Optimal. Leaf size=23 \[ \frac{1}{3} x^3 \tanh ^{-1}(\tanh (a+b x))-\frac{b x^4}{12} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0081557, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2168, 30} \[ \frac{1}{3} x^3 \tanh ^{-1}(\tanh (a+b x))-\frac{b x^4}{12} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2168
Rule 30
Rubi steps
\begin{align*} \int x^2 \tanh ^{-1}(\tanh (a+b x)) \, dx &=\frac{1}{3} x^3 \tanh ^{-1}(\tanh (a+b x))-\frac{1}{3} b \int x^3 \, dx\\ &=-\frac{b x^4}{12}+\frac{1}{3} x^3 \tanh ^{-1}(\tanh (a+b x))\\ \end{align*}
Mathematica [A] time = 0.0163977, size = 20, normalized size = 0.87 \[ -\frac{1}{12} x^3 \left (b x-4 \tanh ^{-1}(\tanh (a+b x))\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.032, size = 20, normalized size = 0.9 \begin{align*} -{\frac{b{x}^{4}}{12}}+{\frac{{x}^{3}{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.15154, size = 26, normalized size = 1.13 \begin{align*} -\frac{1}{12} \, b x^{4} + \frac{1}{3} \, x^{3} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.598, size = 31, normalized size = 1.35 \begin{align*} \frac{1}{4} \, b x^{4} + \frac{1}{3} \, a x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.812558, size = 19, normalized size = 0.83 \begin{align*} - \frac{b x^{4}}{12} + \frac{x^{3} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1336, size = 18, normalized size = 0.78 \begin{align*} \frac{1}{4} \, b x^{4} + \frac{1}{3} \, a x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]