Optimal. Leaf size=95 \[ -\frac{(1-a x)^{5/4}}{x \sqrt [4]{a x+1}}-\frac{10 a \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-5 a \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+5 a \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.039102, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6126, 94, 93, 298, 203, 206} \[ -\frac{(1-a x)^{5/4}}{x \sqrt [4]{a x+1}}-\frac{10 a \sqrt [4]{1-a x}}{\sqrt [4]{a x+1}}-5 a \tan ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right )+5 a \tanh ^{-1}\left (\frac{\sqrt [4]{a x+1}}{\sqrt [4]{1-a x}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6126
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^2} \, dx &=\int \frac{(1-a x)^{5/4}}{x^2 (1+a x)^{5/4}} \, dx\\ &=-\frac{(1-a x)^{5/4}}{x \sqrt [4]{1+a x}}-\frac{1}{2} (5 a) \int \frac{\sqrt [4]{1-a x}}{x (1+a x)^{5/4}} \, dx\\ &=-\frac{10 a \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}-\frac{(1-a x)^{5/4}}{x \sqrt [4]{1+a x}}-\frac{1}{2} (5 a) \int \frac{1}{x (1-a x)^{3/4} \sqrt [4]{1+a x}} \, dx\\ &=-\frac{10 a \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}-\frac{(1-a x)^{5/4}}{x \sqrt [4]{1+a x}}-(10 a) \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{10 a \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}-\frac{(1-a x)^{5/4}}{x \sqrt [4]{1+a x}}+(5 a) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )-(5 a) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ &=-\frac{10 a \sqrt [4]{1-a x}}{\sqrt [4]{1+a x}}-\frac{(1-a x)^{5/4}}{x \sqrt [4]{1+a x}}-5 a \tan ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )+5 a \tanh ^{-1}\left (\frac{\sqrt [4]{1+a x}}{\sqrt [4]{1-a x}}\right )\\ \end{align*}
Mathematica [C] time = 0.0175106, size = 55, normalized size = 0.58 \[ \frac{\sqrt [4]{1-a x} \left (10 a x \text{Hypergeometric2F1}\left (\frac{1}{4},1,\frac{5}{4},\frac{1-a x}{a x+1}\right )-9 a x-1\right )}{x \sqrt [4]{a x+1}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.117, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \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^{2} \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 [B] time = 1.73732, size = 383, normalized size = 4.03 \begin{align*} -\frac{2 \, \sqrt{-a^{2} x^{2} + 1}{\left (9 \, a x + 1\right )} \sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 10 \,{\left (a^{2} x^{2} + a x\right )} \arctan \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}}\right ) - 5 \,{\left (a^{2} x^{2} + a x\right )} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} + 1\right ) + 5 \,{\left (a^{2} x^{2} + a x\right )} \log \left (\sqrt{-\frac{\sqrt{-a^{2} x^{2} + 1}}{a x - 1}} - 1\right )}{2 \,{\left (a x^{2} + x\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^{2} \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]