Optimal. Leaf size=104 \[ \frac {\left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (n-2 p)} \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^p \, _2F_1\left (\frac {1}{2} (n-2 p),-1-p;-p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{1+p} \]
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Rubi [A]
time = 0.08, 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 = {6311, 6316,
134} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 134
Rule 6311
Rule 6316
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 ) \text {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 = 0.03, size = 104, normalized size = 1.00 \begin {gather*} \frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \left (\frac {-1+a x}{1+a x}\right )^{\frac {1}{2} (n-2 p)} (1+a x) (c-a c x)^p \, _2F_1\left (-1-p,\frac {n}{2}-p;-p;\frac {2}{1+a x}\right )}{a (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, 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 \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 \left (- c \left (a x - 1\right )\right )^{p} e^{n \operatorname {acoth}{\left (a x \right )}}\, 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 {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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