Optimal. Leaf size=96 \[ \frac{1}{6} a^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (28 a-\frac{3}{x}\right )+\frac{1}{3} a \sqrt{1-\frac{1}{a^2 x^2}} \left (3 a-\frac{1}{x}\right )^2+\frac{\left (a-\frac{1}{x}\right )^3}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{11}{2} a^3 \csc ^{-1}(a x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.766351, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6169, 1633, 1593, 12, 852, 1635, 1654, 780, 216} \[ \frac{1}{6} a^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (28 a-\frac{3}{x}\right )+\frac{1}{3} a \sqrt{1-\frac{1}{a^2 x^2}} \left (3 a-\frac{1}{x}\right )^2+\frac{\left (a-\frac{1}{x}\right )^3}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{11}{2} a^3 \csc ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6169
Rule 1633
Rule 1593
Rule 12
Rule 852
Rule 1635
Rule 1654
Rule 780
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{x^4} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2 \left (1-\frac{x}{a}\right )^2}{\left (1+\frac{x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}} \left (a x^2-x^3\right )}{\left (1+\frac{x}{a}\right )^2} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(a-x) x^2 \sqrt{1-\frac{x^2}{a^2}}}{\left (1+\frac{x}{a}\right )^2} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{a^2 x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}}{\left (1+\frac{x}{a}\right )^3} \, dx,x,\frac{1}{x}\right )}{a^2}\\ &=-\operatorname{Subst}\left (\int \frac{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}}{\left (1+\frac{x}{a}\right )^3} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \frac{x^2 \left (1-\frac{x}{a}\right )^3}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\left (a-\frac{1}{x}\right )^3}{\sqrt{1-\frac{1}{a^2 x^2}}}+\operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2 \left (3 a^2-a x\right )}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\left (a-\frac{1}{x}\right )^3}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{1}{3} a \sqrt{1-\frac{1}{a^2 x^2}} \left (3 a-\frac{1}{x}\right )^2-\frac{1}{3} \operatorname{Subst}\left (\int \frac{\left (-5+\frac{3 x}{a}\right ) \left (3 a^2-a x\right )}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{6} a^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (28 a-\frac{3}{x}\right )+\frac{\left (a-\frac{1}{x}\right )^3}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{1}{3} a \sqrt{1-\frac{1}{a^2 x^2}} \left (3 a-\frac{1}{x}\right )^2+\frac{1}{2} \left (11 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{6} a^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (28 a-\frac{3}{x}\right )+\frac{\left (a-\frac{1}{x}\right )^3}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{1}{3} a \sqrt{1-\frac{1}{a^2 x^2}} \left (3 a-\frac{1}{x}\right )^2+\frac{11}{2} a^3 \csc ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.101494, size = 66, normalized size = 0.69 \[ \frac{1}{6} a \left (\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left (52 a^3 x^3+19 a^2 x^2-7 a x+2\right )}{x^2 (a x+1)}+33 a^2 \sin ^{-1}\left (\frac{1}{a x}\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.147, size = 666, normalized size = 6.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.52274, size = 212, normalized size = 2.21 \begin{align*} -\frac{1}{3} \,{\left (33 \, a^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) - 12 \, a^{2} \sqrt{\frac{a x - 1}{a x + 1}} - \frac{39 \, a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 52 \, a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 21 \, a^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{3 \,{\left (a x - 1\right )}}{a x + 1} + \frac{3 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.61868, size = 166, normalized size = 1.73 \begin{align*} -\frac{66 \, a^{3} x^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) -{\left (52 \, a^{3} x^{3} + 19 \, a^{2} x^{2} - 7 \, a x + 2\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{6 \, x^{3}} \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*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]