\(\int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx\) [361]

Optimal result

Integrand size = 22, antiderivative size = 36 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx=\frac {2 e^{n \coth ^{-1}(a x)} (1+a x) (c-a c x)^{n/2}}{a (2+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6309} \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx=\frac {2 (a x+1) (c-a c x)^{n/2} e^{n \coth ^{-1}(a x)}}{a (n+2)} \]

[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 6309

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*} \text {integral}& = \frac {2 e^{n \coth ^{-1}(a x)} (1+a x) (c-a c x)^{n/2}}{a (2+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.61 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx=-\frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{1+\frac {n}{2}} x (c-a c x)^{n/2}}{-1-\frac {n}{2}} \]

[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)))

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94

method	result	size
gosper	\(\frac {2 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (a x +1\right ) \left (-a c x +c \right )^{\frac {n}{2}}}{a \left (2+n \right )}\)	\(34\)
parallelrisch	\(-\frac {-2 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} x \left (-a c x +c \right )^{\frac {n}{2}} a -2 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a c x +c \right )^{\frac {n}{2}}}{a \left (2+n \right )}\)	\(54\)
risch	\(\frac {2 \left (a x +1\right ) \left (a x +1\right )^{\frac {n}{2}} \left (a x -1\right )^{-\frac {n}{2}} \left (a x -1\right )^{\frac {n}{2}} c^{\frac {n}{2}} {\mathrm e}^{-\frac {i \pi n \left (-\operatorname {csgn}\left (i \left (a x -1\right )\right ) \operatorname {csgn}\left (i c \left (a x -1\right )\right )^{2}+\operatorname {csgn}\left (i \left (a x -1\right )\right ) \operatorname {csgn}\left (i c \left (a x -1\right )\right ) \operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \left (a x -1\right )\right )^{3}-\operatorname {csgn}\left (i c \left (a x -1\right )\right )^{2} \operatorname {csgn}\left (i c \right )+2 \operatorname {csgn}\left (i c \left (a x -1\right )\right )^{2}-2\right )}{4}}}{a \left (2+n \right )}\)	\(151\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.22 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx=\frac {2 \, {\left (a x + 1\right )} {\left (-a c x + c\right )}^{\frac {1}{2} \, n} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a n + 2 \, a} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx=\begin {cases} - \frac {x}{c} & \text {for}\: a = 0 \wedge n = -2 \\c^{\frac {n}{2}} x e^{\frac {i \pi n}{2}} & \text {for}\: a = 0 \\- \frac {\int \frac {1}{a x e^{2 \operatorname {acoth}{\left (a x \right )}} - e^{2 \operatorname {acoth}{\left (a x \right )}}}\, dx}{c} & \text {for}\: n = -2 \\\frac {2 a x \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n + 2 a} + \frac {2 \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n + 2 a} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx=\frac {2 \, {\left (a \left (-c\right )^{\frac {1}{2} \, n} x + \left (-c\right )^{\frac {1}{2} \, n}\right )} {\left (a x + 1\right )}^{\frac {1}{2} \, n}}{a {\left (n + 2\right )}} \]

[In]

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

[Out]

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

Giac [F]

\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx=\int { {\left (-a c x + c\right )}^{\frac {1}{2} \, n} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 4.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.53 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx=\frac {2\,{\left (\frac {1}{a\,x}+1\right )}^{n/2}\,{\left (c-a\,c\,x\right )}^{n/2}\,\left (a\,x+1\right )}{a\,{\left (1-\frac {1}{a\,x}\right )}^{n/2}\,\left (n+2\right )} \]

[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))