Optimal. Leaf size=278 \[ \frac{2 \left (n^2+14 n+56\right ) \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-2} (c-a c x)^{\frac{n+4}{2}}}{a^2 (n+6) \left (n^2+6 n+8\right ) x}-\frac{\left (n^2+14 n+56\right ) \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-2} (c-a c x)^{\frac{n+4}{2}}}{a (n+4) (n+6)}+\frac{(n+8) x \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-2} (c-a c x)^{\frac{n+4}{2}}}{n+6}-\frac{x \left (a-\frac{1}{x}\right ) \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-2} (c-a c x)^{\frac{n+4}{2}}}{a} \]
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Rubi [A] time = 0.267583, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6176, 6181, 90, 79, 45, 37} \[ \frac{2 \left (n^2+14 n+56\right ) \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-2} (c-a c x)^{\frac{n+4}{2}}}{a^2 (n+6) \left (n^2+6 n+8\right ) x}-\frac{\left (n^2+14 n+56\right ) \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-2} (c-a c x)^{\frac{n+4}{2}}}{a (n+4) (n+6)}+\frac{(n+8) x \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-2} (c-a c x)^{\frac{n+4}{2}}}{n+6}-\frac{x \left (a-\frac{1}{x}\right ) \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-2} (c-a c x)^{\frac{n+4}{2}}}{a} \]
Antiderivative was successfully verified.
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Rule 6176
Rule 6181
Rule 90
Rule 79
Rule 45
Rule 37
Rubi steps
\begin{align*} \int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac{n}{2}} \, dx &=\left (\left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} x^{-2-\frac{n}{2}} (c-a c x)^{2+\frac{n}{2}}\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{2+\frac{n}{2}} x^{2+\frac{n}{2}} \, dx\\ &=-\left (\left (\left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} \left (\frac{1}{x}\right )^{2+\frac{n}{2}} (c-a c x)^{2+\frac{n}{2}}\right ) \operatorname{Subst}\left (\int x^{-4-\frac{n}{2}} \left (1-\frac{x}{a}\right )^2 \left (1+\frac{x}{a}\right )^{n/2} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\frac{\left (a-\frac{1}{x}\right ) \left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x (c-a c x)^{\frac{4+n}{2}}}{a}+\left (a \left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} \left (\frac{1}{x}\right )^{2+\frac{n}{2}} (c-a c x)^{2+\frac{n}{2}}\right ) \operatorname{Subst}\left (\int x^{-4-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2} \left (-\frac{8+n}{2 a}+\frac{(4+n) x}{2 a^2}\right ) \, dx,x,\frac{1}{x}\right )\\ &=\frac{(8+n) \left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x (c-a c x)^{\frac{4+n}{2}}}{6+n}-\frac{\left (a-\frac{1}{x}\right ) \left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x (c-a c x)^{\frac{4+n}{2}}}{a}+\frac{\left (\left (56+14 n+n^2\right ) \left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} \left (\frac{1}{x}\right )^{2+\frac{n}{2}} (c-a c x)^{2+\frac{n}{2}}\right ) \operatorname{Subst}\left (\int x^{-3-\frac{n}{2}} \left (1+\frac{x}{a}\right )^{n/2} \, dx,x,\frac{1}{x}\right )}{2 a (6+n)}\\ &=-\frac{\left (56+14 n+n^2\right ) \left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} (c-a c x)^{\frac{4+n}{2}}}{a (4+n) (6+n)}+\frac{(8+n) \left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x (c-a c x)^{\frac{4+n}{2}}}{6+n}-\frac{\left (a-\frac{1}{x}\right ) \left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x (c-a c x)^{\frac{4+n}{2}}}{a}-\frac{\left (\left (56+14 n+n^2\right ) \left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} \left (\frac{1}{x}\right )^{2+\frac{n}{2}} (c-a c x)^{2+\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^2 (4+n) (6+n)}\\ &=-\frac{\left (56+14 n+n^2\right ) \left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} (c-a c x)^{\frac{4+n}{2}}}{a (4+n) (6+n)}+\frac{2 \left (56+14 n+n^2\right ) \left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} (c-a c x)^{\frac{4+n}{2}}}{a^2 (2+n) (4+n) (6+n) x}+\frac{(8+n) \left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x (c-a c x)^{\frac{4+n}{2}}}{6+n}-\frac{\left (a-\frac{1}{x}\right ) \left (1-\frac{1}{a x}\right )^{-2-\frac{n}{2}} \left (1+\frac{1}{a x}\right )^{\frac{2+n}{2}} x (c-a c x)^{\frac{4+n}{2}}}{a}\\ \end{align*}
Mathematica [A] time = 0.0736152, size = 116, normalized size = 0.42 \[ \frac{2 c^2 (a x+1) \left (1-\frac{1}{a x}\right )^{-n/2} \left (\frac{1}{a x}+1\right )^{n/2} \left (2 n \left (3 a^2 x^2-10 a x+7\right )+8 \left (a^2 x^2-4 a x+7\right )+n^2 (a x-1)^2\right ) (c-a c x)^{n/2}}{a (n+2) (n+4) (n+6)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 104, normalized size = 0.4 \begin{align*} 2\,{\frac{ \left ({a}^{2}{n}^{2}{x}^{2}+6\,{a}^{2}n{x}^{2}+8\,{a}^{2}{x}^{2}-2\,a{n}^{2}x-20\,anx-32\,ax+{n}^{2}+14\,n+56 \right ) \left ( ax+1 \right ){{\rm e}^{n{\rm arccoth} \left (ax\right )}} \left ( -acx+c \right ) ^{2+n/2}}{ \left ( ax-1 \right ) ^{2}a \left ({n}^{3}+12\,{n}^{2}+44\,n+48 \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 + 2} \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 + 2} \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 + 2} \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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