Optimal. Leaf size=90 \[ -\frac {\sqrt {1-a^2 x^2}}{3 x^3}+\frac {a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}+\frac {1}{2} a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6259, 849, 821,
272, 65, 214} \begin {gather*} -\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}+\frac {a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}+\frac {1}{2} a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 849
Rule 6259
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{x^4} \, dx &=\int \frac {1-a x}{x^4 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{3 x^3}-\frac {1}{3} \int \frac {3 a-2 a^2 x}{x^3 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{3 x^3}+\frac {a \sqrt {1-a^2 x^2}}{2 x^2}+\frac {1}{6} \int \frac {4 a^2-3 a^3 x}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{3 x^3}+\frac {a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {1}{2} a^3 \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{3 x^3}+\frac {a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {1}{4} a^3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-a^2 x^2}}{3 x^3}+\frac {a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}+\frac {1}{2} a \text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )\\ &=-\frac {\sqrt {1-a^2 x^2}}{3 x^3}+\frac {a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}+\frac {1}{2} a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 66, normalized size = 0.73 \begin {gather*} \frac {1}{6} \left (\frac {\left (-2+3 a x-4 a^2 x^2\right ) \sqrt {1-a^2 x^2}}{x^3}-3 a^3 \log (x)+3 a^3 \log \left (1+\sqrt {1-a^2 x^2}\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. \(241\) vs.
\(2(74)=148\).
time = 0.80, size = 242, normalized size = 2.69
method | result | size |
risch | \(\frac {4 a^{4} x^{4}-3 a^{3} x^{3}-2 a^{2} x^{2}+3 a x -2}{6 x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {a^{3} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\) | \(67\) |
default | \(-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 x^{3}}+a^{3} \left (\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )-a \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {a^{2} \left (\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )+a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}-2 a^{2} \left (\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}\right )\right )-a^{3} \left (\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(242\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 60, normalized size = 0.67 \begin {gather*} -\frac {3 \, a^{3} x^{3} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (4 \, a^{2} x^{2} - 3 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, x^{3}} \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 {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{4} \left (a x + 1\right )}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 78, normalized size = 0.87 \begin {gather*} \frac {a\,\sqrt {1-a^2\,x^2}}{2\,x^2}-\frac {\sqrt {1-a^2\,x^2}}{3\,x^3}-\frac {2\,a^2\,\sqrt {1-a^2\,x^2}}{3\,x}-\frac {a^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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