Optimal. Leaf size=107 \[ -\frac {\sin ^{-1}(a x)}{a^3 c^3}-\frac {3 \left (1-a^2 x^2\right )^{3/2}}{5 a^3 c^3 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a^3 c^3 (1-a x)^4}+\frac {2 \sqrt {1-a^2 x^2}}{a^3 c^3 (1-a x)} \]
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Rubi [A] time = 0.22, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6128, 1637, 659, 651, 663, 216} \[ -\frac {3 \left (1-a^2 x^2\right )^{3/2}}{5 a^3 c^3 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a^3 c^3 (1-a x)^4}+\frac {2 \sqrt {1-a^2 x^2}}{a^3 c^3 (1-a x)}-\frac {\sin ^{-1}(a x)}{a^3 c^3} \]
Antiderivative was successfully verified.
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Rule 216
Rule 651
Rule 659
Rule 663
Rule 1637
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^2}{(c-a c x)^3} \, dx &=c \int \frac {x^2 \sqrt {1-a^2 x^2}}{(c-a c x)^4} \, dx\\ &=c \int \left (\frac {\sqrt {1-a^2 x^2}}{a^2 c^4 (-1+a x)^4}+\frac {2 \sqrt {1-a^2 x^2}}{a^2 c^4 (-1+a x)^3}+\frac {\sqrt {1-a^2 x^2}}{a^2 c^4 (-1+a x)^2}\right ) \, dx\\ &=\frac {\int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^4} \, dx}{a^2 c^3}+\frac {\int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^2} \, dx}{a^2 c^3}+\frac {2 \int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^3} \, dx}{a^2 c^3}\\ &=\frac {2 \sqrt {1-a^2 x^2}}{a^3 c^3 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a^3 c^3 (1-a x)^4}-\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a^3 c^3 (1-a x)^3}-\frac {\int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^3} \, dx}{5 a^2 c^3}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^2 c^3}\\ &=\frac {2 \sqrt {1-a^2 x^2}}{a^3 c^3 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a^3 c^3 (1-a x)^4}-\frac {3 \left (1-a^2 x^2\right )^{3/2}}{5 a^3 c^3 (1-a x)^3}-\frac {\sin ^{-1}(a x)}{a^3 c^3}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 77, normalized size = 0.72 \[ \frac {\sqrt {a x+1} \left (-a^2 x^2+3 a x+4\right )+20 \sqrt {2} (a x-1) \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};\frac {1}{2} (1-a x)\right )}{15 a^3 c^3 (1-a x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.49, size = 138, normalized size = 1.29 \[ \frac {8 \, a^{3} x^{3} - 24 \, a^{2} x^{2} + 24 \, a x + 10 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (13 \, a^{2} x^{2} - 19 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} - 8}{5 \, {\left (a^{6} c^{3} x^{3} - 3 \, a^{5} c^{3} x^{2} + 3 \, a^{4} c^{3} x - a^{3} c^{3}\right )}} \]
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 = 0.05, size = 167, normalized size = 1.56 \[ -\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c^{3} a^{2} \sqrt {a^{2}}}-\frac {7 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 c^{3} a^{5} \left (x -\frac {1}{a}\right )^{2}}-\frac {13 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 c^{3} a^{4} \left (x -\frac {1}{a}\right )}-\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 c^{3} a^{6} \left (x -\frac {1}{a}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 144, normalized size = 1.35 \[ -\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{5 \, {\left (a^{6} c^{3} x^{3} - 3 \, a^{5} c^{3} x^{2} + 3 \, a^{4} c^{3} x - a^{3} c^{3}\right )}} - \frac {7 \, \sqrt {-a^{2} x^{2} + 1}}{5 \, {\left (a^{5} c^{3} x^{2} - 2 \, a^{4} c^{3} x + a^{3} c^{3}\right )}} - \frac {13 \, \sqrt {-a^{2} x^{2} + 1}}{5 \, {\left (a^{4} c^{3} x - a^{3} c^{3}\right )}} - \frac {\arcsin \left (a x\right )}{a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.82, size = 259, normalized size = 2.42 \[ \frac {4\,a^2\,\sqrt {1-a^2\,x^2}}{15\,\left (a^7\,c^3\,x^2-2\,a^6\,c^3\,x+a^5\,c^3\right )}-\frac {13\,\sqrt {1-a^2\,x^2}}{5\,\left (a\,c^3\,\sqrt {-a^2}-a^2\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {5\,\sqrt {1-a^2\,x^2}}{3\,\left (a^5\,c^3\,x^2-2\,a^4\,c^3\,x+a^3\,c^3\right )}-\frac {2\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (a\,c^3\,\sqrt {-a^2}+3\,a^3\,c^3\,x^2\,\sqrt {-a^2}-a^4\,c^3\,x^3\,\sqrt {-a^2}-3\,a^2\,c^3\,x\,\sqrt {-a^2}\right )}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^2\,c^3\,\sqrt {-a^2}} \]
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 \[ - \frac {\int \frac {x^{2}}{a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{3}}{a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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