Optimal. Leaf size=64 \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{a^2 c (1-a x)^2}+\frac {2 \sqrt {1-a^2 x^2}}{a^2 c}-\frac {2 \sin ^{-1}(a x)}{a^2 c} \]
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Rubi [A] time = 0.08, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6128, 793, 665, 216} \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{a^2 c (1-a x)^2}+\frac {2 \sqrt {1-a^2 x^2}}{a^2 c}-\frac {2 \sin ^{-1}(a x)}{a^2 c} \]
Antiderivative was successfully verified.
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Rule 216
Rule 665
Rule 793
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x}{c-a c x} \, dx &=c \int \frac {x \sqrt {1-a^2 x^2}}{(c-a c x)^2} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{a^2 c (1-a x)^2}-\frac {2 \int \frac {\sqrt {1-a^2 x^2}}{c-a c x} \, dx}{a}\\ &=\frac {2 \sqrt {1-a^2 x^2}}{a^2 c}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a^2 c (1-a x)^2}-\frac {2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a c}\\ &=\frac {2 \sqrt {1-a^2 x^2}}{a^2 c}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a^2 c (1-a x)^2}-\frac {2 \sin ^{-1}(a x)}{a^2 c}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 53, normalized size = 0.83 \[ \frac {\frac {\sqrt {a x+1} (3-a x)}{\sqrt {1-a x}}+4 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{a^2 c} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.43, size = 69, normalized size = 1.08 \[ \frac {3 \, a x + 4 \, {\left (a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a x - 3\right )} - 3}{a^{3} c x - a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 78, normalized size = 1.22 \[ -\frac {2 \, \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{a c {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{2} c} + \frac {4}{a c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 98, normalized size = 1.53 \[ \frac {\sqrt {-a^{2} x^{2}+1}}{a^{2} c}-\frac {2 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c a \sqrt {a^{2}}}-\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{c \,a^{3} \left (x -\frac {1}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 61, normalized size = 0.95 \[ -\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{3} c x - a^{2} c} - \frac {2 \, \arcsin \left (a x\right )}{a^{2} c} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 90, normalized size = 1.41 \[ \frac {\sqrt {1-a^2\,x^2}}{a^2\,c}-\frac {2\,\sqrt {1-a^2\,x^2}}{\left (c\,\sqrt {-a^2}-a\,c\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a\,c\,\sqrt {-a^2}} \]
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 \[ - \frac {\int \frac {x}{a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{2}}{a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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