Optimal. Leaf size=74 \[ \frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^3}-\frac {2 \sqrt {1-a^2 x^2}}{a c (1-a x)}+\frac {\sin ^{-1}(a x)}{a c} \]
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Rubi [A] time = 0.06, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6127, 663, 216} \[ \frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^3}-\frac {2 \sqrt {1-a^2 x^2}}{a c (1-a x)}+\frac {\sin ^{-1}(a x)}{a c} \]
Antiderivative was successfully verified.
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Rule 216
Rule 663
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{c-a c x} \, dx &=c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^4} \, dx\\ &=\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^3}-c \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^2} \, dx\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{a c (1-a x)}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^3}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{a c (1-a x)}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^3}+\frac {\sin ^{-1}(a x)}{a c}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 45, normalized size = 0.61 \[ \frac {4 \sqrt {2} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};\frac {1}{2} (1-a x)\right )}{3 a c (1-a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.44, size = 94, normalized size = 1.27 \[ -\frac {2 \, {\left (2 \, a^{2} x^{2} - 4 \, a x + 3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 2 \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x - 1\right )} + 2\right )}}{3 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 79, normalized size = 1.07 \[ \frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{c {\left | a \right |}} + \frac {8 \, {\left (\frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - 1\right )}}{3 \, c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{3} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 146, normalized size = 1.97 \[ -\frac {8 x}{c \sqrt {-a^{2} x^{2}+1}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c \sqrt {a^{2}}}-\frac {4}{c a \sqrt {-a^{2} x^{2}+1}}-\frac {8}{3 c \,a^{2} \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {16 x}{3 c \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 89, normalized size = 1.20 \[ -\frac {8 \, x}{3 \, \sqrt {-a^{2} x^{2} + 1} c} - \frac {8}{3 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{2} c x - \sqrt {-a^{2} x^{2} + 1} a c\right )}} + \frac {\arcsin \left (a x\right )}{a c} - \frac {4}{\sqrt {-a^{2} x^{2} + 1} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 114, normalized size = 1.54 \[ -\frac {4\,a\,\sqrt {1-a^2\,x^2}+3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}-8\,a^2\,x\,\sqrt {1-a^2\,x^2}+3\,a^2\,x^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}-6\,a\,x\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}{3\,a^2\,c\,{\left (a\,x-1\right )}^2} \]
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 \[ - \frac {\int \frac {3 a x}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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