Optimal. Leaf size=112 \[ -\frac{2 (1-10 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{99 a \left (1-a^2 x^2\right )^{5/2}}-\frac{256 (1-2 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{693 a \sqrt{1-a^2 x^2}}-\frac{32 (1-6 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{693 a \left (1-a^2 x^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.117185, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6136, 6135} \[ -\frac{2 (1-10 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{99 a \left (1-a^2 x^2\right )^{5/2}}-\frac{256 (1-2 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{693 a \sqrt{1-a^2 x^2}}-\frac{32 (1-6 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{693 a \left (1-a^2 x^2\right )^{3/2}} \]
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 )^{7/2}} \, dx &=-\frac{2 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-10 a x)}{99 a \left (1-a^2 x^2\right )^{5/2}}+\frac{80}{99} \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)} (1-10 a x)}{99 a \left (1-a^2 x^2\right )^{5/2}}-\frac{32 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-6 a x)}{693 a \left (1-a^2 x^2\right )^{3/2}}+\frac{128}{231} \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)} (1-10 a x)}{99 a \left (1-a^2 x^2\right )^{5/2}}-\frac{32 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-6 a x)}{693 a \left (1-a^2 x^2\right )^{3/2}}-\frac{256 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-2 a x)}{693 a \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0303329, size = 64, normalized size = 0.57 \[ \frac{2 \left (256 a^5 x^5-128 a^4 x^4-608 a^3 x^3+272 a^2 x^2+422 a x-151\right )}{693 a (1-a x)^{11/4} (a x+1)^{9/4}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.03, size = 86, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,ax-2 \right ) \left ( ax+1 \right ) \left ( 256\,{x}^{5}{a}^{5}-128\,{x}^{4}{a}^{4}-608\,{x}^{3}{a}^{3}+272\,{a}^{2}{x}^{2}+422\,ax-151 \right ) }{693\,a}\sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{7}{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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.56681, size = 234, normalized size = 2.09 \begin{align*} -\frac{2 \,{\left (256 \, a^{5} x^{5} - 128 \, a^{4} x^{4} - 608 \, a^{3} x^{3} + 272 \, a^{2} x^{2} + 422 \, a x - 151\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}}{693 \,{\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, 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{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]