Optimal. Leaf size=148 \[ -\frac {a \sqrt {1-a^2 x^2}}{x \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a^2 c x^2}}+\frac {a^2 \sqrt {1-a^2 x^2} \log (x)}{\sqrt {c-a^2 c x^2}}-\frac {a^2 \sqrt {1-a^2 x^2} \log (1-a x)}{\sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.20, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6153, 6150, 44} \[ -\frac {a \sqrt {1-a^2 x^2}}{x \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a^2 c x^2}}+\frac {a^2 \sqrt {1-a^2 x^2} \log (x)}{\sqrt {c-a^2 c x^2}}-\frac {a^2 \sqrt {1-a^2 x^2} \log (1-a x)}{\sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 44
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{\tanh ^{-1}(a x)}}{x^3 \sqrt {1-a^2 x^2}} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {1}{x^3 (1-a x)} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \left (\frac {1}{x^3}+\frac {a}{x^2}+\frac {a^2}{x}-\frac {a^3}{-1+a x}\right ) \, dx}{\sqrt {c-a^2 c x^2}}\\ &=-\frac {\sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a^2 c x^2}}-\frac {a \sqrt {1-a^2 x^2}}{x \sqrt {c-a^2 c x^2}}+\frac {a^2 \sqrt {1-a^2 x^2} \log (x)}{\sqrt {c-a^2 c x^2}}-\frac {a^2 \sqrt {1-a^2 x^2} \log (1-a x)}{\sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 62, normalized size = 0.42 \[ \frac {\sqrt {1-a^2 x^2} \left (a^2 \log (x)-a^2 \log (1-a x)-\frac {a}{x}-\frac {1}{2 x^2}\right )}{\sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 453, normalized size = 3.06 \[ \left [\frac {{\left (a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt {c} \log \left (-\frac {4 \, a^{5} c x^{5} - {\left (2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right )} c x^{6} - {\left (4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right )} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x + {\left (4 \, a^{3} x^{3} - {\left (4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right )} x^{4} - 6 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right ) - \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a + 1\right )} x^{2} - 2 \, a x - 1\right )}}{2 \, {\left (a^{2} c x^{4} - c x^{2}\right )}}, -\frac {2 \, {\left (a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt {-c} \arctan \left (-\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a^{2} - 2 \, a + 1\right )} x^{2} - 2 \, a x + 1\right )} \sqrt {-c}}{2 \, a^{3} c x^{3} - {\left (2 \, a^{3} - a^{2}\right )} c x^{4} - {\left (a^{2} - 2 \, a + 1\right )} c x^{2} - 2 \, a c x + c}\right ) + \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a + 1\right )} x^{2} - 2 \, a x - 1\right )}}{2 \, {\left (a^{2} c x^{4} - c x^{2}\right )}}\right ] \]
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 \[ \int \frac {a x + 1}{\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 76, normalized size = 0.51 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 a^{2} \ln \relax (x ) x^{2}-2 \ln \left (a x -1\right ) x^{2} a^{2}-2 a x -1\right )}{2 \left (a^{2} x^{2}-1\right ) c \,x^{2}} \]
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 \[ -\frac {\left (-1\right )^{-2 \, a^{2} c x^{2} + 2 \, c} a^{2} \log \left (-2 \, a^{2} c + \frac {2 \, c}{x^{2}}\right )}{2 \, \sqrt {c}} + \frac {\frac {1}{2} \, {\left (a \log \left (a x + 1\right ) - a \log \left (a x - 1\right ) - \frac {2}{x}\right )} a}{\sqrt {c}} - \frac {\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}}{2 \, c x^{2}} \]
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 \[ \int \frac {a\,x+1}{x^3\,\sqrt {c-a^2\,c\,x^2}\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {a x + 1}{x^{3} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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