Optimal. Leaf size=127 \[ \frac {\left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (n-2 p)} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}+p} \left (1+\frac {1}{a x}\right )^{1+\frac {n}{2}+p} x \left (c-a^2 c x^2\right )^p \, _2F_1\left (-1-2 p,\frac {1}{2} (n-2 p);-2 p;\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{1+2 p} \]
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Rubi [A]
time = 0.09, antiderivative size = 127, 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 (c-a^2 c x^2\right )^p \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {1}{2} (n-2 p)} \left (1-\frac {1}{a x}\right )^{p-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n}{2}+p+1} \, _2F_1\left (-2 p-1,\frac {1}{2} (n-2 p);-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^{n \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^{n \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 {n}{2}+p} \left (1+\frac {x}{a}\right )^{\frac {n}{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} (n-2 p)} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}+p} \left (1+\frac {1}{a x}\right )^{1+\frac {n}{2}+p} x \left (c-a^2 c x^2\right )^p \, _2F_1\left (-1-2 p,\frac {1}{2} (n-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 = 0.13, size = 83, normalized size = 0.65 \begin {gather*} -\frac {e^{(-2+n) \coth ^{-1}(a x)} \left (-1+e^{2 \coth ^{-1}(a x)}\right ) \left (-1+a^2 x^2\right ) \left (c-a^2 c x^2\right )^p \, _2F_1\left (1,-\frac {n}{2}-p;2-\frac {n}{2}+p;e^{-2 \coth ^{-1}(a x)}\right )}{a (n-2 (1+p))} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{p}\, dx\]
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 ) \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^2\,c\,x^2\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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