Optimal. Leaf size=104 \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{a c^2 (1-a x)^2}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a c^2 (1-a x)^3}-\frac {6 \sqrt {1-a^2 x^2}}{a c^2 (1-a x)}+\frac {3 \sin ^{-1}(a x)}{a c^2} \]
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Rubi [A] time = 0.19, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6131, 6128, 1639, 793, 663, 216} \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{a c^2 (1-a x)^2}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a c^2 (1-a x)^3}-\frac {6 \sqrt {1-a^2 x^2}}{a c^2 (1-a x)}+\frac {3 \sin ^{-1}(a x)}{a c^2} \]
Antiderivative was successfully verified.
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Rule 216
Rule 663
Rule 793
Rule 1639
Rule 6128
Rule 6131
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx &=\frac {a^2 \int \frac {e^{\tanh ^{-1}(a x)} x^2}{(1-a x)^2} \, dx}{c^2}\\ &=\frac {a^2 \int \frac {x^2 \sqrt {1-a^2 x^2}}{(1-a x)^3} \, dx}{c^2}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{a c^2 (1-a x)^2}-\frac {\int \frac {\left (2 a^2-3 a^3 x\right ) \sqrt {1-a^2 x^2}}{(1-a x)^3} \, dx}{a^2 c^2}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a c^2 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a c^2 (1-a x)^2}-\frac {3 \int \frac {\sqrt {1-a^2 x^2}}{(1-a x)^2} \, dx}{c^2}\\ &=-\frac {6 \sqrt {1-a^2 x^2}}{a c^2 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a c^2 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a c^2 (1-a x)^2}+\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^2}\\ &=-\frac {6 \sqrt {1-a^2 x^2}}{a c^2 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a c^2 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a c^2 (1-a x)^2}+\frac {3 \sin ^{-1}(a x)}{a c^2}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 53, normalized size = 0.51 \[ \frac {\frac {\sqrt {1-a^2 x^2} \left (-3 a^2 x^2+19 a x-14\right )}{(a x-1)^2}+9 \sin ^{-1}(a x)}{3 a c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 107, normalized size = 1.03 \[ -\frac {14 \, a^{2} x^{2} - 28 \, a x + 18 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{2} x^{2} - 19 \, a x + 14\right )} \sqrt {-a^{2} x^{2} + 1} + 14}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \]
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 = 0.04, size = 140, normalized size = 1.35 \[ -\frac {\sqrt {-a^{2} x^{2}+1}}{a \,c^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c^{2} \sqrt {a^{2}}}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a^{3} c^{2} \left (x -\frac {1}{a}\right )^{2}}+\frac {13 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a^{2} c^{2} \left (x -\frac {1}{a}\right )} \]
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 \[ \int \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a x}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.82, size = 140, normalized size = 1.35 \[ \frac {2\,a\,\sqrt {1-a^2\,x^2}}{3\,\left (a^4\,c^2\,x^2-2\,a^3\,c^2\,x+a^2\,c^2\right )}+\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^2\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a\,c^2}-\frac {13\,\sqrt {1-a^2\,x^2}}{3\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}\right )} \]
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 \[ \frac {a^{2} \left (\int \frac {x^{2}}{a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{3}}{a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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