Optimal. Leaf size=75 \[ -\frac{1}{3} a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+a^3 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 x}-\frac{1}{2} a^3 \csc ^{-1}(a x) \]
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Rubi [A] time = 0.0618059, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6169, 797, 641, 195, 216} \[ -\frac{1}{3} a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+a^3 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 x}-\frac{1}{2} a^3 \csc ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6169
Rule 797
Rule 641
Rule 195
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{x^4} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2 \left (1+\frac{x}{a}\right )}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\left (a^2 \operatorname{Subst}\left (\int \frac{1+\frac{x}{a}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\right )+a^2 \operatorname{Subst}\left (\int \left (1+\frac{x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}} \, dx,x,\frac{1}{x}\right )\\ &=a^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{1}{3} a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}-a^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )+a^2 \operatorname{Subst}\left (\int \sqrt{1-\frac{x^2}{a^2}} \, dx,x,\frac{1}{x}\right )\\ &=a^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{1}{3} a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 x}-a^3 \csc ^{-1}(a x)+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=a^3 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{1}{3} a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 x}-\frac{1}{2} a^3 \csc ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0783049, size = 51, normalized size = 0.68 \[ \frac{1}{6} a \left (\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left (4 a^2 x^2+3 a x+2\right )}{x^2}-3 a^2 \sin ^{-1}\left (\frac{1}{a x}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.135, size = 284, normalized size = 3.8 \begin{align*} -{\frac{ax-1}{6\,{x}^{3}} \left ( -6\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{4}{a}^{4}+6\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+3\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+6\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{3}{a}^{4}+3\,{a}^{3}{x}^{3}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) -6\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{3}{a}^{3}-6\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{3}{a}^{4}+3\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa+2\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}} \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.55237, size = 184, normalized size = 2.45 \begin{align*} \frac{1}{3} \,{\left (3 \, a^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + \frac{3 \, a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 4 \, a^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 9 \, 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.
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Fricas [A] time = 1.55917, size = 161, normalized size = 2.15 \begin{align*} \frac{6 \, a^{3} x^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) +{\left (4 \, a^{3} x^{3} + 7 \, a^{2} x^{2} + 5 \, 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.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4} \sqrt{\frac{a x - 1}{a x + 1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17312, size = 176, normalized size = 2.35 \begin{align*} \frac{1}{3} \,{\left (3 \, a^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + \frac{\frac{4 \,{\left (a x - 1\right )} a^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} + \frac{3 \,{\left (a x - 1\right )}^{2} a^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + 9 \, a^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (\frac{a x - 1}{a x + 1} + 1\right )}^{3}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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