Optimal. Leaf size=101 \[ \frac {c^2 (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}+\frac {5 c^2 (1-a x) \sqrt {1-a^2 x^2}}{6 a}+\frac {5 c^2 \sqrt {1-a^2 x^2}}{2 a}+\frac {5 c^2 \sin ^{-1}(a x)}{2 a} \]
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Rubi [A] time = 0.07, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6127, 671, 641, 216} \[ \frac {c^2 (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}+\frac {5 c^2 (1-a x) \sqrt {1-a^2 x^2}}{6 a}+\frac {5 c^2 \sqrt {1-a^2 x^2}}{2 a}+\frac {5 c^2 \sin ^{-1}(a x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 671
Rule 6127
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} (c-a c x)^2 \, dx &=\frac {\int \frac {(c-a c x)^3}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {c^2 (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}+\frac {5}{3} \int \frac {(c-a c x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {5 c^2 (1-a x) \sqrt {1-a^2 x^2}}{6 a}+\frac {c^2 (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}+\frac {1}{2} (5 c) \int \frac {c-a c x}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {5 c^2 \sqrt {1-a^2 x^2}}{2 a}+\frac {5 c^2 (1-a x) \sqrt {1-a^2 x^2}}{6 a}+\frac {c^2 (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}+\frac {1}{2} \left (5 c^2\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {5 c^2 \sqrt {1-a^2 x^2}}{2 a}+\frac {5 c^2 (1-a x) \sqrt {1-a^2 x^2}}{6 a}+\frac {c^2 (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}+\frac {5 c^2 \sin ^{-1}(a x)}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 72, normalized size = 0.71 \[ \frac {c^2 \left (\frac {\sqrt {a x+1} \left (-2 a^3 x^3+11 a^2 x^2-31 a x+22\right )}{\sqrt {1-a x}}-30 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{6 a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.52, size = 71, normalized size = 0.70 \[ -\frac {30 \, c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (2 \, a^{2} c^{2} x^{2} - 9 \, a c^{2} x + 22 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.76, size = 54, normalized size = 0.53 \[ \frac {5 \, c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, {\left | a \right |}} + \frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a c^{2} x - 9 \, c^{2}\right )} x + \frac {22 \, c^{2}}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 142, normalized size = 1.41 \[ -\frac {c^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 a}-\frac {3 c^{2} x \sqrt {-a^{2} x^{2}+1}}{2}-\frac {3 c^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}+\frac {4 c^{2} \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{a}+\frac {4 c^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 71, normalized size = 0.70 \[ -\frac {3}{2} \, \sqrt {-a^{2} x^{2} + 1} c^{2} x - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}{3 \, a} + \frac {5 \, c^{2} \arcsin \left (a x\right )}{2 \, a} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 82, normalized size = 0.81 \[ \frac {5\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}-\frac {3\,c^2\,x\,\sqrt {1-a^2\,x^2}}{2}+\frac {11\,c^2\,\sqrt {1-a^2\,x^2}}{3\,a}+\frac {a\,c^2\,x^2\,\sqrt {1-a^2\,x^2}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ c^{2} \left (\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a x + 1}\, dx + \int \left (- \frac {2 a x \sqrt {- a^{2} x^{2} + 1}}{a x + 1}\right )\, dx + \int \frac {a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a x + 1}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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