Optimal. Leaf size=87 \[ -\frac {9}{2} a^2 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{\left (a+\frac {1}{x}\right )^3}-\frac {3 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 \left (a+\frac {1}{x}\right )}-\frac {9}{2} a^2 \csc ^{-1}(a x) \]
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Rubi [A]
time = 0.28, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6304, 1647,
1607, 12, 807, 679, 222} \begin {gather*} -\frac {9}{2} a^2 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {9}{2} a^2 \csc ^{-1}(a x)-\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{\left (a+\frac {1}{x}\right )^3}-\frac {3 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 \left (a+\frac {1}{x}\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 222
Rule 679
Rule 807
Rule 1607
Rule 1647
Rule 6304
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{x^3} \, dx &=-\text {Subst}\left (\int \frac {x \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 {\text {Subst}\left (\int \frac {\left (a x-x^2\right ) \sqrt {1-\frac {x^2}{a^2}}}{\left (1+\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {\text {Subst}\left (\int \frac {(a-x) x \sqrt {1-\frac {x^2}{a^2}}}{\left (1+\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {\text {Subst}\left (\int \frac {a^2 x \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}\\ &=-\text {Subst}\left (\int \frac {x \left (1-\frac {x^2}{a^2}\right )^{3/2}}{\left (1+\frac {x}{a}\right )^3} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{\left (a+\frac {1}{x}\right )^3}-(3 a) \text {Subst}\left (\int \frac {\left (1-\frac {x^2}{a^2}\right )^{3/2}}{\left (1+\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{\left (a+\frac {1}{x}\right )^3}-\frac {3 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 \left (a+\frac {1}{x}\right )}-\frac {1}{2} (9 a) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{1+\frac {x}{a}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {9}{2} a^2 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{\left (a+\frac {1}{x}\right )^3}-\frac {3 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 \left (a+\frac {1}{x}\right )}-\frac {1}{2} (9 a) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {9}{2} a^2 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{\left (a+\frac {1}{x}\right )^3}-\frac {3 a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{2 \left (a+\frac {1}{x}\right )}-\frac {9}{2} a^2 \csc ^{-1}(a x)\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 56, normalized size = 0.64 \begin {gather*} \frac {1}{2} a \left (\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (1-5 a x-14 a^2 x^2\right )}{x (1+a x)}-9 a \text {ArcSin}\left (\frac {1}{a x}\right )\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. \(640\) vs.
\(2(75)=150\).
time = 0.11, size = 641, normalized size = 7.37
method | result | size |
risch | \(-\frac {\left (a x +1\right ) \left (6 a x -1\right ) \sqrt {\frac {a x -1}{a x +1}}}{2 x^{2}}+\frac {\left (-\frac {4 a \sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{x +\frac {1}{a}}-\frac {9 a^{2} \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )}{2}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{a x -1}\) | \(119\) |
default | \(-\frac {\left (6 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{5} x^{5}+6 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+21 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{4} x^{4}+9 a^{4} x^{4} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-6 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{4} x^{4}-6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}+12 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}-11 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+24 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{3} x^{3}+18 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )-4 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a^{2} x^{2}-12 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} x^{3}-12 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+6 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-4 \sqrt {a^{2}}\, \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} a x +9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+9 \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a^{2} x^{2}-6 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{2 x^{2} \sqrt {a^{2}}\, \left (a x -1\right ) \sqrt {\left (a x +1\right ) \left (a x -1\right )}}\) | \(641\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 112, normalized size = 1.29 \begin {gather*} {\left (9 \, a \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 4 \, a \sqrt {\frac {a x - 1}{a x + 1}} - \frac {7 \, a \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 5 \, 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 = 61, normalized size = 0.70 \begin {gather*} \frac {18 \, a^{2} x^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (14 \, a^{2} x^{2} + 5 \, 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 {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{3}}\, 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 = 0.06, size = 118, normalized size = 1.36 \begin {gather*} 9\,a^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )-\frac {5\,a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}+7\,a^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{\frac {{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {2\,\left (a\,x-1\right )}{a\,x+1}+1}-4\,a^2\,\sqrt {\frac {a\,x-1}{a\,x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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