Optimal. Leaf size=65 \[ -\frac {(1-a x)^2}{a c \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2}}{a c}-\frac {2 \sin ^{-1}(a x)}{a c} \]
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Rubi [A] time = 0.10, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6131, 6128, 789, 641, 216} \[ -\frac {(1-a x)^2}{a c \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2}}{a c}-\frac {2 \sin ^{-1}(a x)}{a c} \]
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 789
Rule 6128
Rule 6131
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx &=-\frac {a \int \frac {e^{-3 \tanh ^{-1}(a x)} x}{1-a x} \, dx}{c}\\ &=-\frac {a \int \frac {x (1-a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c}\\ &=-\frac {(1-a x)^2}{a c \sqrt {1-a^2 x^2}}-\frac {2 \int \frac {1-a x}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {(1-a x)^2}{a c \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2}}{a c}-\frac {2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=-\frac {(1-a x)^2}{a c \sqrt {1-a^2 x^2}}-\frac {2 \sqrt {1-a^2 x^2}}{a c}-\frac {2 \sin ^{-1}(a x)}{a c}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 67, normalized size = 1.03 \[ \frac {a^2 x^2+4 \sqrt {1-a^2 x^2} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )+2 a x-3}{a c \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.47, size = 67, normalized size = 1.03 \[ -\frac {3 \, a x - 4 \, {\left (a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (a x + 3\right )} + 3}{a^{2} c x + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.82, size = 73, normalized size = 1.12 \[ -\frac {2 \, \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{c {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a c} + \frac {4}{c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 292, normalized size = 4.49 \[ \frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{24 a c}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}\, x}{16 c}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{16 c \sqrt {a^{2}}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{2 a^{4} c \left (x +\frac {1}{a}\right )^{3}}-\frac {5 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{4 c \,a^{3} \left (x +\frac {1}{a}\right )^{2}}-\frac {31 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{24 c a}-\frac {31 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x}{16 c}-\frac {31 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{16 c \sqrt {a^{2}}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a x}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 90, normalized size = 1.38 \[ \frac {2\,\sqrt {1-a^2\,x^2}}{c\,\left (x\,\sqrt {-a^2}+\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a\,c}-\frac {2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c\,\sqrt {-a^2}} \]
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 \[ \frac {a \left (\int \frac {x \sqrt {- a^{2} x^{2} + 1}}{a^{4} x^{4} + 2 a^{3} x^{3} - 2 a x - 1}\, dx + \int \left (- \frac {a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1}}{a^{4} x^{4} + 2 a^{3} x^{3} - 2 a x - 1}\right )\, dx\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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