Optimal. Leaf size=150 \[ -\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{5 x^5}-\frac {a \sqrt {1-a^2 x^2}}{20 x^4}+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 x^3}+\frac {11}{120} a^5 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\frac {2 a^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 x}-\frac {a^3 \sqrt {1-a^2 x^2}}{24 x^2} \]
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Rubi [A] time = 0.37, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6010, 6026, 266, 51, 63, 208, 6008} \[ -\frac {a^3 \sqrt {1-a^2 x^2}}{24 x^2}-\frac {a \sqrt {1-a^2 x^2}}{20 x^4}+\frac {11}{120} a^5 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\frac {2 a^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 x}+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 x^3}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rule 6008
Rule 6010
Rule 6026
Rubi steps
\begin {align*} \int \frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x^6} \, dx &=-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{4 x^5}-\frac {1}{4} \int \frac {\tanh ^{-1}(a x)}{x^6 \sqrt {1-a^2 x^2}} \, dx+\frac {1}{4} a \int \frac {1}{x^5 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{5 x^5}-\frac {1}{20} a \int \frac {1}{x^5 \sqrt {1-a^2 x^2}} \, dx+\frac {1}{8} a \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1-a^2 x}} \, dx,x,x^2\right )-\frac {1}{5} a^2 \int \frac {\tanh ^{-1}(a x)}{x^4 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2}}{16 x^4}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{5 x^5}+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 x^3}-\frac {1}{40} a \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1-a^2 x}} \, dx,x,x^2\right )-\frac {1}{15} a^3 \int \frac {1}{x^3 \sqrt {1-a^2 x^2}} \, dx+\frac {1}{32} \left (3 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-a^2 x}} \, dx,x,x^2\right )-\frac {1}{15} \left (2 a^4\right ) \int \frac {\tanh ^{-1}(a x)}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2}}{20 x^4}-\frac {3 a^3 \sqrt {1-a^2 x^2}}{32 x^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{5 x^5}+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 x^3}+\frac {2 a^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 x}-\frac {1}{160} \left (3 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-a^2 x}} \, dx,x,x^2\right )-\frac {1}{30} a^3 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-a^2 x}} \, dx,x,x^2\right )+\frac {1}{64} \left (3 a^5\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )-\frac {1}{15} \left (2 a^5\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2}}{20 x^4}-\frac {a^3 \sqrt {1-a^2 x^2}}{24 x^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{5 x^5}+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 x^3}+\frac {2 a^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 x}-\frac {1}{32} \left (3 a^3\right ) \operatorname {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}{320} \left (3 a^5\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )-\frac {1}{60} a^5 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )-\frac {1}{15} a^5 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {a \sqrt {1-a^2 x^2}}{20 x^4}-\frac {a^3 \sqrt {1-a^2 x^2}}{24 x^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{5 x^5}+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 x^3}+\frac {2 a^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 x}-\frac {3}{32} a^5 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\frac {1}{160} \left (3 a^3\right ) \operatorname {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}{30} a^3 \operatorname {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}{15} \left (2 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )\\ &=-\frac {a \sqrt {1-a^2 x^2}}{20 x^4}-\frac {a^3 \sqrt {1-a^2 x^2}}{24 x^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{5 x^5}+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 x^3}+\frac {2 a^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 x}+\frac {11}{120} a^5 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 104, normalized size = 0.69 \[ \frac {1}{120} \left (-11 a^5 \log (x)-\frac {a \sqrt {1-a^2 x^2} \left (5 a^2 x^2+6\right )}{x^4}+11 a^5 \log \left (\sqrt {1-a^2 x^2}+1\right )+\frac {8 \sqrt {1-a^2 x^2} \left (2 a^4 x^4+a^2 x^2-3\right ) \tanh ^{-1}(a x)}{x^5}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 93, normalized size = 0.62 \[ -\frac {11 \, a^{5} x^{5} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + {\left (5 \, a^{3} x^{3} + 6 \, a x - 4 \, {\left (2 \, a^{4} x^{4} + a^{2} x^{2} - 3\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {-a^{2} x^{2} + 1}}{120 \, x^{5}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 116, normalized size = 0.77 \[ \frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (16 a^{4} x^{4} \arctanh \left (a x \right )-5 x^{3} a^{3}+8 a^{2} x^{2} \arctanh \left (a x \right )-6 a x -24 \arctanh \left (a x \right )\right )}{120 x^{5}}+\frac {11 a^{5} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{120}-\frac {11 a^{5} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )}{120} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 204, normalized size = 1.36 \[ \frac {1}{120} \, {\left (3 \, a^{4} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - 3 \, \sqrt {-a^{2} x^{2} + 1} a^{4} + 8 \, {\left (a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \sqrt {-a^{2} x^{2} + 1} a^{2} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{x^{2}}\right )} a^{2} - \frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{x^{2}} - \frac {6 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{x^{4}}\right )} a - \frac {1}{15} \, {\left (\frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{x^{3}} + \frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{x^{5}}\right )} \operatorname {artanh}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}}{x^6} \,d x \]
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 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}{\left (a x \right )}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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