Optimal. Leaf size=127 \[ \frac{2 x \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} (c-a c x)^{\frac{n+2}{2}}}{n+4}-\frac{2 (n+6) \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} (c-a c x)^{\frac{n+2}{2}}}{a (n+2) (n+4)} \]
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Rubi [A] time = 0.157131, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6176, 6181, 79, 37} \[ \frac{2 x \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} (c-a c x)^{\frac{n+2}{2}}}{n+4}-\frac{2 (n+6) \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} (c-a c x)^{\frac{n+2}{2}}}{a (n+2) (n+4)} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 79
Rule 37
Rubi steps
\begin{align*} \int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac{n}{2}} \, dx &=\left (\left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} x^{-1-\frac{n}{2}} (c-a c x)^{1+\frac{n}{2}}\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{1+\frac{n}{2}} x^{1+\frac{n}{2}} \, dx\\ &=-\left (\left (\left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (\frac{1}{x}\right )^{1+\frac{n}{2}} (c-a c x)^{1+\frac{n}{2}}\right ) \operatorname{Subst}\left (\int x^{-3-\frac{n}{2}} \left (1-\frac{x}{a}\right ) \left (1+\frac{x}{a}\right )^{n/2} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{2 \left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x (c-a c x)^{\frac{2+n}{2}}}{4+n}+\frac{\left ((6+n) \left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (\frac{1}{x}\right )^{1+\frac{n}{2}} (c-a c x)^{1+\frac{n}{2}}\right ) \operatorname{Subst}\left (\int x^{-2-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2} \, dx,x,\frac{1}{x}\right )}{a (4+n)}\\ &=-\frac{2 (6+n) \left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} (c-a c x)^{\frac{2+n}{2}}}{a (2+n) (4+n)}+\frac{2 \left (1-\frac{1}{a x}\right )^{-1-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x (c-a c x)^{\frac{2+n}{2}}}{4+n}\\ \end{align*}
Mathematica [A] time = 0.0482707, size = 78, normalized size = 0.61 \[ -\frac{2 c (a x+1) \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{n/2} (n (a x-1)+2 a x-6) (c-a c x)^{n/2}}{a (n+2) (n+4)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 61, normalized size = 0.5 \begin{align*} 2\,{\frac{{{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( -acx+c \right ) ^{1+n/2} \left ( anx+2\,ax-n-6 \right ) \left ( ax+1 \right ) }{ \left ( ax-1 \right ) a \left ({n}^{2}+6\,n+8 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a c x + c\right )}^{\frac{1}{2} \, n + 1} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (-a c x + c\right )}^{\frac{1}{2} \, n + 1} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a c x + c\right )}^{\frac{1}{2} \, n + 1} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{1}{2} \, n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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