Optimal. Leaf size=104 \[ \frac {11 \sin ^{-1}(a x)}{2 a^4 c^2}+\frac {(a x+1)^3}{3 a^4 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {3 (a x+1)^2}{a^4 c^2 \sqrt {1-a^2 x^2}}-\frac {(a x+12) \sqrt {1-a^2 x^2}}{2 a^4 c^2} \]
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Rubi [A] time = 0.30, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6128, 852, 1635, 780, 216} \[ \frac {(a x+1)^3}{3 a^4 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {3 (a x+1)^2}{a^4 c^2 \sqrt {1-a^2 x^2}}-\frac {(a x+12) \sqrt {1-a^2 x^2}}{2 a^4 c^2}+\frac {11 \sin ^{-1}(a x)}{2 a^4 c^2} \]
Antiderivative was successfully verified.
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Rule 216
Rule 780
Rule 852
Rule 1635
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^3}{(c-a c x)^2} \, dx &=c \int \frac {x^3 \sqrt {1-a^2 x^2}}{(c-a c x)^3} \, dx\\ &=\frac {\int \frac {x^3 (c+a c x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^5}\\ &=\frac {(1+a x)^3}{3 a^4 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {(c+a c x)^2 \left (\frac {3}{a^3}+\frac {3 x}{a^2}+\frac {3 x^2}{a}\right )}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^4}\\ &=\frac {(1+a x)^3}{3 a^4 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {3 (1+a x)^2}{a^4 c^2 \sqrt {1-a^2 x^2}}+\frac {\int \frac {\left (\frac {15}{a^3}+\frac {3 x}{a^2}\right ) (c+a c x)}{\sqrt {1-a^2 x^2}} \, dx}{3 c^3}\\ &=\frac {(1+a x)^3}{3 a^4 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {3 (1+a x)^2}{a^4 c^2 \sqrt {1-a^2 x^2}}-\frac {(12+a x) \sqrt {1-a^2 x^2}}{2 a^4 c^2}+\frac {11 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 a^3 c^2}\\ &=\frac {(1+a x)^3}{3 a^4 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {3 (1+a x)^2}{a^4 c^2 \sqrt {1-a^2 x^2}}-\frac {(12+a x) \sqrt {1-a^2 x^2}}{2 a^4 c^2}+\frac {11 \sin ^{-1}(a x)}{2 a^4 c^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 72, normalized size = 0.69 \[ -\frac {\frac {\sqrt {a x+1} \left (3 a^3 x^3+12 a^2 x^2-71 a x+52\right )}{(1-a x)^{3/2}}+66 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{6 a^4 c^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.43, size = 117, normalized size = 1.12 \[ -\frac {52 \, a^{2} x^{2} - 104 \, a x + 66 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 71 \, a x + 52\right )} \sqrt {-a^{2} x^{2} + 1} + 52}{6 \, {\left (a^{6} c^{2} x^{2} - 2 \, a^{5} c^{2} x + a^{4} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 176, normalized size = 1.69 \[ -\frac {4 \, a^{3} c^{6} {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} + 132 \, a^{3} c^{6} \arctan \left (\sqrt {-\frac {2 \, c}{a c x - c} - 1}\right ) - 72 \, a^{3} c^{6} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - \frac {3 \, {\left (7 \, a^{3} c^{6} {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} + 5 \, a^{3} c^{6} \sqrt {-\frac {2 \, c}{a c x - c} - 1}\right )} {\left (a c x - c\right )}^{2}}{c^{2}}}{12 \, a^{6} c^{8} {\left | a \right |} \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\relax (a) \mathrm {sgn}\relax (c)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 164, normalized size = 1.58 \[ -\frac {x \sqrt {-a^{2} x^{2}+1}}{2 c^{2} a^{3}}+\frac {11 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 c^{2} a^{3} \sqrt {a^{2}}}-\frac {3 \sqrt {-a^{2} x^{2}+1}}{c^{2} a^{4}}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 c^{2} a^{6} \left (x -\frac {1}{a}\right )^{2}}+\frac {19 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 c^{2} a^{5} \left (x -\frac {1}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 130, normalized size = 1.25 \[ \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{6} c^{2} x^{2} - 2 \, a^{5} c^{2} x + a^{4} c^{2}\right )}} + \frac {19 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{5} c^{2} x - a^{4} c^{2}\right )}} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{2 \, a^{3} c^{2}} + \frac {11 \, \arcsin \left (a x\right )}{2 \, a^{4} c^{2}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 166, normalized size = 1.60 \[ \frac {2\,\sqrt {1-a^2\,x^2}}{3\,\left (a^6\,c^2\,x^2-2\,a^5\,c^2\,x+a^4\,c^2\right )}+\frac {19\,\sqrt {1-a^2\,x^2}}{3\,\left (a^2\,c^2\,\sqrt {-a^2}-a^3\,c^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {3\,\sqrt {1-a^2\,x^2}}{a^4\,c^2}-\frac {x\,\sqrt {1-a^2\,x^2}}{2\,a^3\,c^2}+\frac {11\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^3\,c^2\,\sqrt {-a^2}} \]
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 \[ \frac {\int \frac {x^{3}}{a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{4}}{a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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