Optimal. Leaf size=46 \[ \frac {(1-a n x) e^{n \tanh ^{-1}(a x)}}{a^2 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {6144} \[ \frac {(1-a n x) e^{n \tanh ^{-1}(a x)}}{a^2 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6144
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac {e^{n \tanh ^{-1}(a x)} (1-a n x)}{a^2 c \left (1-n^2\right ) \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 81, normalized size = 1.76 \[ \frac {\sqrt {1-a^2 x^2} (1-a x)^{-\frac {n}{2}-\frac {1}{2}} (a x+1)^{\frac {n-1}{2}} (a n x-1)}{a^2 c (n-1) (n+1) \sqrt {c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.59, size = 82, normalized size = 1.78 \[ \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a n x - 1\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c^{2} n^{2} - a^{2} c^{2} - {\left (a^{4} c^{2} n^{2} - a^{4} c^{2}\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 49, normalized size = 1.07 \[ -\frac {\left (a x -1\right ) \left (a x +1\right ) \left (n a x -1\right ) {\mathrm e}^{n \arctanh \left (a x \right )}}{a^{2} \left (n^{2}-1\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}} \]
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 \[ \int \frac {x \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 68, normalized size = 1.48 \[ -\frac {{\mathrm {e}}^{\frac {n\,\ln \left (a\,x+1\right )}{2}-\frac {n\,\ln \left (1-a\,x\right )}{2}}\,\left (\frac {1}{a^2\,c\,\left (n^2-1\right )}-\frac {n\,x}{a\,c\,\left (n^2-1\right )}\right )}{\sqrt {c-a^2\,c\,x^2}} \]
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 \[ \int \frac {x e^{n \operatorname {atanh}{\left (a x \right )}}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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