3.361 \(\int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx\)

Optimal. Leaf size=36 \[ \frac {2 (a x+1) (c-a c x)^{n/2} e^{n \coth ^{-1}(a x)}}{a (n+2)} \]

[Out]

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

________________________________________________________________________________________

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.

[In]

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

[Out]

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

Rule 6174

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

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*}

________________________________________________________________________________________

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.

[In]

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

[Out]

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

________________________________________________________________________________________

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.

[In]

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

[Out]

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

________________________________________________________________________________________

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.

[In]

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

[Out]

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

________________________________________________________________________________________

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.

[In]

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

[Out]

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

________________________________________________________________________________________

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.

[In]

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

[Out]

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

________________________________________________________________________________________

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.

[In]

int(exp(n*acoth(a*x))*(c - a*c*x)^(n/2),x)

[Out]

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

________________________________________________________________________________________

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.

[In]

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

[Out]

Piecewise((-x/c, Eq(a, 0) & Eq(n, -2)), (c**(n/2)*x*exp(I*pi*n/2), Eq(a, 0)), (-Integral(1/(a*x*exp(2*acoth(a*
x)) - exp(2*acoth(a*x))), x)/c, Eq(n, -2)), (2*a*x*(-a*c*x + c)**(n/2)*exp(n*acoth(a*x))/(a*n + 2*a) + 2*(-a*c
*x + c)**(n/2)*exp(n*acoth(a*x))/(a*n + 2*a), True))

________________________________________________________________________________________