Optimal. Leaf size=107 \[ -\frac {\sqrt {1-a^2 x^2}}{x \sqrt {c-a^2 c x^2}}+\frac {a \sqrt {1-a^2 x^2} \log (x)}{\sqrt {c-a^2 c x^2}}-\frac {a \sqrt {1-a^2 x^2} \log (1-a x)}{\sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.19, antiderivative size = 107, 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} \[ -\frac {\sqrt {1-a^2 x^2}}{x \sqrt {c-a^2 c x^2}}+\frac {a \sqrt {1-a^2 x^2} \log (x)}{\sqrt {c-a^2 c x^2}}-\frac {a \sqrt {1-a^2 x^2} \log (1-a x)}{\sqrt {c-a^2 c x^2}} \]
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^2 \sqrt {c-a^2 c x^2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{\tanh ^{-1}(a x)}}{x^2 \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^2 (1-a x)} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \left (\frac {1}{x^2}+\frac {a}{x}-\frac {a^2}{-1+a x}\right ) \, dx}{\sqrt {c-a^2 c x^2}}\\ &=-\frac {\sqrt {1-a^2 x^2}}{x \sqrt {c-a^2 c x^2}}+\frac {a \sqrt {1-a^2 x^2} \log (x)}{\sqrt {c-a^2 c x^2}}-\frac {a \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 = 0.03, size = 50, normalized size = 0.47 \[ \frac {\sqrt {1-a^2 x^2} \left (a \log (x)-a \log (1-a x)-\frac {1}{x}\right )}{\sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 416, normalized size = 3.89 \[ \left [\frac {{\left (a^{3} x^{3} - a x\right )} \sqrt {c} \log \left (-\frac {4 \, a^{5} c x^{5} - {\left (2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right )} c x^{6} - {\left (4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right )} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x + {\left (4 \, a^{3} x^{3} - {\left (4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right )} x^{4} - 6 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right ) - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left (x - 1\right )}}{2 \, {\left (a^{2} c x^{3} - c x\right )}}, -\frac {{\left (a^{3} x^{3} - a x\right )} \sqrt {-c} \arctan \left (-\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a^{2} - 2 \, a + 1\right )} x^{2} - 2 \, a x + 1\right )} \sqrt {-c}}{2 \, a^{3} c x^{3} - {\left (2 \, a^{3} - a^{2}\right )} c x^{4} - {\left (a^{2} - 2 \, a + 1\right )} c x^{2} - 2 \, a c x + c}\right ) + \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left (x - 1\right )}}{a^{2} c x^{3} - c x}\right ] \]
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 \[ \int \frac {a x + 1}{\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 62, normalized size = 0.58 \[ \frac {\left (-a \ln \relax (x ) x +\ln \left (a x -1\right ) x a +1\right ) \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}}{\left (a^{2} x^{2}-1\right ) c x} \]
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 \[ \int \frac {a x + 1}{\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} x^{2}}\,{d x} \]
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 \[ \int \frac {a\,x+1}{x^2\,\sqrt {c-a^2\,c\,x^2}\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {a x + 1}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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