frac{x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{2 b^2}+\frac{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}{b^3}-\frac{\left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^3 \int \frac{1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b^3}\\ &=\frac{x^3}{3 b}+\frac{x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{2 b^2}+\frac{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}{b^3}-\frac{\left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^3 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b^4}\\ &=\frac{x^3}{3 b}+\frac{x^2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{2 b^2}+\frac{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2}{b^3}+\frac{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^3 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0481871, size = 79, normalized size = 0.98 \[ -\frac{x^2 \left (\coth ^{-1}(\tanh (a+b x))-b x\right )}{2 b^2}+\frac{x \left (\coth ^{-1}(\tanh (a+b x))-b x\right )^2}{b^3}-\frac{\left (\coth ^{-1}(\tanh (a+b x))-b x\right )^3 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^4}+\frac{x^3}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 4.901, size = 130774, normalized size = 1614.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 1.80687, size = 116, normalized size = 1.43 \begin{align*} \frac{4 \, b^{2} x^{3} +{\left (3 i \, \pi b - 6 \, a b\right )} x^{2} -{\left (3 \, \pi ^{2} + 12 i \, \pi a - 12 \, a^{2}\right )} x}{12 \, b^{3}} - \frac{{\left (i \, \pi ^{3} - 6 \, \pi ^{2} a - 12 i \, \pi a^{2} + 8 \, a^{3}\right )} \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{8 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.5517, size = 293, normalized size = 3.62 \begin{align*} \frac{8 \, b^{3} x^{3} - 12 \, a b^{2} x^{2} - 6 \,{\left (\pi ^{2} b - 4 \, a^{2} b\right )} x - 6 \,{\left (\pi ^{3} - 12 \, \pi a^{2}\right )} \arctan \left (-\frac{2 \, b x + 2 \, a - \sqrt{4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}}}{\pi }\right ) + 3 \,{\left (3 \, \pi ^{2} a - 4 \, a^{3}\right )} \log \left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right )}{24 \, b^{4}} \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{x^{3}}{\operatorname{acoth}{\left (\tanh{\left (a + b x \right )} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]