Optimal. Leaf size=63 \[ \frac {c (1-a x) \sqrt {1-a^2 x^2}}{2 a}+\frac {3 c \sqrt {1-a^2 x^2}}{2 a}+\frac {3 c \sin ^{-1}(a x)}{2 a} \]
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Rubi [A] time = 0.04, antiderivative size = 63, 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, 671, 641, 216} \[ \frac {c (1-a x) \sqrt {1-a^2 x^2}}{2 a}+\frac {3 c \sqrt {1-a^2 x^2}}{2 a}+\frac {3 c \sin ^{-1}(a x)}{2 a} \]
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) \, dx &=\frac {\int \frac {(c-a c x)^2}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {c (1-a x) \sqrt {1-a^2 x^2}}{2 a}+\frac {3}{2} \int \frac {c-a c x}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3 c \sqrt {1-a^2 x^2}}{2 a}+\frac {c (1-a x) \sqrt {1-a^2 x^2}}{2 a}+\frac {1}{2} (3 c) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3 c \sqrt {1-a^2 x^2}}{2 a}+\frac {c (1-a x) \sqrt {1-a^2 x^2}}{2 a}+\frac {3 c \sin ^{-1}(a x)}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 61, normalized size = 0.97 \[ \frac {c \left (\frac {\sqrt {a x+1} \left (a^2 x^2-5 a x+4\right )}{\sqrt {1-a x}}-6 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{2 a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.43, size = 52, normalized size = 0.83 \[ -\frac {6 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a c x - 4 \, c\right )}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 38, normalized size = 0.60 \[ \frac {3 \, c \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, {\left | a \right |}} - \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} {\left (c x - \frac {4 \, c}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 114, normalized size = 1.81 \[ -\frac {c x \sqrt {-a^{2} x^{2}+1}}{2}-\frac {c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}+\frac {2 c \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{a}+\frac {2 c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 45, normalized size = 0.71 \[ -\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} c x + \frac {3 \, c \arcsin \left (a x\right )}{2 \, a} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.80, size = 59, normalized size = 0.94 \[ \frac {\frac {3\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2}-\sqrt {1-a^2\,x^2}\,\left (\frac {2\,a\,c}{\sqrt {-a^2}}+\frac {c\,x\,\sqrt {-a^2}}{2}\right )}{\sqrt {-a^2}} \]
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 \[ - c \left (\int \left (- \frac {\sqrt {- a^{2} x^{2} + 1}}{a x + 1}\right )\, dx + \int \frac {a x \sqrt {- a^{2} x^{2} + 1}}{a x + 1}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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