\(\int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx\) [299]

Optimal result

Integrand size = 24, antiderivative size = 127 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx=-\frac {2 (6+n) \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} (c-a c x)^{\frac {2+n}{2}}}{a (2+n) (4+n)}+\frac {2 \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {2+n}{2}}}{4+n} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.61 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx=-\frac {2 c \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} (1+a x) (c-a c x)^{n/2} (-6+2 a x+n (-1+a x))}{a (2+n) (4+n)} \] Input:

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

Output:

(-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))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6727, 27, 88, 48}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6727

\(\displaystyle \left (\frac {1}{x}\right )^{\frac {n+2}{2}} \left (-\left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1}\right ) (c-a c x)^{\frac {n+2}{2}} \int \frac {\left (a-\frac {1}{x}\right ) \left (1+\frac {1}{a x}\right )^{n/2} \left (\frac {1}{x}\right )^{-\frac {n}{2}-3}}{a}d\frac {1}{x}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} (c-a c x)^{\frac {n+2}{2}} \int \left (a-\frac {1}{x}\right ) \left (1+\frac {1}{a x}\right )^{n/2} \left (\frac {1}{x}\right )^{-\frac {n}{2}-3}d\frac {1}{x}}{a}\)

\(\Big \downarrow \) 88

\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{\frac {n+2}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} (c-a c x)^{\frac {n+2}{2}} \left (-\frac {(n+6) \int \left (1+\frac {1}{a x}\right )^{n/2} \left (\frac {1}{x}\right )^{-\frac {n}{2}-2}d\frac {1}{x}}{n+4}-\frac {2 a \left (\frac {1}{x}\right )^{-\frac {n}{2}-2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{n+4}\right )}{a}\)

\(\Big \downarrow \) 48

\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{\frac {n+2}{2}} \left (\frac {2 (n+6) \left (\frac {1}{x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{(n+2) (n+4)}-\frac {2 a \left (\frac {1}{x}\right )^{-\frac {n}{2}-2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{n+4}\right ) \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} (c-a c x)^{\frac {n+2}{2}}}{a}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 48

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] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 88

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(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] && SumSimpl 
erQ[p, 1]
 

rule 6727

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

Maple [A] (verified)

Time = 0.99 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.48

method	result	size
gosper	\(\frac {2 \left (-a c x +c \right )^{1+\frac {n}{2}} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (n a x +2 a x -n -6\right ) \left (a x +1\right )}{\left (a x -1\right ) a \left (n^{2}+6 n +8\right )}\)	\(61\)
orering	\(\frac {2 \left (-a c x +c \right )^{1+\frac {n}{2}} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (n a x +2 a x -n -6\right ) \left (a x +1\right )}{\left (a x -1\right ) a \left (n^{2}+6 n +8\right )}\)	\(61\)
parallelrisch	\(-\frac {12 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a c x +c \right )^{1+\frac {n}{2}}-2 x^{2} \left (-a c x +c \right )^{1+\frac {n}{2}} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{2} n -4 x^{2} \left (-a c x +c \right )^{1+\frac {n}{2}} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{2}+8 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} x \left (-a c x +c \right )^{1+\frac {n}{2}} a +2 \left (-a c x +c \right )^{1+\frac {n}{2}} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} n}{\left (a x -1\right ) a \left (n^{2}+6 n +8\right )}\)	\(150\)

Input:

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

Output:

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)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.73 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx=-\frac {2 \, {\left ({\left (a^{2} n + 2 \, a^{2}\right )} x^{2} - 4 \, a x - n - 6\right )} {\left (-a c x + c\right )}^{\frac {1}{2} \, n + 1} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a n^{2} + 6 \, a n - {\left (a^{2} n^{2} + 6 \, a^{2} n + 8 \, a^{2}\right )} x + 8 \, a} \] Input:

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

Output:

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

Sympy [F]

