Optimal. Leaf size=91 \[ -\frac {2 c (1-a x)^3}{a \sqrt {1-a^2 x^2}}-\frac {5 c \sqrt {1-a^2 x^2} (1-a x)}{2 a}-\frac {15 c \sqrt {1-a^2 x^2}}{2 a}-\frac {15 c \sin ^{-1}(a x)}{2 a} \]
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Rubi [A] time = 0.06, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6127, 669, 671, 641, 216} \[ -\frac {2 c (1-a x)^3}{a \sqrt {1-a^2 x^2}}-\frac {5 c \sqrt {1-a^2 x^2} (1-a x)}{2 a}-\frac {15 c \sqrt {1-a^2 x^2}}{2 a}-\frac {15 c \sin ^{-1}(a x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 669
Rule 671
Rule 6127
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} (c-a c x) \, dx &=\frac {\int \frac {(c-a c x)^4}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=-\frac {2 c (1-a x)^3}{a \sqrt {1-a^2 x^2}}-\frac {5 \int \frac {(c-a c x)^2}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {2 c (1-a x)^3}{a \sqrt {1-a^2 x^2}}-\frac {5 c (1-a x) \sqrt {1-a^2 x^2}}{2 a}-\frac {15}{2} \int \frac {c-a c x}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 c (1-a x)^3}{a \sqrt {1-a^2 x^2}}-\frac {15 c \sqrt {1-a^2 x^2}}{2 a}-\frac {5 c (1-a x) \sqrt {1-a^2 x^2}}{2 a}-\frac {1}{2} (15 c) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 c (1-a x)^3}{a \sqrt {1-a^2 x^2}}-\frac {15 c \sqrt {1-a^2 x^2}}{2 a}-\frac {5 c (1-a x) \sqrt {1-a^2 x^2}}{2 a}-\frac {15 c \sin ^{-1}(a x)}{2 a}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 43, normalized size = 0.47 \[ -\frac {c (1-a x)^{7/2} \, _2F_1\left (\frac {3}{2},\frac {7}{2};\frac {9}{2};\frac {1}{2} (1-a x)\right )}{7 \sqrt {2} a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.45, size = 81, normalized size = 0.89 \[ -\frac {24 \, a c x - 30 \, {\left (a c x + c\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (a^{2} c x^{2} - 7 \, a c x - 24 \, c\right )} \sqrt {-a^{2} x^{2} + 1} + 24 \, c}{2 \, {\left (a^{2} x + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 73, normalized size = 0.80 \[ -\frac {15 \, c \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, {\left | a \right |}} + \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} {\left (c x - \frac {8 \, c}{a}\right )} + \frac {16 \, c}{{\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 169, normalized size = 1.86 \[ -\frac {5 c \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{3} \left (x +\frac {1}{a}\right )^{2}}-\frac {5 c \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{a}-\frac {15 c \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x}{2}-\frac {15 c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}-\frac {2 c \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a^{4} \left (x +\frac {1}{a}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 109, normalized size = 1.20 \[ \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{a^{3} x^{2} + 2 \, a^{2} x + a} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{2 \, {\left (a^{2} x + a\right )}} - \frac {15 \, c \arcsin \left (a x\right )}{2 \, a} - \frac {12 \, \sqrt {-a^{2} x^{2} + 1} c}{a^{2} x + a} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} c}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.80, size = 96, normalized size = 1.05 \[ \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {4\,a\,c}{\sqrt {-a^2}}+\frac {c\,x\,\sqrt {-a^2}}{2}\right )-\frac {15\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2}+\frac {8\,c\,\sqrt {1-a^2\,x^2}}{x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}}}{\sqrt {-a^2}} \]
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 \[ - c \left (\int \left (- \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx + \int \frac {a x \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \frac {a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \left (- \frac {a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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