### 3.360 $$\int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac{n}{2}} \, dx$$

Optimal. Leaf size=127 $\frac{2 x \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} (c-a c x)^{\frac{n+2}{2}}}{n+4}-\frac{2 (n+6) \left (1-\frac{1}{a x}\right )^{-\frac{n}{2}-1} \left (\frac{1}{a x}+1\right )^{\frac{n+2}{2}} (c-a c x)^{\frac{n+2}{2}}}{a (n+2) (n+4)}$

[Out]

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

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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.

[In]

Int[E^(n*ArcCoth[a*x])*(c - a*c*x)^(1 + n/2),x]

[Out]

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

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 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 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)^{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.

[In]

Integrate[E^(n*ArcCoth[a*x])*(c - a*c*x)^(1 + n/2),x]

[Out]

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

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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.

[In]

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

[Out]

2*(-a*c*x+c)^(1+1/2*n)*exp(n*arccoth(a*x))*(a*n*x+2*a*x-n-6)*(a*x+1)/(a*x-1)/a/(n^2+6*n+8)

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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.

[In]

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

[Out]

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

[In]

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

[Out]

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

[In]

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

[Out]

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