Optimal. Leaf size=74 \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^2 c^2 (1-a x)^3}-\frac {2 \sqrt {1-a^2 x^2}}{a^2 c^2 (1-a x)}+\frac {\sin ^{-1}(a x)}{a^2 c^2} \]
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Rubi [A] time = 0.08, antiderivative size = 74, 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, 663, 216} \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^2 c^2 (1-a x)^3}-\frac {2 \sqrt {1-a^2 x^2}}{a^2 c^2 (1-a x)}+\frac {\sin ^{-1}(a x)}{a^2 c^2} \]
Antiderivative was successfully verified.
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Rule 216
Rule 663
Rule 793
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x}{(c-a c x)^2} \, dx &=c \int \frac {x \sqrt {1-a^2 x^2}}{(c-a c x)^3} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^2 c^2 (1-a x)^3}-\frac {\int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^2} \, dx}{a}\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{a^2 c^2 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^2 c^2 (1-a x)^3}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a c^2}\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{a^2 c^2 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^2 c^2 (1-a x)^3}+\frac {\sin ^{-1}(a x)}{a^2 c^2}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 57, normalized size = 0.77 \[ -\frac {(a x+1)^{3/2}-4 \sqrt {2} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};\frac {1}{2} (1-a x)\right )}{3 a^2 c^2 (1-a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.43, size = 102, normalized size = 1.38 \[ -\frac {5 \, a^{2} x^{2} - 10 \, a x + 6 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt {-a^{2} x^{2} + 1} {\left (7 \, a x - 5\right )} + 5}{3 \, {\left (a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x + a^{2} c^{2}\right )}} \]
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 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 122, normalized size = 1.65 \[ \frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c^{2} a \sqrt {a^{2}}}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 c^{2} a^{4} \left (x -\frac {1}{a}\right )^{2}}+\frac {7 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 c^{2} a^{3} \left (x -\frac {1}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 88, normalized size = 1.19 \[ \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x + a^{2} c^{2}\right )}} + \frac {7 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{3} c^{2} x - a^{2} c^{2}\right )}} + \frac {\arcsin \left (a x\right )}{a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.82, size = 108, normalized size = 1.46 \[ \frac {4}{3\,a^2\,c^2\,{\left (1-a^2\,x^2\right )}^{3/2}}-\frac {3}{a^2\,c^2\,\sqrt {1-a^2\,x^2}}-\frac {7\,x}{3\,a\,c^2\,\sqrt {1-a^2\,x^2}}+\frac {4\,x}{3\,a\,c^2\,{\left (1-a^2\,x^2\right )}^{3/2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}{a^3\,c^2} \]
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 \[ \frac {\int \frac {x}{a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{2}}{a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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