Optimal. Leaf size=104 \[ \frac {x \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} (c-a c x)^p \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (n-2 p)} \, _2F_1\left (\frac {1}{2} (n-2 p),-p-1;-p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{p+1} \]
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Rubi [A] time = 0.13, antiderivative size = 104, 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 = {6176, 6181, 132} \[ \frac {x \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} (c-a c x)^p \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (n-2 p)} \, _2F_1\left (\frac {1}{2} (n-2 p),-p-1;-p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{p+1} \]
Antiderivative was successfully verified.
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Rule 132
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{n \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^{n \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 ) \operatorname {Subst}\left (\int x^{-2-p} \left (1-\frac {x}{a}\right )^{-\frac {n}{2}+p} \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {\left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (n-2 p)} \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^p \, _2F_1\left (\frac {1}{2} (n-2 p),-1-p;-p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{1+p}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 104, normalized size = 1.00 \[ \frac {(a x+1) \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{n/2} (c-a c x)^p \left (\frac {a x-1}{a x+1}\right )^{\frac {1}{2} (n-2 p)} \, _2F_1\left (-p-1,\frac {n}{2}-p;-p;\frac {2}{a x+1}\right )}{a (p+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-a c x + c\right )}^{p} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \int {\left (-a c x + c\right )}^{p} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (-a c x +c \right )^{p}\, dx \]
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 \[ \int {\left (-a c x + c\right )}^{p} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^p \,d x \]
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 \[ \int \left (- c \left (a x - 1\right )\right )^{p} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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