Optimal. Leaf size=104 \[ -\frac {6 \sqrt {1-a^2 x^2}}{a^3 c^2 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^3 c^2 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 c^2 (1-a x)^2}+\frac {3 \text {ArcSin}(a x)}{a^3 c^2} \]
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Rubi [A]
time = 0.13, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6263, 1653,
807, 677, 222} \begin {gather*} \frac {3 \text {ArcSin}(a x)}{a^3 c^2}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 c^2 (1-a x)^2}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^3 c^2 (1-a x)^3}-\frac {6 \sqrt {1-a^2 x^2}}{a^3 c^2 (1-a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 677
Rule 807
Rule 1653
Rule 6263
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^2}{(c-a c x)^2} \, dx &=c \int \frac {x^2 \sqrt {1-a^2 x^2}}{(c-a c x)^3} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 c^2 (1-a x)^2}-\frac {\int \frac {\left (2 a^2 c^2-3 a^3 c^2 x\right ) \sqrt {1-a^2 x^2}}{(c-a c x)^3} \, dx}{a^4 c}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^3 c^2 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 c^2 (1-a x)^2}-\frac {3 \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^2} \, dx}{a^2}\\ &=-\frac {6 \sqrt {1-a^2 x^2}}{a^3 c^2 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^3 c^2 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 c^2 (1-a x)^2}+\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^2 c^2}\\ &=-\frac {6 \sqrt {1-a^2 x^2}}{a^3 c^2 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^3 c^2 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 c^2 (1-a x)^2}+\frac {3 \sin ^{-1}(a x)}{a^3 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 64, normalized size = 0.62 \begin {gather*} \frac {\frac {\sqrt {1+a x} \left (-14+19 a x-3 a^2 x^2\right )}{(1-a x)^{3/2}}-18 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{3 a^3 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.77, size = 179, normalized size = 1.72
method | result | size |
risch | \(\frac {a^{2} x^{2}-1}{a^{3} \sqrt {-a^{2} x^{2}+1}\, c^{2}}+\frac {\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a^{5} \left (x -\frac {1}{a}\right )^{2}}+\frac {13 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a^{4} \left (x -\frac {1}{a}\right )}}{c^{2}}\) | \(147\) |
default | \(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{3}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}+\frac {\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a^{4}}+\frac {5 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{4} \left (x -\frac {1}{a}\right )}}{c^{2}}\) | \(179\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 109, normalized size = 1.05 \begin {gather*} \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{5} c^{2} x^{2} - 2 \, a^{4} c^{2} x + a^{3} c^{2}\right )}} + \frac {13 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{4} c^{2} x - a^{3} c^{2}\right )}} + \frac {3 \, \arcsin \left (a x\right )}{a^{3} c^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{3} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 109, normalized size = 1.05 \begin {gather*} -\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^{5} c^{2} x^{2} - 2 \, a^{4} c^{2} x + a^{3} c^{2}\right )}} \end {gather*}
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 \begin {gather*} \frac {\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}{c^{2}} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.81, size = 143, normalized size = 1.38 \begin {gather*} \frac {2\,\sqrt {1-a^2\,x^2}}{3\,\left (a^5\,c^2\,x^2-2\,a^4\,c^2\,x+a^3\,c^2\right )}+\frac {13\,\sqrt {1-a^2\,x^2}}{3\,\left (a\,c^2\,\sqrt {-a^2}-a^2\,c^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a^3\,c^2}+\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^2\,c^2\,\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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