Optimal. Leaf size=136 \[ \frac{25 a^2 \sqrt [4]{1-a x}}{2 \sqrt [4]{a x+1}}+\frac{25}{4} a^2 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{25}{4} a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{(1-a x)^{9/4}}{2 x^2 \sqrt [4]{a x+1}}+\frac{5 a (1-a x)^{5/4}}{4 x \sqrt [4]{a x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0536025, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6126, 96, 94, 93, 298, 203, 206} \[ \frac{25 a^2 \sqrt [4]{1-a x}}{2 \sqrt [4]{a x+1}}+\frac{25}{4} a^2 \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{25}{4} a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )-\frac{(1-a x)^{9/4}}{2 x^2 \sqrt [4]{a x+1}}+\frac{5 a (1-a x)^{5/4}}{4 x \sqrt [4]{a x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6126
Rule 96
Rule 94
Rule 93
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-\frac{5}{2} \tanh ^{-1}(a x)}}{x^3} \, dx &=\int \frac{(1-a x)^{5/4}}{x^3 (1+a x)^{5/4}} \, dx\\ &=-\frac{(1-a x)^{9/4}}{2 x^2 \sqrt [4]{1+a x}}-\frac{1}{4} (5 a) \int \frac{(1-a x)^{5/4}}{x^2 (1+a x)^{5/4}} \, dx\\ &=\frac{5 a (1-a x)^{5/4}}{4 x \sqrt [4]{1+a x}}-\frac{(1-a x)^{9/4}}{2 x^2 \sqrt [4]{1+a x}}+\frac{1}{8} \left (25 a^2\right ) \int \frac{\sqrt [4]{1-a x}}{x (1+a x)^{5/4}} \, dx\\ &=\frac{25 a^2 \sqrt [4]{1-a x}}{2 \sqrt [4]{1+a x}}+\frac{5 a (1-a x)^{5/4}}{4 x \sqrt [4]{1+a x}}-\frac{(1-a x)^{9/4}}{2 x^2 \sqrt [4]{1+a x}}+\frac{1}{8} \left (25 a^2\right ) \int \frac{1}{x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=\frac{25 a^2 \sqrt [4]{1-a x}}{2 \sqrt [4]{1+a x}}+\frac{5 a (1-a x)^{5/4}}{4 x \sqrt [4]{1+a x}}-\frac{(1-a x)^{9/4}}{2 x^2 \sqrt [4]{1+a x}}+\frac{1}{2} \left (25 a^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac{25 a^2 \sqrt [4]{1-a x}}{2 \sqrt [4]{1+a x}}+\frac{5 a (1-a x)^{5/4}}{4 x \sqrt [4]{1+a x}}-\frac{(1-a x)^{9/4}}{2 x^2 \sqrt [4]{1+a x}}-\frac{1}{4} \left (25 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+\frac{1}{4} \left (25 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=\frac{25 a^2 \sqrt [4]{1-a x}}{2 \sqrt [4]{1+a x}}+\frac{5 a (1-a x)^{5/4}}{4 x \sqrt [4]{1+a x}}-\frac{(1-a x)^{9/4}}{2 x^2 \sqrt [4]{1+a x}}+\frac{25}{4} a^2 \tan ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-\frac{25}{4} a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0226725, size = 70, normalized size = 0.51 \[ \frac{\sqrt [4]{1-a x} \left (-50 a^2 x^2 \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},\frac{1-a x}{a x+1}\right )+43 a^2 x^2+9 a x-2\right )}{4 x^2 \sqrt [4]{a x+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.117, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ({(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) ^{-{\frac{5}{2}}}}\, dx \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{1}{x^{3} \left (\frac{a x + 1}{\sqrt{-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 = 1.76074, size = 421, normalized size = 3.1 \begin{align*} \frac{2 \,{\left (43 \, a^{2} x^{2} + 9 \, a x - 2\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 50 \,{\left (a^{3} x^{3} + a^{2} x^{2}\right )} \arctan \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 25 \,{\left (a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + 25 \,{\left (a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right )}{8 \,{\left (a x^{3} + x^{2}\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{1}{x^{3} \left (\frac{a x + 1}{\sqrt{-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]