Optimal. Leaf size=79 \[ -\frac {8 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-4+n)} \, _2F_1\left (3,2-\frac {n}{2};3-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)} \]
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Rubi [A]
time = 0.05, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6310, 6315,
133} \begin {gather*} -\frac {8 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-4}{2}} \, _2F_1\left (3,2-\frac {n}{2};3-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 133
Rule 6310
Rule 6315
Rubi steps
\begin {align*} \int e^{n \coth ^{-1}(a x)} (c-a c x) \, dx &=-\left ((a c) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right ) x \, dx\right )\\ &=(a c) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{1-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {8 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-4+n)} \, _2F_1\left (3,2-\frac {n}{2};3-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 104, normalized size = 1.32 \begin {gather*} -\frac {c e^{n \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)} (-2+n) n \, _2F_1\left (1,1+\frac {n}{2};2+\frac {n}{2};e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (-1+a (-2+n) x+a^2 x^2+(-2+n) \, _2F_1\left (1,\frac {n}{2};1+\frac {n}{2};e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{2 a (2+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (-a c x +c \right )\, dx\]
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*} - c \left (\int a x e^{n \operatorname {acoth}{\left (a x \right )}}\, dx + \int \left (- e^{n \operatorname {acoth}{\left (a x \right )}}\right )\, dx\right ) \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 ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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