Optimal. Leaf size=94 \[ \frac {\left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-\frac {1}{2}-p} \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x (c-a c x)^p \, _2F_1\left (-1-p,-\frac {1}{2}-p;-p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{1+p} \]
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Rubi [A]
time = 0.08, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6311, 6316,
134} \begin {gather*} \frac {x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-p-\frac {1}{2}} (c-a c x)^p \, _2F_1\left (-p-1,-p-\frac {1}{2};-p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{p+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 134
Rule 6311
Rule 6316
Rubi steps
\begin {align*} \int e^{-\coth ^{-1}(a x)} (c-a c x)^p \, dx &=\left (\left (1-\frac {1}{a x}\right )^{-p} x^{-p} (c-a c x)^p\right ) \int e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^p x^p \, dx\\ &=-\left (\left (\left (1-\frac {1}{a x}\right )^{-p} \left (\frac {1}{x}\right )^p (c-a c x)^p\right ) \text {Subst}\left (\int \frac {x^{-2-p} \left (1-\frac {x}{a}\right )^{\frac {1}{2}+p}}{\sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {\left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{-\frac {1}{2}-p} \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x (c-a c x)^p \, _2F_1\left (-1-p,-\frac {1}{2}-p;-p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{1+p}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 76, normalized size = 0.81 \begin {gather*} \frac {\sqrt {1-\frac {1}{a^2 x^2}} x \left (\frac {-1+a x}{1+a x}\right )^{-\frac {1}{2}-p} (c-a c x)^p \, _2F_1\left (-1-p,-\frac {1}{2}-p;-p;\frac {2}{1+a x}\right )}{1+p} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (-a c x +c \right )^{p} \sqrt {\frac {a x -1}{a x +1}}\, dx\]
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 \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\frac {a x - 1}{a x + 1}} \left (- c \left (a x - 1\right )\right )^{p}\, dx \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\,c\,x\right )}^p\,\sqrt {\frac {a\,x-1}{a\,x+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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