Optimal. Leaf size=38 \[ \frac {1}{2} a \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {1}{x}\right )-\frac {1}{2} a^2 \csc ^{-1}(a x) \]
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Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6304, 794, 222}
\begin {gather*} \frac {1}{2} a \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {1}{x}\right )-\frac {1}{2} a^2 \csc ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 794
Rule 6304
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(a x)}}{x^3} \, dx &=-\text {Subst}\left (\int \frac {x \left (1+\frac {x}{a}\right )}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} a \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {1}{x}\right )-\frac {1}{2} a \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} a \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {1}{x}\right )-\frac {1}{2} a^2 \csc ^{-1}(a x)\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 42, normalized size = 1.11 \begin {gather*} \frac {a \left (\sqrt {1-\frac {1}{a^2 x^2}} (1+2 a x)-a x \text {ArcSin}\left (\frac {1}{a x}\right )\right )}{2 x} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(259\) vs.
\(2(32)=64\).
time = 0.10, size = 260, normalized size = 6.84
method | result | size |
risch | \(\frac {\left (a x -1\right ) \left (2 a x +1\right )}{2 x^{2} \sqrt {\frac {a x -1}{a x +1}}}-\frac {a^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(85\) |
default | \(\frac {\left (a x -1\right ) \left (2 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}+2 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-2 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x -\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}-2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-a^{2} x^{2} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+2 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{2 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, x^{2} \sqrt {a^{2}}}\) | \(260\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 91 vs.
\(2 (32) = 64\).
time = 0.46, size = 91, normalized size = 2.39 \begin {gather*} {\left (a \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \frac {a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 3 \, a \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {2 \, {\left (a x - 1\right )}}{a x + 1} + \frac {{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}\right )} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 60, normalized size = 1.58 \begin {gather*} \frac {2 \, a^{2} x^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + {\left (2 \, a^{2} x^{2} + 3 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, x^{2}} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{3} \sqrt {\frac {a x - 1}{a x + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs.
\(2 (32) = 64\).
time = 0.40, size = 143, normalized size = 3.76 \begin {gather*} \frac {a^{2} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{\mathrm {sgn}\left (a x + 1\right )} - \frac {{\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} a^{2} - 2 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a {\left | a \right |} - {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} a^{2} - 2 \, a {\left | a \right |}}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{2} \mathrm {sgn}\left (a x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 81, normalized size = 2.13 \begin {gather*} a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}+\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,x^2}+a^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )+\frac {3\,a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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