Optimal. Leaf size=53 \[ 3 a \sqrt {1-\frac {1}{a^2 x^2}}+\frac {2 \left (a-\frac {1}{x}\right )^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}+3 a \csc ^{-1}(a x) \]
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Rubi [A]
time = 0.05, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6304, 867, 683,
655, 222} \begin {gather*} \frac {2 \left (a-\frac {1}{x}\right )^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}+3 a \sqrt {1-\frac {1}{a^2 x^2}}+3 a \csc ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 655
Rule 683
Rule 867
Rule 6304
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^2} \, dx &=-\text {Subst}\left (\int \frac {\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 )\\ &=-\text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^3}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {2 \left (a-\frac {1}{x}\right )^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}+3 \text {Subst}\left (\int \frac {1-\frac {x}{a}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=3 a \sqrt {1-\frac {1}{a^2 x^2}}+\frac {2 \left (a-\frac {1}{x}\right )^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}+3 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=3 a \sqrt {1-\frac {1}{a^2 x^2}}+\frac {2 \left (a-\frac {1}{x}\right )^2}{a \sqrt {1-\frac {1}{a^2 x^2}}}+3 a \csc ^{-1}(a x)\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 41, normalized size = 0.77 \begin {gather*} \frac {a \sqrt {1-\frac {1}{a^2 x^2}} (1+5 a x)}{1+a x}+3 a \text {ArcSin}\left (\frac {1}{a x}\right ) \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. \(593\) vs.
\(2(49)=98\).
time = 0.10, size = 594, normalized size = 11.21
method | result | size |
risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{x}+\frac {\left (\frac {4 \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{x +\frac {1}{a}}+3 a \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a x -1}\) | \(109\) |
default | \(\frac {\left (\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}+\ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+5 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}+3 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} x^{3}-\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+2 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-2 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x +7 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+6 \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a^{2} x^{2}-2 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a x -2 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-2 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+\ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}+3 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x +3 \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a x -\sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x -\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x \right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{x \sqrt {a^{2}}\, \left (a x -1\right ) \sqrt {\left (a x +1\right ) \left (a x -1\right )}}\) | \(594\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 72, normalized size = 1.36 \begin {gather*} 2 \, a {\left (2 \, \sqrt {\frac {a x - 1}{a x + 1}} + \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\frac {a x - 1}{a x + 1} + 1} - 3 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 49, normalized size = 0.92 \begin {gather*} -\frac {6 \, a x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (5 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{x} \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 {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 59, normalized size = 1.11 \begin {gather*} \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}+5\,a\,x\,\sqrt {\frac {a\,x-1}{a\,x+1}}-6\,a\,x\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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