Optimal. Leaf size=135 \[ -\frac {\sqrt {1-a^2 x^2}}{4 x^4}+\frac {a \sqrt {1-a^2 x^2}}{x^3}-\frac {19 a^2 \sqrt {1-a^2 x^2}}{8 x^2}+\frac {6 a^3 \sqrt {1-a^2 x^2}}{x}+\frac {4 a^4 \sqrt {1-a^2 x^2}}{1+a x}-\frac {51}{8} a^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
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Rubi [A]
time = 0.59, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6259, 6874,
272, 44, 65, 214, 277, 270, 665} \begin {gather*} -\frac {19 a^2 \sqrt {1-a^2 x^2}}{8 x^2}-\frac {\sqrt {1-a^2 x^2}}{4 x^4}+\frac {a \sqrt {1-a^2 x^2}}{x^3}+\frac {4 a^4 \sqrt {1-a^2 x^2}}{a x+1}-\frac {51}{8} a^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\frac {6 a^3 \sqrt {1-a^2 x^2}}{x} \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^5} \, dx &=\int \frac {(1-a x)^2}{x^5 (1+a x) \sqrt {1-a^2 x^2}} \, dx\\ &=\int \left (\frac {1}{x^5 \sqrt {1-a^2 x^2}}-\frac {3 a}{x^4 \sqrt {1-a^2 x^2}}+\frac {4 a^2}{x^3 \sqrt {1-a^2 x^2}}-\frac {4 a^3}{x^2 \sqrt {1-a^2 x^2}}+\frac {4 a^4}{x \sqrt {1-a^2 x^2}}-\frac {4 a^5}{(1+a x) \sqrt {1-a^2 x^2}}\right ) \, dx\\ &=-\left ((3 a) \int \frac {1}{x^4 \sqrt {1-a^2 x^2}} \, dx\right )+\left (4 a^2\right ) \int \frac {1}{x^3 \sqrt {1-a^2 x^2}} \, dx-\left (4 a^3\right ) \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx+\left (4 a^4\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx-\left (4 a^5\right ) \int \frac {1}{(1+a x) \sqrt {1-a^2 x^2}} \, dx+\int \frac {1}{x^5 \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {a \sqrt {1-a^2 x^2}}{x^3}+\frac {4 a^3 \sqrt {1-a^2 x^2}}{x}+\frac {4 a^4 \sqrt {1-a^2 x^2}}{1+a x}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^3 \sqrt {1-a^2 x}} \, dx,x,x^2\right )+\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-a^2 x}} \, dx,x,x^2\right )-\left (2 a^3\right ) \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx+\left (2 a^4\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}}{4 x^4}+\frac {a \sqrt {1-a^2 x^2}}{x^3}-\frac {2 a^2 \sqrt {1-a^2 x^2}}{x^2}+\frac {6 a^3 \sqrt {1-a^2 x^2}}{x}+\frac {4 a^4 \sqrt {1-a^2 x^2}}{1+a x}+\frac {1}{8} \left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-a^2 x}} \, dx,x,x^2\right )-\left (4 a^2\right ) \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^4 \text {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-a^2 x^2}}{4 x^4}+\frac {a \sqrt {1-a^2 x^2}}{x^3}-\frac {19 a^2 \sqrt {1-a^2 x^2}}{8 x^2}+\frac {6 a^3 \sqrt {1-a^2 x^2}}{x}+\frac {4 a^4 \sqrt {1-a^2 x^2}}{1+a x}-4 a^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\left (2 a^2\right ) \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}{16} \left (3 a^4\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}}{4 x^4}+\frac {a \sqrt {1-a^2 x^2}}{x^3}-\frac {19 a^2 \sqrt {1-a^2 x^2}}{8 x^2}+\frac {6 a^3 \sqrt {1-a^2 x^2}}{x}+\frac {4 a^4 \sqrt {1-a^2 x^2}}{1+a x}-6 a^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {1}{8} \left (3 a^2\right ) \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}}{4 x^4}+\frac {a \sqrt {1-a^2 x^2}}{x^3}-\frac {19 a^2 \sqrt {1-a^2 x^2}}{8 x^2}+\frac {6 a^3 \sqrt {1-a^2 x^2}}{x}+\frac {4 a^4 \sqrt {1-a^2 x^2}}{1+a x}-\frac {51}{8} a^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 89, normalized size = 0.66 \begin {gather*} \frac {1}{8} \left (\frac {\sqrt {1-a^2 x^2} \left (-2+6 a x-11 a^2 x^2+29 a^3 x^3+80 a^4 x^4\right )}{x^4 (1+a x)}+51 a^4 \log (x)-51 a^4 \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. \(785\) vs.
\(2(117)=234\).
time = 0.83, size = 786, normalized size = 5.82
method | result | size |
risch | \(-\frac {48 a^{5} x^{5}-19 a^{4} x^{4}-40 a^{3} x^{3}+17 a^{2} x^{2}-8 a x +2}{8 x^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {a^{4} \left (-51 \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {32 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{a \left (x +\frac {1}{a}\right )}\right )}{8}\) | \(113\) |
default | \(-3 a \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{3 x^{3}}-\frac {2 a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}-4 a^{2} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} x}{4}+\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}\right )\right )}{3}\right )-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{4 x^{4}}+\frac {23 a^{2} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{2 x^{2}}-\frac {3 a^{2} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}\right )}{4}-15 a^{4} \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )-a^{2} \left (-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{3}}-2 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )\right )\right )-5 a^{3} \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\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 )}{2 \sqrt {a^{2}}}\right )\right )\right )-10 a^{3} \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}-4 a^{2} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} x}{4}+\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 \sqrt {a^{2}}}\right )\right )+15 a^{4} \left (\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {-a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )\) | \(786\) |
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 = 0.35, size = 109, normalized size = 0.81 \begin {gather*} \frac {32 \, a^{5} x^{5} + 32 \, a^{4} x^{4} + 51 \, {\left (a^{5} x^{5} + a^{4} x^{4}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (80 \, a^{4} x^{4} + 29 \, a^{3} x^{3} - 11 \, a^{2} x^{2} + 6 \, a x - 2\right )} \sqrt {-a^{2} x^{2} + 1}}{8 \, {\left (a x^{5} + x^{4}\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 (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{5} \left (a x + 1\right )^{3}}\, 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. 326 vs.
\(2 (117) = 234\).
time = 0.41, size = 326, normalized size = 2.41 \begin {gather*} \frac {{\left (a^{5} - \frac {7 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{3}}{x} + \frac {32 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a}{x^{2}} - \frac {160 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a x^{3}} - \frac {712 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{3} x^{4}}\right )} a^{8} x^{4}}{64 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} - \frac {51 \, a^{5} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{8 \, {\left | a \right |}} + \frac {\frac {200 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{5} {\left | a \right |}}{x} - \frac {40 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{3} {\left | a \right |}}{x^{2}} + \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} a {\left | a \right |}}{x^{3}} - \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} {\left | a \right |}}{a x^{4}}}{64 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 144, normalized size = 1.07 \begin {gather*} \frac {a\,\sqrt {1-a^2\,x^2}}{x^3}-\frac {\sqrt {1-a^2\,x^2}}{4\,x^4}-\frac {19\,a^2\,\sqrt {1-a^2\,x^2}}{8\,x^2}+\frac {6\,a^3\,\sqrt {1-a^2\,x^2}}{x}-\frac {4\,a^5\,\sqrt {1-a^2\,x^2}}{\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}+\frac {a^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,51{}\mathrm {i}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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