Optimal. Leaf size=36 \[ \frac {2 (a x+1) (c-a c x)^{n/2} e^{n \coth ^{-1}(a x)}}{a (n+2)} \]
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Rubi [A] time = 0.03, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6174} \[ \frac {2 (a x+1) (c-a c x)^{n/2} e^{n \coth ^{-1}(a x)}}{a (n+2)} \]
Antiderivative was successfully verified.
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Rule 6174
Rubi steps
\begin {align*} \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx &=\frac {2 e^{n \coth ^{-1}(a x)} (1+a x) (c-a c x)^{n/2}}{a (2+n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 58, normalized size = 1.61 \[ -\frac {x \left (1-\frac {1}{a x}\right )^{-n/2} \left (\frac {1}{a x}+1\right )^{\frac {n}{2}+1} (c-a c x)^{n/2}}{-\frac {n}{2}-1} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-a c x + c\right )}^{\frac {1}{2} \, n} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \int {\left (-a c x + c\right )}^{\frac {1}{2} \, n} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 34, normalized size = 0.94 \[ \frac {2 \,{\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (a x +1\right ) \left (-a c x +c \right )^{\frac {n}{2}}}{a \left (2+n \right )} \]
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 \[ \int {\left (-a c x + c\right )}^{\frac {1}{2} \, n} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 55, normalized size = 1.53 \[ \frac {2\,{\left (\frac {1}{a\,x}+1\right )}^{n/2}\,{\left (c-a\,c\,x\right )}^{n/2}\,\left (a\,x+1\right )}{a\,{\left (1-\frac {1}{a\,x}\right )}^{n/2}\,\left (n+2\right )} \]
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 \[ \begin {cases} - \frac {x}{c} & \text {for}\: a = 0 \wedge n = -2 \\c^{\frac {n}{2}} x e^{\frac {i \pi n}{2}} & \text {for}\: a = 0 \\- \frac {\int \frac {1}{a x e^{2 \operatorname {acoth}{\left (a x \right )}} - e^{2 \operatorname {acoth}{\left (a x \right )}}}\, dx}{c} & \text {for}\: n = -2 \\\frac {2 a x \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n + 2 a} + \frac {2 \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n + 2 a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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