Optimal. Leaf size=95 \[ \frac {3 \sin ^{-1}(a x)}{a^3 c}+\frac {(a x+1)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (a x+1)^2}{a^3 c \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{a^3 c} \]
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Rubi [A] time = 0.19, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6148, 1635, 21, 669, 641, 216} \[ \frac {(a x+1)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (a x+1)^2}{a^3 c \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{a^3 c}+\frac {3 \sin ^{-1}(a x)}{a^3 c} \]
Antiderivative was successfully verified.
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Rule 21
Rule 216
Rule 641
Rule 669
Rule 1635
Rule 6148
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)} x^2}{c-a^2 c x^2} \, dx &=\frac {\int \frac {x^2 (1+a x)^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c}\\ &=\frac {(1+a x)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {\left (\frac {3}{a^2}+\frac {3 x}{a}\right ) (1+a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c}\\ &=\frac {(1+a x)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {(1+a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a^2 c}\\ &=\frac {(1+a x)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1+a x)^2}{a^3 c \sqrt {1-a^2 x^2}}+\frac {3 \int \frac {1+a x}{\sqrt {1-a^2 x^2}} \, dx}{a^2 c}\\ &=\frac {(1+a x)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1+a x)^2}{a^3 c \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{a^3 c}+\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^2 c}\\ &=\frac {(1+a x)^3}{3 a^3 c \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (1+a x)^2}{a^3 c \sqrt {1-a^2 x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{a^3 c}+\frac {3 \sin ^{-1}(a x)}{a^3 c}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 78, normalized size = 0.82 \[ \frac {3 a^3 x^3-16 a^2 x^2+9 (a x-1) \sqrt {1-a^2 x^2} \sin ^{-1}(a x)-5 a x+14}{3 a^3 c (a x-1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 103, normalized size = 1.08 \[ -\frac {14 \, a^{2} x^{2} - 28 \, a x + 18 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{2} x^{2} - 19 \, a x + 14\right )} \sqrt {-a^{2} x^{2} + 1} + 14}{3 \, {\left (a^{5} c x^{2} - 2 \, a^{4} c x + a^{3} c\right )}} \]
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 = 178, normalized size = 1.87 \[ \frac {x^{2}}{c a \sqrt {-a^{2} x^{2}+1}}-\frac {6}{c \,a^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {7 x}{c \,a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c \,a^{2} \sqrt {a^{2}}}-\frac {4}{3 c \,a^{4} \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {8 x}{3 c \,a^{2} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 364, normalized size = 3.83 \[ \frac {1}{6} \, {\left (\frac {a^{3} c^{3}}{\sqrt {-a^{2} x^{2} + 1} a^{8} c^{4} x + \sqrt {-a^{2} x^{2} + 1} a^{7} c^{4}} - \frac {a^{3} c^{3}}{\sqrt {-a^{2} x^{2} + 1} a^{8} c^{4} x - \sqrt {-a^{2} x^{2} + 1} a^{7} c^{4}} + \frac {3 \, a c}{\sqrt {-a^{2} x^{2} + 1} a^{6} c^{2} x + \sqrt {-a^{2} x^{2} + 1} a^{5} c^{2}} - \frac {3 \, a c}{\sqrt {-a^{2} x^{2} + 1} a^{6} c^{2} x - \sqrt {-a^{2} x^{2} + 1} a^{5} c^{2}} - \frac {4 \, c}{\sqrt {-a^{2} x^{2} + 1} a^{5} c^{2} x + \sqrt {-a^{2} x^{2} + 1} a^{4} c^{2}} - \frac {4 \, c}{\sqrt {-a^{2} x^{2} + 1} a^{5} c^{2} x - \sqrt {-a^{2} x^{2} + 1} a^{4} c^{2}} + \frac {6 \, x^{2}}{\sqrt {-a^{2} x^{2} + 1} a^{2} c} - \frac {26 \, x}{\sqrt {-a^{2} x^{2} + 1} a^{3} c} + \frac {18 \, \arcsin \left (a x\right )}{a^{4} c} - \frac {36}{\sqrt {-a^{2} x^{2} + 1} a^{4} c}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 133, normalized size = 1.40 \[ \frac {2\,\sqrt {1-a^2\,x^2}}{3\,\left (c\,a^5\,x^2-2\,c\,a^4\,x+c\,a^3\right )}+\frac {13\,\sqrt {1-a^2\,x^2}}{3\,\left (a\,c\,\sqrt {-a^2}-a^2\,c\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a^3\,c}+\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^2\,c\,\sqrt {-a^2}} \]
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 \[ \frac {\int \frac {x^{2}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a x^{3}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{4}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{5}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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