Optimal. Leaf size=41 \[ -\frac {2 (1-a x)}{a c \sqrt {1-a^2 x^2}}-\frac {\sin ^{-1}(a x)}{a c} \]
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Rubi [A] time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6127, 653, 216} \[ -\frac {2 (1-a x)}{a c \sqrt {1-a^2 x^2}}-\frac {\sin ^{-1}(a x)}{a c} \]
Antiderivative was successfully verified.
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Rule 216
Rule 653
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{c-a c x} \, dx &=\frac {\int \frac {(c-a c x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=-\frac {2 (1-a x)}{a c \sqrt {1-a^2 x^2}}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {2 (1-a x)}{a c \sqrt {1-a^2 x^2}}-\frac {\sin ^{-1}(a x)}{a c}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 59, normalized size = 1.44 \[ \frac {2 \left (\sqrt {1-a^2 x^2} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )+a x-1\right )}{a c \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.46, size = 60, normalized size = 1.46 \[ -\frac {2 \, {\left (a x - {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} + 1\right )}}{a^{2} c x + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 53, normalized size = 1.29 \[ -\frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{c {\left | a \right |}} + \frac {4}{c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 292, normalized size = 7.12 \[ -\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{24 c a}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}\, x}{16 c}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{16 c \sqrt {a^{2}}}-\frac {3 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{4 c \,a^{3} \left (x +\frac {1}{a}\right )^{2}}-\frac {17 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{24 c a}-\frac {17 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x}{16 c}-\frac {17 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{16 c \sqrt {a^{2}}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{2 c \,a^{4} \left (x +\frac {1}{a}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a c x - c\right )} {\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 70, normalized size = 1.71 \[ \frac {2\,\sqrt {1-a^2\,x^2}}{c\,\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c\,\sqrt {-a^2}} \]
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 \[ - \frac {\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{4} x^{4} + 2 a^{3} x^{3} - 2 a x - 1}\, dx + \int \left (- \frac {a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{4} x^{4} + 2 a^{3} x^{3} - 2 a x - 1}\right )\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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