Optimal. Leaf size=118 \[ \frac {x \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-p-\frac {1}{2}} \left (1-\frac {1}{a x}\right )^{p+\frac {1}{2}} \left (\frac {1}{a x}+1\right )^{p+\frac {1}{2}} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-2 p-1,-p-\frac {1}{2};-2 p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{2 p+1} \]
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Rubi [A] time = 0.14, 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 = {6192, 6196, 132} \[ \frac {x \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-p-\frac {1}{2}} \left (1-\frac {1}{a x}\right )^{p+\frac {1}{2}} \left (\frac {1}{a x}+1\right )^{p+\frac {1}{2}} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-2 p-1,-p-\frac {1}{2};-2 p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{2 p+1} \]
Antiderivative was successfully verified.
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Rule 132
Rule 6192
Rule 6196
Rubi steps
\begin {align*} \int e^{-\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^{-\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 ) \operatorname {Subst}\left (\int x^{-2-2 p} \left (1-\frac {x}{a}\right )^{\frac {1}{2}+p} \left (1+\frac {x}{a}\right )^{-\frac {1}{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 {1}{2}-p} \left (1-\frac {1}{a x}\right )^{\frac {1}{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 {1}{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 = 0.24, size = 118, normalized size = 1.00 \[ -\frac {4^{p+1} \left (a x \sqrt {1-\frac {1}{a^2 x^2}}\right )^{-2 p} e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \left (1-e^{2 \coth ^{-1}(a x)}\right )^{2 p} \left (\frac {e^{\coth ^{-1}(a x)}}{e^{2 \coth ^{-1}(a x)}-1}\right )^{2 p} \, _2F_1\left (p+\frac {1}{2},2 p+2;p+\frac {3}{2};e^{2 \coth ^{-1}(a x)}\right )}{2 a p+a} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-a^{2} c x^{2} + c\right )}^{p} \sqrt {\frac {a x - 1}{a x + 1}}, x\right ) \]
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 \[ \int {\left (-a^{2} c x^{2} + c\right )}^{p} \sqrt {\frac {a x - 1}{a x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \left (-a^{2} c \,x^{2}+c \right )^{p} \sqrt {\frac {a x -1}{a x +1}}\, dx \]
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 \[ \int {\left (-a^{2} c x^{2} + c\right )}^{p} \sqrt {\frac {a x - 1}{a x + 1}}\,{d x} \]
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 \[ \int {\left (c-a^2\,c\,x^2\right )}^p\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \]
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 \[ \int \sqrt {\frac {a x - 1}{a x + 1}} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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