Optimal. Leaf size=75 \[ -\frac{2 (1-6 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{35 a \left (1-a^2 x^2\right )^{3/2}}-\frac{16 (1-2 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{35 a \sqrt{1-a^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0793326, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6136, 6135} \[ -\frac{2 (1-6 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{35 a \left (1-a^2 x^2\right )^{3/2}}-\frac{16 (1-2 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{35 a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6136
Rule 6135
Rubi steps
\begin{align*} \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=-\frac{2 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-6 a x)}{35 a \left (1-a^2 x^2\right )^{3/2}}+\frac{24}{35} \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{2 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-6 a x)}{35 a \left (1-a^2 x^2\right )^{3/2}}-\frac{16 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-2 a x)}{35 a \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0198521, size = 48, normalized size = 0.64 \[ -\frac{2 \left (16 a^3 x^3-8 a^2 x^2-22 a x+9\right )}{35 a (1-a x)^{7/4} (a x+1)^{5/4}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.033, size = 70, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2\,ax-2 \right ) \left ( ax+1 \right ) \left ( 16\,{x}^{3}{a}^{3}-8\,{a}^{2}{x}^{2}-22\,ax+9 \right ) }{35\,a}\sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.59857, size = 170, normalized size = 2.27 \begin{align*} -\frac{2 \,{\left (16 \, a^{3} x^{3} - 8 \, a^{2} x^{2} - 22 \, a x + 9\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{35 \,{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]