Optimal. Leaf size=74 \[ \frac {\sin ^{-1}(a x)}{a^4 c^2}+\frac {x^2 (a x+1)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {3 a x+2}{3 a^4 c^2 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.10, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6148, 819, 778, 216} \[ \frac {x^2 (a x+1)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {3 a x+2}{3 a^4 c^2 \sqrt {1-a^2 x^2}}+\frac {\sin ^{-1}(a x)}{a^4 c^2} \]
Antiderivative was successfully verified.
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Rule 216
Rule 778
Rule 819
Rule 6148
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {x^3 (1+a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac {x^2 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {x (2+3 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 a^2 c^2}\\ &=\frac {x^2 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {2+3 a x}{3 a^4 c^2 \sqrt {1-a^2 x^2}}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^3 c^2}\\ &=\frac {x^2 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {2+3 a x}{3 a^4 c^2 \sqrt {1-a^2 x^2}}+\frac {\sin ^{-1}(a x)}{a^4 c^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 69, normalized size = 0.93 \[ \frac {-4 a^2 x^2+3 (a x-1) \sqrt {1-a^2 x^2} \sin ^{-1}(a x)+a x+2}{3 a^4 c^2 (a x-1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 137, normalized size = 1.85 \[ -\frac {2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 2 \, a x + 6 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (4 \, a^{2} x^{2} - a x - 2\right )} \sqrt {-a^{2} x^{2} + 1} + 2}{3 \, {\left (a^{7} c^{2} x^{3} - a^{6} c^{2} x^{2} - a^{5} c^{2} x + a^{4} c^{2}\right )}} \]
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 [B] time = 0.04, size = 160, normalized size = 2.16 \[ \frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c^{2} a^{3} \sqrt {a^{2}}}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{6 c^{2} a^{6} \left (x -\frac {1}{a}\right )^{2}}+\frac {13 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{12 c^{2} a^{5} \left (x -\frac {1}{a}\right )}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 c^{2} a^{5} \left (x +\frac {1}{a}\right )} \]
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 \[ a \int \frac {x^{4}}{{\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}\,{d x} + \frac {3 \, a^{2} x^{2} - 2}{3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{4} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 176, normalized size = 2.38 \[ \frac {\sqrt {1-a^2\,x^2}}{6\,\left (a^6\,c^2\,x^2-2\,a^5\,c^2\,x+a^4\,c^2\right )}-\frac {\sqrt {1-a^2\,x^2}}{4\,\left (a^2\,c^2\,\sqrt {-a^2}+a^3\,c^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {13\,\sqrt {1-a^2\,x^2}}{12\,\left (a^2\,c^2\,\sqrt {-a^2}-a^3\,c^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^3\,c^2\,\sqrt {-a^2}} \]
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 \[ \frac {\int \frac {x^{3}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{4}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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