Optimal. Leaf size=35 \[ \frac{2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0140935, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2167} \[ \frac{2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2167
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x^{7/2}} \, dx &=\frac{2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{5 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}
Mathematica [A] time = 0.0363926, size = 34, normalized size = 0.97 \[ \frac{2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{x^{5/2} \left (5 b x-5 \tanh ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.146, size = 29, normalized size = 0.8 \begin{align*} -{\frac{2}{5\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -5\,bx} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{{\frac{5}{2}}}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.48283, size = 20, normalized size = 0.57 \begin{align*} -\frac{2 \,{\left (b x + a\right )}^{\frac{5}{2}}}{5 \, a x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.30816, size = 78, normalized size = 2.23 \begin{align*} -\frac{2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x + a}}{5 \, a x^{\frac{5}{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 = 1.33761, size = 45, normalized size = 1.29 \begin{align*} -\frac{2 \,{\left (b x + a\right )}^{\frac{5}{2}} b^{6}}{5 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{5}{2}} a{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]