\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx=\begin {cases} c^{\frac {n}{2} + 1} x e^{\frac {i \pi n}{2}} & \text {for}\: a = 0 \\0^{\frac {n}{2} + 1} x e^{\infty n} & \text {for}\: a = \frac {1}{x} \\- \frac {\int \frac {1}{a x e^{4 \operatorname {acoth}{\left (a x \right )}} - e^{4 \operatorname {acoth}{\left (a x \right )}}}\, dx}{c} & \text {for}\: n = -4 \\\int e^{- 2 \operatorname {acoth}{\left (a x \right )}}\, dx & \text {for}\: n = -2 \\\frac {2 a^{2} n x^{2} \left (- a c x + c\right )^{\frac {n}{2} + 1} e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{2} n^{2} x + 6 a^{2} n x + 8 a^{2} x - a n^{2} - 6 a n - 8 a} + \frac {4 a^{2} x^{2} \left (- a c x + c\right )^{\frac {n}{2} + 1} e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{2} n^{2} x + 6 a^{2} n x + 8 a^{2} x - a n^{2} - 6 a n - 8 a} - \frac {8 a x \left (- a c x + c\right )^{\frac {n}{2} + 1} e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{2} n^{2} x + 6 a^{2} n x + 8 a^{2} x - a n^{2} - 6 a n - 8 a} - \frac {2 n \left (- a c x + c\right )^{\frac {n}{2} + 1} e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{2} n^{2} x + 6 a^{2} n x + 8 a^{2} x - a n^{2} - 6 a n - 8 a} - \frac {12 \left (- a c x + c\right )^{\frac {n}{2} + 1} e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{2} n^{2} x + 6 a^{2} n x + 8 a^{2} x - a n^{2} - 6 a n - 8 a} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((c**(n/2 + 1)*x*exp(I*pi*n/2), Eq(a, 0)), (0**(n/2 + 1)*x*exp(oo 
*n), Eq(a, 1/x)), (-Integral(1/(a*x*exp(4*acoth(a*x)) - exp(4*acoth(a*x))) 
, x)/c, Eq(n, -4)), (Integral(exp(-2*acoth(a*x)), x), Eq(n, -2)), (2*a**2* 
n*x**2*(-a*c*x + c)**(n/2 + 1)*exp(n*acoth(a*x))/(a**2*n**2*x + 6*a**2*n*x 
 + 8*a**2*x - a*n**2 - 6*a*n - 8*a) + 4*a**2*x**2*(-a*c*x + c)**(n/2 + 1)* 
exp(n*acoth(a*x))/(a**2*n**2*x + 6*a**2*n*x + 8*a**2*x - a*n**2 - 6*a*n - 
8*a) - 8*a*x*(-a*c*x + c)**(n/2 + 1)*exp(n*acoth(a*x))/(a**2*n**2*x + 6*a* 
*2*n*x + 8*a**2*x - a*n**2 - 6*a*n - 8*a) - 2*n*(-a*c*x + c)**(n/2 + 1)*ex 
p(n*acoth(a*x))/(a**2*n**2*x + 6*a**2*n*x + 8*a**2*x - a*n**2 - 6*a*n - 8* 
a) - 12*(-a*c*x + c)**(n/2 + 1)*exp(n*acoth(a*x))/(a**2*n**2*x + 6*a**2*n* 
x + 8*a**2*x - a*n**2 - 6*a*n - 8*a), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.54 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx=-\frac {2 \, {\left (a^{2} \left (-c\right )^{\frac {1}{2} \, n} c {\left (n + 2\right )} x^{2} - 4 \, a \left (-c\right )^{\frac {1}{2} \, n} c x - \left (-c\right )^{\frac {1}{2} \, n} c {\left (n + 6\right )}\right )} {\left (a x + 1\right )}^{\frac {1}{2} \, n}}{{\left (n^{2} + 6 \, n + 8\right )} a} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 13.78 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.10 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx=-\frac {\left (\frac {\left (2\,n+12\right )\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+1}}{a^2\,\left (n^2+6\,n+8\right )}-\frac {x^2\,\left (2\,n+4\right )\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+1}}{n^2+6\,n+8}+\frac {8\,x\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+1}}{a\,\left (n^2+6\,n+8\right )}\right )\,{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}}{\left (x-\frac {1}{a}\right )\,{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}} \] Input:

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

Output:

-((((2*n + 12)*(c - a*c*x)^(n/2 + 1))/(a^2*(6*n + n^2 + 8)) - (x^2*(2*n + 
4)*(c - a*c*x)^(n/2 + 1))/(6*n + n^2 + 8) + (8*x*(c - a*c*x)^(n/2 + 1))/(a 
*(6*n + n^2 + 8)))*((a*x + 1)/(a*x))^(n/2))/((x - 1/a)*((a*x - 1)/(a*x))^( 
n/2))
 

Reduce [F]

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

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

Output:

c*( - int(e**(acoth(a*x)*n)*( - a*c*x + c)**(n/2)*x,x)*a + int(e**(acoth(a 
*x)*n)*( - a*c*x + c)**(n/2),x))