Optimal. Leaf size=118 \[ \frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-\frac {3}{2}-p} \left (1-\frac {1}{a x}\right )^{\frac {3}{2}+p} \left (1+\frac {1}{a x}\right )^{-\frac {1}{2}+p} x \left (c-a^2 c x^2\right )^p \, _2F_1\left (-1-2 p,-\frac {3}{2}-p;-2 p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{1+2 p} \]
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Rubi [A]
time = 0.10, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6327, 6331,
134} \begin {gather*} \frac {x \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-p-\frac {3}{2}} \left (1-\frac {1}{a x}\right )^{p+\frac {3}{2}} \left (\frac {1}{a x}+1\right )^{p-\frac {1}{2}} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-2 p-1,-p-\frac {3}{2};-2 p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{2 p+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 134
Rule 6327
Rule 6331
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx &=\left (\left (1-\frac {1}{a^2 x^2}\right )^{-p} x^{-2 p} \left (c-a^2 c x^2\right )^p\right ) \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^p x^{2 p} \, dx\\ &=-\left (\left (\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {1}{x}\right )^{2 p} \left (c-a^2 c x^2\right )^p\right ) \text {Subst}\left (\int x^{-2-2 p} \left (1-\frac {x}{a}\right )^{\frac {3}{2}+p} \left (1+\frac {x}{a}\right )^{-\frac {3}{2}+p} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-\frac {3}{2}-p} \left (1-\frac {1}{a x}\right )^{\frac {3}{2}+p} \left (1+\frac {1}{a x}\right )^{-\frac {1}{2}+p} x \left (c-a^2 c x^2\right )^p \, _2F_1\left (-1-2 p,-\frac {3}{2}-p;-2 p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{1+2 p}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 119, normalized size = 1.01 \begin {gather*} \frac {4^{1+p} e^{-\coth ^{-1}(a x)} \left (1-e^{2 \coth ^{-1}(a x)}\right )^{2 p} \left (\frac {e^{\coth ^{-1}(a x)}}{-1+e^{2 \coth ^{-1}(a x)}}\right )^{2 p} \left (a \sqrt {1-\frac {1}{a^2 x^2}} x\right )^{-2 p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-\frac {1}{2}+p,2+2 p;\frac {1}{2}+p;e^{2 \coth ^{-1}(a x)}\right )}{a-2 a p} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (-a^{2} c \,x^{2}+c \right )^{p} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (c-a^2\,c\,x^2\right )}^p\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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