Optimal. Leaf size=51 \[ \frac {x}{3 c \sqrt {c-a^2 c x^2}}+\frac {2 (a x+1)}{3 a \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6141, 653, 191} \[ \frac {x}{3 c \sqrt {c-a^2 c x^2}}+\frac {2 (a x+1)}{3 a \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 653
Rule 6141
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=c \int \frac {(1+a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac {2 (1+a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}}+\frac {1}{3} \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac {2 (1+a x)}{3 a \left (c-a^2 c x^2\right )^{3/2}}+\frac {x}{3 c \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 63, normalized size = 1.24 \[ -\frac {(a x-2) \sqrt {a x+1} \sqrt {1-a^2 x^2}}{3 a c (1-a x)^{3/2} \sqrt {c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.79, size = 47, normalized size = 0.92 \[ -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x - 2\right )}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 148, normalized size = 2.90 \[ -\frac {{\left (a c - 3 \, \sqrt {-a^{2} c} \sqrt {c}\right )} \mathrm {sgn}\relax (x)}{3 \, {\left (a^{2} c^{\frac {5}{2}} - \sqrt {-a^{2} c} a c^{2}\right )}} + \frac {2 \, {\left (2 \, a^{2} c + 3 \, a \sqrt {c} {\left (\sqrt {-a^{2} c + \frac {c}{x^{2}}} - \frac {\sqrt {c}}{x}\right )} + 3 \, {\left (\sqrt {-a^{2} c + \frac {c}{x^{2}}} - \frac {\sqrt {c}}{x}\right )}^{2}\right )}}{3 \, {\left (a \sqrt {c} + \sqrt {-a^{2} c + \frac {c}{x^{2}}} - \frac {\sqrt {c}}{x}\right )}^{3} c \mathrm {sgn}\relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 31, normalized size = 0.61 \[ -\frac {\left (a x -2\right ) \left (a x +1\right )^{2}}{3 \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 196, normalized size = 3.84 \[ \frac {1}{3} \, a {\left (\frac {a}{\sqrt {-a^{2} c x^{2} + c} a^{4} c x + \sqrt {-a^{2} c x^{2} + c} a^{3} c} - \frac {a}{\sqrt {-a^{2} c x^{2} + c} a^{4} c x - \sqrt {-a^{2} c x^{2} + c} a^{3} c} - \frac {1}{\sqrt {-a^{2} c x^{2} + c} a^{3} c x + \sqrt {-a^{2} c x^{2} + c} a^{2} c} - \frac {1}{\sqrt {-a^{2} c x^{2} + c} a^{3} c x - \sqrt {-a^{2} c x^{2} + c} a^{2} c} + \frac {x}{\sqrt {-a^{2} c x^{2} + c} a c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.95, size = 33, normalized size = 0.65 \[ -\frac {\sqrt {c-a^2\,c\,x^2}\,\left (a\,x-2\right )}{3\,a\,c^2\,{\left (a\,x-1\right )}^2} \]
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 \[ - \int \frac {a x}{- a^{3} c x^{3} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt {- a^{2} c x^{2} + c} + a c x \sqrt {- a^{2} c x^{2} + c} - c \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {1}{- a^{3} c x^{3} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt {- a^{2} c x^{2} + c} + a c x \sqrt {- a^{2} c x^{2} + c} - c \sqrt {- a^{2} c x^{2} + c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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