Optimal. Leaf size=63 \[ -\frac {\sqrt {1-a^2 x^2}}{x}+\frac {4 a \sqrt {1-a^2 x^2}}{1-a x}-3 a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
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Rubi [A]
time = 0.49, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6259, 6874,
270, 272, 65, 214, 665} \begin {gather*} \frac {4 a \sqrt {1-a^2 x^2}}{1-a x}-\frac {\sqrt {1-a^2 x^2}}{x}-3 a \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 270
Rule 272
Rule 665
Rule 6259
Rule 6874
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{x^2} \, dx &=\int \frac {(1+a x)^2}{x^2 (1-a x) \sqrt {1-a^2 x^2}} \, dx\\ &=\int \left (\frac {1}{x^2 \sqrt {1-a^2 x^2}}+\frac {3 a}{x \sqrt {1-a^2 x^2}}-\frac {4 a^2}{(-1+a x) \sqrt {1-a^2 x^2}}\right ) \, dx\\ &=(3 a) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx-\left (4 a^2\right ) \int \frac {1}{(-1+a x) \sqrt {1-a^2 x^2}} \, dx+\int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{x}+\frac {4 a \sqrt {1-a^2 x^2}}{1-a x}+\frac {1}{2} (3 a) \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-a^2 x^2}}{x}+\frac {4 a \sqrt {1-a^2 x^2}}{1-a x}-\frac {3 \text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a}\\ &=-\frac {\sqrt {1-a^2 x^2}}{x}+\frac {4 a \sqrt {1-a^2 x^2}}{1-a x}-3 a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 57, normalized size = 0.90 \begin {gather*} \sqrt {1-a^2 x^2} \left (-\frac {1}{x}-\frac {4 a}{-1+a x}\right )+3 a \log (x)-3 a \log \left (1+\sqrt {1-a^2 x^2}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.92, size = 82, normalized size = 1.30
method | result | size |
default | \(\frac {a}{\sqrt {-a^{2} x^{2}+1}}+\frac {5 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+3 a \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(82\) |
risch | \(\frac {a^{2} x^{2}-1}{x \sqrt {-a^{2} x^{2}+1}}+a \left (-\frac {4 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )}-3 \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(86\) |
meijerg | \(-\frac {-2 a^{2} x^{2}+1}{x \sqrt {-a^{2} x^{2}+1}}-\frac {a \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {\pi }}+\frac {3 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}+\frac {3 a \left (-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}\right )}{\sqrt {\pi }}\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 80, normalized size = 1.27 \begin {gather*} \frac {5 \, a^{2} x}{\sqrt {-a^{2} x^{2} + 1}} - 3 \, a \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {4 \, a}{\sqrt {-a^{2} x^{2} + 1}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 78, normalized size = 1.24 \begin {gather*} \frac {4 \, a^{2} x^{2} - 4 \, a x + 3 \, {\left (a^{2} x^{2} - a x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - \sqrt {-a^{2} x^{2} + 1} {\left (5 \, a x - 1\right )}}{a x^{2} - 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 (a x + 1\right )^{3}}{x^{2} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, 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. 150 vs.
\(2 (56) = 112\).
time = 0.45, size = 150, normalized size = 2.38 \begin {gather*} -\frac {3 \, a^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} - \frac {{\left (a^{2} - \frac {17 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{x}\right )} a^{2} x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{2 \, x {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.80, size = 82, normalized size = 1.30 \begin {gather*} \frac {4\,a^2\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{x}-3\,a\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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