Optimal. Leaf size=93 \[ \frac {c \sqrt {1-a^2 x^2} (1-a x)^2}{3 a}+\frac {5 c \sqrt {1-a^2 x^2} (1-a x)}{6 a}+\frac {5 c \sqrt {1-a^2 x^2}}{2 a}+\frac {5 c \sin ^{-1}(a x)}{2 a} \]
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Rubi [A] time = 0.06, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6139, 671, 641, 216} \[ \frac {c \sqrt {1-a^2 x^2} (1-a x)^2}{3 a}+\frac {5 c \sqrt {1-a^2 x^2} (1-a x)}{6 a}+\frac {5 c \sqrt {1-a^2 x^2}}{2 a}+\frac {5 c \sin ^{-1}(a x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 671
Rule 6139
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx &=c \int \frac {(1-a x)^3}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {c (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}+\frac {1}{3} (5 c) \int \frac {(1-a x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {5 c (1-a x) \sqrt {1-a^2 x^2}}{6 a}+\frac {c (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}+\frac {1}{2} (5 c) \int \frac {1-a x}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {5 c \sqrt {1-a^2 x^2}}{2 a}+\frac {5 c (1-a x) \sqrt {1-a^2 x^2}}{6 a}+\frac {c (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}+\frac {1}{2} (5 c) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {5 c \sqrt {1-a^2 x^2}}{2 a}+\frac {5 c (1-a x) \sqrt {1-a^2 x^2}}{6 a}+\frac {c (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}+\frac {5 c \sin ^{-1}(a x)}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 70, normalized size = 0.75 \[ \frac {c \left (\frac {\sqrt {a x+1} \left (-2 a^3 x^3+11 a^2 x^2-31 a x+22\right )}{\sqrt {1-a x}}-30 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{6 a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.61, size = 63, normalized size = 0.68 \[ -\frac {30 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (2 \, a^{2} c x^{2} - 9 \, a c x + 22 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 46, normalized size = 0.49 \[ \frac {5 \, c \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, {\left | a \right |}} + \frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a c x - 9 \, c\right )} x + \frac {22 \, c}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 133, normalized size = 1.43 \[ \frac {2 c \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{3} \left (x +\frac {1}{a}\right )^{2}}+\frac {5 c \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3 a}+\frac {5 c \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x}{2}+\frac {5 c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.50, size = 122, normalized size = 1.31 \[ -\frac {1}{2} \, \sqrt {a^{2} x^{2} + 4 \, a x + 3} c x + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{a^{2} x + a} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{3 \, a} + \frac {i \, c \arcsin \left (a x + 2\right )}{2 \, a} + \frac {3 \, c \arcsin \left (a x\right )}{a} - \frac {\sqrt {a^{2} x^{2} + 4 \, a x + 3} c}{a} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} c}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 74, normalized size = 0.80 \[ \frac {11\,c\,\sqrt {1-a^2\,x^2}}{3\,a}-\frac {3\,c\,x\,\sqrt {1-a^2\,x^2}}{2}+\frac {5\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}+\frac {a\,c\,x^2\,\sqrt {1-a^2\,x^2}}{3} \]
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 \[ c \left (\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \left (- \frac {2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx + \int \frac {a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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