Optimal. Leaf size=149 \[ -\frac {3 a \sqrt {c-a^2 c x^2}}{x \sqrt {1-a^2 x^2}}-\frac {\sqrt {c-a^2 c x^2}}{2 x^2 \sqrt {1-a^2 x^2}}+\frac {4 a^2 \log (x) \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}}-\frac {4 a^2 \sqrt {c-a^2 c x^2} \log (1-a x)}{\sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.21, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6153, 6150, 88} \[ -\frac {3 a \sqrt {c-a^2 c x^2}}{x \sqrt {1-a^2 x^2}}-\frac {\sqrt {c-a^2 c x^2}}{2 x^2 \sqrt {1-a^2 x^2}}+\frac {4 a^2 \log (x) \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}}-\frac {4 a^2 \sqrt {c-a^2 c x^2} \log (1-a x)}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^3} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int \frac {e^{3 \tanh ^{-1}(a x)} \sqrt {1-a^2 x^2}}{x^3} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \frac {(1+a x)^2}{x^3 (1-a x)} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \left (\frac {1}{x^3}+\frac {3 a}{x^2}+\frac {4 a^2}{x}-\frac {4 a^3}{-1+a x}\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {\sqrt {c-a^2 c x^2}}{2 x^2 \sqrt {1-a^2 x^2}}-\frac {3 a \sqrt {c-a^2 c x^2}}{x \sqrt {1-a^2 x^2}}+\frac {4 a^2 \sqrt {c-a^2 c x^2} \log (x)}{\sqrt {1-a^2 x^2}}-\frac {4 a^2 \sqrt {c-a^2 c x^2} \log (1-a x)}{\sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 63, normalized size = 0.42 \[ \frac {\sqrt {c-a^2 c x^2} \left (4 a^2 \log (x)-4 a^2 \log (1-a x)-\frac {3 a}{x}-\frac {1}{2 x^2}\right )}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 450, normalized size = 3.02 \[ \left [\frac {4 \, {\left (a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt {c} \log \left (-\frac {4 \, a^{5} c x^{5} - {\left (2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right )} c x^{6} - {\left (4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right )} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x + {\left (4 \, a^{3} x^{3} - {\left (4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right )} x^{4} - 6 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right ) - \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left ({\left (6 \, a + 1\right )} x^{2} - 6 \, a x - 1\right )}}{2 \, {\left (a^{2} x^{4} - x^{2}\right )}}, -\frac {8 \, {\left (a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt {-c} \arctan \left (-\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a^{2} - 2 \, a + 1\right )} x^{2} - 2 \, a x + 1\right )} \sqrt {-c}}{2 \, a^{3} c x^{3} - {\left (2 \, a^{3} - a^{2}\right )} c x^{4} - {\left (a^{2} - 2 \, a + 1\right )} c x^{2} - 2 \, a c x + c}\right ) + \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left ({\left (6 \, a + 1\right )} x^{2} - 6 \, a x - 1\right )}}{2 \, {\left (a^{2} x^{4} - x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 73, normalized size = 0.49 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 a^{2} \ln \relax (x ) x^{2}-8 \ln \left (a x -1\right ) x^{2} a^{2}-6 a x -1\right )}{2 \left (a^{2} x^{2}-1\right ) x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 160, normalized size = 1.07 \[ -2 \, \left (-1\right )^{-2 \, a^{2} c x^{2} + 2 \, c} a^{2} \sqrt {c} \log \left (-2 \, a^{2} c + \frac {2 \, c}{x^{2}}\right ) + \frac {1}{2} \, a^{3} {\left (\frac {\sqrt {c} \log \left (a x + 1\right )}{a} - \frac {\sqrt {c} \log \left (a x - 1\right )}{a}\right )} + \frac {a^{2} c}{2 \, \sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac {3}{2} \, {\left (a \sqrt {c} \log \left (a x + 1\right ) - a \sqrt {c} \log \left (a x - 1\right ) - \frac {2 \, \sqrt {c}}{x}\right )} a - \frac {c}{2 \, \sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c} x^{2}} \]
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 \[ \int \frac {\sqrt {c-a^2\,c\,x^2}\,{\left (a\,x+1\right )}^3}{x^3\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )^{3}}{x^{3} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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