Optimal. Leaf size=61 \[ \frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {1}{2} c^2 x \sqrt {1-a^2 x^2}+\frac {c^2 \sin ^{-1}(a x)}{2 a} \]
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Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6127, 641, 195, 216} \[ \frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {1}{2} c^2 x \sqrt {1-a^2 x^2}+\frac {c^2 \sin ^{-1}(a x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 641
Rule 6127
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} (c-a c x)^2 \, dx &=c \int (c-a c x) \sqrt {1-a^2 x^2} \, dx\\ &=\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+c^2 \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {1}{2} c^2 x \sqrt {1-a^2 x^2}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {1}{2} c^2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {1}{2} c^2 x \sqrt {1-a^2 x^2}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {c^2 \sin ^{-1}(a x)}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 59, normalized size = 0.97 \[ -\frac {c^2 \left (\sqrt {1-a^2 x^2} \left (2 a^2 x^2-3 a x-2\right )+6 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{6 a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.17, size = 70, normalized size = 1.15 \[ -\frac {6 \, c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{2} c^{2} x^{2} - 3 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 54, normalized size = 0.89 \[ \frac {c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, {\left | a \right |}} - \frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a c^{2} x - 3 \, c^{2}\right )} x - \frac {2 \, c^{2}}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 91, normalized size = 1.49 \[ -\frac {c^{2} a \,x^{2} \sqrt {-a^{2} x^{2}+1}}{3}+\frac {c^{2} \sqrt {-a^{2} x^{2}+1}}{3 a}+\frac {c^{2} x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 72, normalized size = 1.18 \[ -\frac {1}{3} \, \sqrt {-a^{2} x^{2} + 1} a c^{2} x^{2} + \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} c^{2} x + \frac {c^{2} \arcsin \left (a x\right )}{2 \, a} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 82, normalized size = 1.34 \[ \frac {c^2\,x\,\sqrt {1-a^2\,x^2}}{2}+\frac {c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}+\frac {c^2\,\sqrt {1-a^2\,x^2}}{3\,a}-\frac {a\,c^2\,x^2\,\sqrt {1-a^2\,x^2}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.38, size = 102, normalized size = 1.67 \[ \begin {cases} \frac {c^{2} \sqrt {- a^{2} x^{2} + 1} - c^{2} \left (\begin {cases} - \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\operatorname {asin}{\left (a x \right )}}{2} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + c^{2} \left (\begin {cases} \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} - \sqrt {- a^{2} x^{2} + 1} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + c^{2} \operatorname {asin}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\c^{2} x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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