Optimal. Leaf size=99 \[ -\frac {\left (1-a^2 x^2\right )^{5/2}}{3 a c (1-a x)^4}+\frac {8 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^2}+\frac {4 \sqrt {1-a^2 x^2}}{a c}-\frac {4 \sin ^{-1}(a x)}{a c} \]
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Rubi [A] time = 0.12, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6131, 6128, 793, 663, 665, 216} \[ -\frac {\left (1-a^2 x^2\right )^{5/2}}{3 a c (1-a x)^4}+\frac {8 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^2}+\frac {4 \sqrt {1-a^2 x^2}}{a c}-\frac {4 \sin ^{-1}(a x)}{a c} \]
Antiderivative was successfully verified.
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Rule 216
Rule 663
Rule 665
Rule 793
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 \left (1-a^2 x^2\right )^{3/2}}{(1-a x)^4} \, dx}{c}\\ &=-\frac {\left (1-a^2 x^2\right )^{5/2}}{3 a c (1-a x)^4}+\frac {4 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(1-a x)^3} \, dx}{3 c}\\ &=\frac {8 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^2}-\frac {\left (1-a^2 x^2\right )^{5/2}}{3 a c (1-a x)^4}-\frac {4 \int \frac {\sqrt {1-a^2 x^2}}{1-a x} \, dx}{c}\\ &=\frac {4 \sqrt {1-a^2 x^2}}{a c}+\frac {8 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^2}-\frac {\left (1-a^2 x^2\right )^{5/2}}{3 a c (1-a x)^4}-\frac {4 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c}\\ &=\frac {4 \sqrt {1-a^2 x^2}}{a c}+\frac {8 \left (1-a^2 x^2\right )^{3/2}}{3 a c (1-a x)^2}-\frac {\left (1-a^2 x^2\right )^{5/2}}{3 a c (1-a x)^4}-\frac {4 \sin ^{-1}(a x)}{a c}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 62, normalized size = 0.63 \[ -\frac {16 \sqrt {2} (a x-1) \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};\frac {1}{2} (1-a x)\right )+(a x+1)^{5/2}}{3 a c (1-a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.47, size = 101, normalized size = 1.02 \[ \frac {19 \, a^{2} x^{2} - 38 \, a x + 24 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{2} x^{2} - 26 \, a x + 19\right )} \sqrt {-a^{2} x^{2} + 1} + 19}{3 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 126, normalized size = 1.27 \[ -\frac {4 \, \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{c {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a c} - \frac {8 \, {\left (\frac {9 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} - 4\right )}}{3 \, c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{3} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 168, normalized size = 1.70 \[ -\frac {a \,x^{2}}{c \sqrt {-a^{2} x^{2}+1}}+\frac {9}{c a \sqrt {-a^{2} x^{2}+1}}+\frac {12 x}{c \sqrt {-a^{2} x^{2}+1}}-\frac {4 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c \sqrt {a^{2}}}+\frac {8}{3 c \,a^{2} \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {16 x}{3 c \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a x}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 129, normalized size = 1.30 \[ \frac {\sqrt {1-a^2\,x^2}}{a\,c}-\frac {20\,\sqrt {1-a^2\,x^2}}{3\,\left (\frac {c\,\sqrt {-a^2}}{a}-c\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c\,\sqrt {-a^2}}-\frac {4\,a\,\sqrt {1-a^2\,x^2}}{3\,\left (c\,a^4\,x^2-2\,c\,a^3\,x+c\,a^2\right )} \]
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 \[ \frac {a \left (\int \frac {x}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a x^{2}}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{3}}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{4}}{- a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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