Optimal. Leaf size=76 \[ -\frac{4 x^2 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{b^2}-\frac{32 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{35 b^4}+\frac{16 x \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b^3}+\frac{2 x^3 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0481367, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2168, 2157, 30} \[ -\frac{4 x^2 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{b^2}-\frac{32 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{35 b^4}+\frac{16 x \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b^3}+\frac{2 x^3 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2168
Rule 2157
Rule 30
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx &=\frac{2 x^3 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{b}-\frac{6 \int x^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b}\\ &=\frac{2 x^3 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{b}-\frac{4 x^2 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{b^2}+\frac{8 \int x \tanh ^{-1}(\tanh (a+b x))^{3/2} \, dx}{b^2}\\ &=\frac{2 x^3 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{b}-\frac{4 x^2 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{b^2}+\frac{16 x \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b^3}-\frac{16 \int \tanh ^{-1}(\tanh (a+b x))^{5/2} \, dx}{5 b^3}\\ &=\frac{2 x^3 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{b}-\frac{4 x^2 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{b^2}+\frac{16 x \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b^3}-\frac{16 \operatorname{Subst}\left (\int x^{5/2} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{5 b^4}\\ &=\frac{2 x^3 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{b}-\frac{4 x^2 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{b^2}+\frac{16 x \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 b^3}-\frac{32 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{35 b^4}\\ \end{align*}
Mathematica [A] time = 0.0371401, size = 66, normalized size = 0.87 \[ \frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (-70 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+56 b x \tanh ^{-1}(\tanh (a+b x))^2-16 \tanh ^{-1}(\tanh (a+b x))^3+35 b^3 x^3\right )}{35 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.063, size = 123, normalized size = 1.6 \begin{align*} 2\,{\frac{1/7\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{7/2}+1/5\, \left ( -3\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) +3\,bx \right ) \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{5/2}+1/3\, \left ( \left ( bx-{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) \left ( -2\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) +2\,bx \right ) + \left ( bx-{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2} \right ) \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3/2}+ \left ( bx-{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}{{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.80731, size = 72, normalized size = 0.95 \begin{align*} \frac{2 \,{\left (5 \, b^{4} x^{4} - a b^{3} x^{3} + 2 \, a^{2} b^{2} x^{2} - 8 \, a^{3} b x - 16 \, a^{4}\right )}}{35 \, \sqrt{b x + a} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.0453, size = 96, normalized size = 1.26 \begin{align*} \frac{2 \,{\left (5 \, b^{3} x^{3} - 6 \, a b^{2} x^{2} + 8 \, a^{2} b x - 16 \, a^{3}\right )} \sqrt{b x + a}}{35 \, 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}}{\sqrt{\operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16973, size = 66, normalized size = 0.87 \begin{align*} \frac{2 \,{\left (5 \,{\left (b x + a\right )}^{\frac{7}{2}} - 21 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} - 35 \, \sqrt{b x + a} a^{3}\right )}}{35 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]