Optimal. Leaf size=42 \[ -\frac{1}{10} b x^5 \tanh ^{-1}(\tanh (a+b x))+\frac{1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^2+\frac{b^2 x^6}{60} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0234797, antiderivative size = 42, 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 = {2168, 30} \[ -\frac{1}{10} b x^5 \tanh ^{-1}(\tanh (a+b x))+\frac{1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^2+\frac{b^2 x^6}{60} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2168
Rule 30
Rubi steps
\begin{align*} \int x^3 \tanh ^{-1}(\tanh (a+b x))^2 \, dx &=\frac{1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^2-\frac{1}{2} b \int x^4 \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac{1}{10} b x^5 \tanh ^{-1}(\tanh (a+b x))+\frac{1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^2+\frac{1}{10} b^2 \int x^5 \, dx\\ &=\frac{b^2 x^6}{60}-\frac{1}{10} b x^5 \tanh ^{-1}(\tanh (a+b x))+\frac{1}{4} x^4 \tanh ^{-1}(\tanh (a+b x))^2\\ \end{align*}
Mathematica [A] time = 0.0295993, size = 37, normalized size = 0.88 \[ \frac{1}{60} x^4 \left (-6 b x \tanh ^{-1}(\tanh (a+b x))+15 \tanh ^{-1}(\tanh (a+b x))^2+b^2 x^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.036, size = 38, normalized size = 0.9 \begin{align*}{\frac{{x}^{4} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{4}}-{\frac{b}{2} \left ({\frac{{x}^{5}{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{5}}-{\frac{{x}^{6}b}{30}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.34876, size = 49, normalized size = 1.17 \begin{align*} \frac{1}{60} \, b^{2} x^{6} - \frac{1}{10} \, b x^{5} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right ) + \frac{1}{4} \, x^{4} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.95857, size = 55, normalized size = 1.31 \begin{align*} \frac{1}{6} \, b^{2} x^{6} + \frac{2}{5} \, a b x^{5} + \frac{1}{4} \, a^{2} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.62917, size = 37, normalized size = 0.88 \begin{align*} \frac{b^{2} x^{6}}{60} - \frac{b x^{5} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{10} + \frac{x^{4} \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15066, size = 32, normalized size = 0.76 \begin{align*} \frac{1}{6} \, b^{2} x^{6} + \frac{2}{5} \, a b x^{5} + \frac{1}{4} \, a^{2} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]