Optimal. Leaf size=76 \[ -\frac {1}{2} a^3 \csc ^{-1}(a x)+\frac {a^2 \sqrt {1-\frac {1}{a^2 x^2}}}{2 x}+\frac {1}{3} a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}-a^3 \sqrt {1-\frac {1}{a^2 x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, 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 195
Rule 216
Rule 641
Rule 797
Rule 6169
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*}
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Mathematica [A] time = 0.08, size = 52, normalized size = 0.68 \[ -\frac {1}{2} a^3 \sin ^{-1}\left (\frac {1}{a x}\right )-\frac {a \sqrt {1-\frac {1}{a^2 x^2}} \left (4 a^2 x^2-3 a x+2\right )}{6 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.52, size = 68, normalized size = 0.89 \[ \frac {6 \, a^{3} x^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (4 \, a^{3} x^{3} + a^{2} x^{2} - a x + 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 284, normalized size = 3.74 \[ \frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (-6 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{4} a^{4}+6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}-3 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{3} a^{3}-3 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}+6 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}-6 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}-3 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a +2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{6 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 137, normalized size = 1.80 \[ \frac {1}{3} \, {\left (3 \, a^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - \frac {9 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 4 \, a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 3 \, a^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {3 \, {\left (a x - 1\right )}}{a x + 1} + \frac {3 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 105, normalized size = 1.38 \[ a^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )-\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{3\,x^3}-\frac {2\,a^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{3}-\frac {a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{6\,x}+\frac {a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{6\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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