Optimal. Leaf size=117 \[ -\frac {\sqrt {1-a^2 x^2}}{3 x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {14 a^2 \sqrt {1-a^2 x^2}}{3 x}+\frac {4 a^3 \sqrt {1-a^2 x^2}}{1-a x}-\frac {11}{2} a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
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Rubi [A]
time = 0.54, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6259, 6874,
277, 270, 272, 44, 65, 214, 665} \begin {gather*} -\frac {14 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {3 a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {\sqrt {1-a^2 x^2}}{3 x^3}+\frac {4 a^3 \sqrt {1-a^2 x^2}}{1-a x}-\frac {11}{2} a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 65
Rule 214
Rule 270
Rule 272
Rule 277
Rule 665
Rule 6259
Rule 6874
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{x^4} \, dx &=\int \frac {(1+a x)^2}{x^4 (1-a x) \sqrt {1-a^2 x^2}} \, dx\\ &=\int \left (\frac {1}{x^4 \sqrt {1-a^2 x^2}}+\frac {3 a}{x^3 \sqrt {1-a^2 x^2}}+\frac {4 a^2}{x^2 \sqrt {1-a^2 x^2}}+\frac {4 a^3}{x \sqrt {1-a^2 x^2}}-\frac {4 a^4}{(-1+a x) \sqrt {1-a^2 x^2}}\right ) \, dx\\ &=(3 a) \int \frac {1}{x^3 \sqrt {1-a^2 x^2}} \, dx+\left (4 a^2\right ) \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx+\left (4 a^3\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx-\left (4 a^4\right ) \int \frac {1}{(-1+a x) \sqrt {1-a^2 x^2}} \, dx+\int \frac {1}{x^4 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2}}{3 x^3}-\frac {4 a^2 \sqrt {1-a^2 x^2}}{x}+\frac {4 a^3 \sqrt {1-a^2 x^2}}{1-a x}+\frac {1}{2} (3 a) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-a^2 x}} \, dx,x,x^2\right )+\frac {1}{3} \left (2 a^2\right ) \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx+\left (2 a^3\right ) \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 {3 a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {14 a^2 \sqrt {1-a^2 x^2}}{3 x}+\frac {4 a^3 \sqrt {1-a^2 x^2}}{1-a x}-(4 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 {1}{4} \left (3 a^3\right ) \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 {3 a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {14 a^2 \sqrt {1-a^2 x^2}}{3 x}+\frac {4 a^3 \sqrt {1-a^2 x^2}}{1-a x}-4 a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {1}{2} (3 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 {3 a \sqrt {1-a^2 x^2}}{2 x^2}-\frac {14 a^2 \sqrt {1-a^2 x^2}}{3 x}+\frac {4 a^3 \sqrt {1-a^2 x^2}}{1-a x}-\frac {11}{2} a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 81, normalized size = 0.69 \begin {gather*} \frac {1}{6} \left (\frac {\sqrt {1-a^2 x^2} \left (2+7 a x+19 a^2 x^2-52 a^3 x^3\right )}{x^3 (-1+a x)}+33 a^3 \log (x)-33 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 [A]
time = 1.05, size = 146, normalized size = 1.25
method | result | size |
risch | \(\frac {28 a^{4} x^{4}+9 a^{3} x^{3}-26 a^{2} x^{2}-9 a x -2}{6 x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {a^{3} \left (-\frac {8 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )}-11 \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\) | \(111\) |
default | \(-\frac {1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {13 a^{2} \left (-\frac {1}{x \sqrt {-a^{2} x^{2}+1}}+\frac {2 a^{2} x}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}+3 a \left (-\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )+a^{3} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(146\) |
meijerg | \(-\frac {-8 a^{4} x^{4}+4 a^{2} x^{2}+1}{3 x^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {a^{3} \left (-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )+\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{2}\right )}{\sqrt {\pi }}-\frac {3 a^{2} \left (-2 a^{2} x^{2}+1\right )}{x \sqrt {-a^{2} x^{2}+1}}-\frac {3 a^{3} \left (-\frac {\sqrt {\pi }\, \left (-20 a^{2} x^{2}+8\right )}{16 a^{2} x^{2}}+\frac {\sqrt {\pi }\, \left (-24 a^{2} x^{2}+8\right )}{16 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-a^{2} x^{2}+1}}{2}\right )}{2}-\frac {3 \left (\frac {5}{3}-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (-a^{2}\right )\right ) \sqrt {\pi }}{4}+\frac {\sqrt {\pi }}{2 x^{2} a^{2}}\right )}{\sqrt {\pi }}\) | \(256\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 122, normalized size = 1.04 \begin {gather*} \frac {26 \, a^{4} x}{3 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {11}{2} \, a^{3} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {11 \, a^{3}}{2 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {13 \, a^{2}}{3 \, \sqrt {-a^{2} x^{2} + 1} x} - \frac {3 \, a}{2 \, \sqrt {-a^{2} x^{2} + 1} x^{2}} - \frac {1}{3 \, \sqrt {-a^{2} x^{2} + 1} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 105, normalized size = 0.90 \begin {gather*} \frac {24 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 33 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (52 \, a^{3} x^{3} - 19 \, a^{2} x^{2} - 7 \, a x - 2\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a x^{4} - x^{3}\right )}} \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^{4} \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. 265 vs.
\(2 (98) = 196\).
time = 0.44, size = 265, normalized size = 2.26 \begin {gather*} -\frac {{\left (a^{4} + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2}}{x} + \frac {48 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{x^{2}} - \frac {249 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{2} x^{3}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} - \frac {11 \, a^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, {\left | a \right |}} - \frac {\frac {57 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4}}{x} + \frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2}}{x^{2}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{x^{3}}}{24 \, a^{2} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.81, size = 126, normalized size = 1.08 \begin {gather*} \frac {4\,a^4\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {3\,a\,\sqrt {1-a^2\,x^2}}{2\,x^2}-\frac {14\,a^2\,\sqrt {1-a^2\,x^2}}{3\,x}-\frac {\sqrt {1-a^2\,x^2}}{3\,x^3}+\frac {a^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,11{}\mathrm {i}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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