Optimal. Leaf size=148 \[ \frac{16 b^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{35 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}+\frac{32 b^3 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{35 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^4}+\frac{12 b \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{35 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{7 x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0830278, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2171, 2167} \[ \frac{16 b^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{35 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}+\frac{32 b^3 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{35 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^4}+\frac{12 b \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{35 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{7 x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2171
Rule 2167
Rubi steps
\begin{align*} \int \frac{1}{x^{9/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx &=\frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{7 x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{(6 b) \int \frac{1}{x^{7/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx}{7 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac{12 b \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{35 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{7 x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{\left (24 b^2\right ) \int \frac{1}{x^{5/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx}{35 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}\\ &=\frac{16 b^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{35 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}+\frac{12 b \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{35 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{7 x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{\left (16 b^3\right ) \int \frac{1}{x^{3/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx}{35 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}\\ &=\frac{32 b^3 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{35 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^4}+\frac{16 b^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{35 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}+\frac{12 b \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{35 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{7 x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}
Mathematica [A] time = 0.0522516, size = 82, normalized size = 0.55 \[ \frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (-35 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+21 b x \tanh ^{-1}(\tanh (a+b x))^2-5 \tanh ^{-1}(\tanh (a+b x))^3+35 b^3 x^3\right )}{35 x^{7/2} \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.169, size = 151, normalized size = 1. \begin{align*} -{\frac{2}{7\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -7\,bx}\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{x}^{-{\frac{7}{2}}}}-{\frac{12\,b}{7\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -7\,bx} \left ( -{\frac{1}{5\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -5\,bx}\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{x}^{-{\frac{5}{2}}}}-{\frac{4\,b}{5\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -5\,bx} \left ( -{\frac{1}{3\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -3\,bx}\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{x}^{-{\frac{3}{2}}}}+{\frac{2\,b}{3\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) ^{2}}\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{\frac{1}{\sqrt{x}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.50141, size = 74, normalized size = 0.5 \begin{align*} \frac{2 \,{\left (16 \, b^{4} x^{4} + 8 \, a b^{3} x^{3} - 2 \, a^{2} b^{2} x^{2} + a^{3} b x - 5 \, a^{4}\right )}}{35 \, \sqrt{b x + a} a^{4} x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.09876, size = 109, normalized size = 0.74 \begin{align*} \frac{2 \,{\left (16 \, b^{3} x^{3} - 8 \, a b^{2} x^{2} + 6 \, a^{2} b x - 5 \, a^{3}\right )} \sqrt{b x + a}}{35 \, a^{4} x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17013, size = 139, normalized size = 0.94 \begin{align*} \frac{64 \,{\left (35 \,{\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{6} - 21 \, a{\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{4} + 7 \, a^{2}{\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{2} - a^{3}\right )} b^{\frac{7}{2}}}{35 \,{\left ({\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{2} - a\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]