### 3.359 $$\int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac{n}{2}} \, dx$$

Optimal. Leaf size=278 $\frac{2 \left (n^2+14 n+56\right ) \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-2} (c-a c x)^{\frac{n+4}{2}}}{a^2 (n+6) \left (n^2+6 n+8\right ) x}-\frac{\left (n^2+14 n+56\right ) \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-2} (c-a c x)^{\frac{n+4}{2}}}{a (n+4) (n+6)}+\frac{(n+8) x \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-2} (c-a c x)^{\frac{n+4}{2}}}{n+6}-\frac{x \left (a-\frac{1}{x}\right ) \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-2} (c-a c x)^{\frac{n+4}{2}}}{a}$

[Out]

-(((56 + 14*n + n^2)*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*(c - a*c*x)^((4 + n)/2))/(a*(4 + n)*(6
+ n))) + (2*(56 + 14*n + n^2)*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*(c - a*c*x)^((4 + n)/2))/(a^
2*(6 + n)*(8 + 6*n + n^2)*x) + ((8 + n)*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^((4 +
n)/2))/(6 + n) - ((a - x^(-1))*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^((4 + n)/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.

[In]

Int[E^(n*ArcCoth[a*x])*(c - a*c*x)^(2 + n/2),x]

[Out]

-(((56 + 14*n + n^2)*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*(c - a*c*x)^((4 + n)/2))/(a*(4 + n)*(6
+ n))) + (2*(56 + 14*n + n^2)*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*(c - a*c*x)^((4 + n)/2))/(a^
2*(6 + n)*(8 + 6*n + n^2)*x) + ((8 + n)*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^((4 +
n)/2))/(6 + n) - ((a - x^(-1))*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^((4 + n)/2))/
a

Rule 6176

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_), x_Symbol] :> Dist[(c + d*x)^p/(x^p*(1 + c/(d
*x))^p), Int[u*x^p*(1 + c/(d*x))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^
2, 0] &&  !IntegerQ[n/2] &&  !IntegerQ[p]

Rule 6181

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> -Dist[c^p*x^m*(1/x)^m, Sub
st[Int[((1 + (d*x)/c)^p*(1 + x/a)^(n/2))/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x], x] /; FreeQ[{a, c, d, m, n,
p}, x] && EqQ[c^2 - a^2*d^2, 0] &&  !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) &&  !IntegerQ[m]

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c,
d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimplerQ[p, 1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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.

[In]

Integrate[E^(n*ArcCoth[a*x])*(c - a*c*x)^(2 + n/2),x]

[Out]

(2*c^2*(1 + 1/(a*x))^(n/2)*(1 + a*x)*(c - a*c*x)^(n/2)*(n^2*(-1 + a*x)^2 + 8*(7 - 4*a*x + a^2*x^2) + 2*n*(7 -
10*a*x + 3*a^2*x^2)))/(a*(2 + n)*(4 + n)*(6 + n)*(1 - 1/(a*x))^(n/2))

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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.

[In]

int(exp(n*arccoth(a*x))*(-a*c*x+c)^(2+1/2*n),x)

[Out]

2*(a*x+1)*(a^2*n^2*x^2+6*a^2*n*x^2+8*a^2*x^2-2*a*n^2*x-20*a*n*x-32*a*x+n^2+14*n+56)*exp(n*arccoth(a*x))*(-a*c*
x+c)^(2+1/2*n)/(a*x-1)^2/a/(n^3+12*n^2+44*n+48)

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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.

[In]

integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^(2+1/2*n),x, algorithm="maxima")

[Out]

integrate((-a*c*x + c)^(1/2*n + 2)*((a*x - 1)/(a*x + 1))^(1/2*n), x)

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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.

[In]

integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^(2+1/2*n),x, algorithm="fricas")

[Out]

integral((-a*c*x + c)^(1/2*n + 2)*((a*x - 1)/(a*x + 1))^(1/2*n), x)

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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.

[In]

integrate(exp(n*acoth(a*x))*(-a*c*x+c)**(2+1/2*n),x)

[Out]

Timed out

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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.

[In]

integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^(2+1/2*n),x, algorithm="giac")

[Out]

integrate((-a*c*x + c)^(1/2*n + 2)*((a*x - 1)/(a*x + 1))^(1/2*n), x)