\(\int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac {n}{2}} \, dx\) [298]

Optimal result

Integrand size = 24, antiderivative size = 278 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac {n}{2}} \, dx=-\frac {\left (56+14 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} (c-a c x)^{\frac {4+n}{2}}}{a (4+n) (6+n)}+\frac {2 \left (56+14 n+n^2\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} (c-a c x)^{\frac {4+n}{2}}}{a^2 (6+n) \left (8+6 n+n^2\right ) x}+\frac {(8+n) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{6+n}-\frac {\left (a-\frac {1}{x}\right ) \left (1-\frac {1}{a x}\right )^{-2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {4+n}{2}}}{a} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.42 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac {n}{2}} \, dx=\frac {2 c^2 \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} (1+a x) (c-a c x)^{n/2} \left (n^2 (-1+a x)^2+8 \left (7-4 a x+a^2 x^2\right )+2 n \left (7-10 a x+3 a^2 x^2\right )\right )}{a (2+n) (4+n) (6+n)} \] Input:

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

Output:

(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))
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.81, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6727, 27, 101, 27, 88, 55, 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.

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

\(\Big \downarrow \) 6727

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 101

\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{\frac {n+4}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}} \left (a \left (a-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{-\frac {n}{2}-3} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}-a \int -\frac {1}{2} \left (1+\frac {1}{a x}\right )^{n/2} \left (a (n+8)-\frac {n+4}{x}\right ) \left (\frac {1}{x}\right )^{-\frac {n}{2}-4}d\frac {1}{x}\right )}{a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{\frac {n+4}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}} \left (\frac {1}{2} a \int \left (1+\frac {1}{a x}\right )^{n/2} \left (a (n+8)-\frac {n+4}{x}\right ) \left (\frac {1}{x}\right )^{-\frac {n}{2}-4}d\frac {1}{x}+a \left (a-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{-\frac {n}{2}-3} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}\right )}{a^2}\)

\(\Big \downarrow \) 88

\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{\frac {n+4}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}} \left (\frac {1}{2} a \left (-\frac {\left (n^2+14 n+56\right ) \int \left (1+\frac {1}{a x}\right )^{n/2} \left (\frac {1}{x}\right )^{-\frac {n}{2}-3}d\frac {1}{x}}{n+6}-\frac {2 a (n+8) \left (\frac {1}{x}\right )^{-\frac {n}{2}-3} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{n+6}\right )+a \left (a-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{-\frac {n}{2}-3} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}\right )}{a^2}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{\frac {n+4}{2}} \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}} \left (\frac {1}{2} a \left (-\frac {\left (n^2+14 n+56\right ) \left (-\frac {2 \int \left (1+\frac {1}{a x}\right )^{n/2} \left (\frac {1}{x}\right )^{-\frac {n}{2}-2}d\frac {1}{x}}{a (n+4)}-\frac {2 \left (\frac {1}{x}\right )^{-\frac {n}{2}-2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{n+4}\right )}{n+6}-\frac {2 a (n+8) \left (\frac {1}{x}\right )^{-\frac {n}{2}-3} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{n+6}\right )+a \left (a-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{-\frac {n}{2}-3} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}\right )}{a^2}\)

\(\Big \downarrow \) 48

\(\displaystyle -\frac {\left (\frac {1}{x}\right )^{\frac {n+4}{2}} \left (\frac {1}{2} a \left (-\frac {\left (n^2+14 n+56\right ) \left (\frac {4 \left (\frac {1}{x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{a (n+2) (n+4)}-\frac {2 \left (\frac {1}{x}\right )^{-\frac {n}{2}-2} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{n+4}\right )}{n+6}-\frac {2 a (n+8) \left (\frac {1}{x}\right )^{-\frac {n}{2}-3} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}}{n+6}\right )+a \left (a-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{-\frac {n}{2}-3} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}}\right ) \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-2} (c-a c x)^{\frac {n+4}{2}}}{a^2}\)

Input:

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

Output:

-((((a*(-(((56 + 14*n + n^2)*((-2*(1 + 1/(a*x))^((2 + n)/2)*(x^(-1))^(-2 - 
 n/2))/(4 + n) + (4*(1 + 1/(a*x))^((2 + n)/2)*(x^(-1))^(-1 - n/2))/(a*(2 + 
 n)*(4 + n))))/(6 + n)) - (2*a*(8 + n)*(1 + 1/(a*x))^((2 + n)/2)*(x^(-1))^ 
(-3 - n/2))/(6 + n)))/2 + a*(a - x^(-1))*(1 + 1/(a*x))^((2 + n)/2)*(x^(-1) 
)^(-3 - n/2))*(1 - 1/(a*x))^(-2 - n/2)*(x^(-1))^((4 + n)/2)*(c - a*c*x)^(( 
4 + n)/2))/a^2)
 

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 55

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] - Simp[d*(S 
implify[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] && 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] ||  !SumSimp 
lerQ[n, 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 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[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 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 = 1.48 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.37

method	result	size
gosper	\(\frac {2 \left (a x +1\right ) \left (a^{2} n^{2} x^{2}+6 n \,x^{2} a^{2}+8 a^{2} x^{2}-2 n^{2} x a -20 n a x -32 a x +n^{2}+14 n +56\right ) {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a c x +c \right )^{2+\frac {n}{2}}}{\left (a x -1\right )^{2} a \left (n^{3}+12 n^{2}+44 n +48\right )}\)	\(104\)
orering	\(\frac {2 \left (a x +1\right ) \left (a^{2} n^{2} x^{2}+6 n \,x^{2} a^{2}+8 a^{2} x^{2}-2 n^{2} x a -20 n a x -32 a x +n^{2}+14 n +56\right ) {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a c x +c \right )^{2+\frac {n}{2}}}{\left (a x -1\right )^{2} a \left (n^{3}+12 n^{2}+44 n +48\right )}\)	\(104\)

Input:

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

Output:

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)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.67 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac {n}{2}} \, dx=\frac {2 \, {\left ({\left (a^{3} n^{2} + 6 \, a^{3} n + 8 \, a^{3}\right )} x^{3} - {\left (a^{2} n^{2} + 14 \, a^{2} n + 24 \, a^{2}\right )} x^{2} + n^{2} - {\left (a n^{2} + 6 \, a n - 24 \, a\right )} x + 14 \, n + 56\right )} {\left (-a c x + c\right )}^{\frac {1}{2} \, n + 2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a n^{3} + 12 \, a n^{2} + {\left (a^{3} n^{3} + 12 \, a^{3} n^{2} + 44 \, a^{3} n + 48 \, a^{3}\right )} x^{2} + 44 \, a n - 2 \, {\left (a^{2} n^{3} + 12 \, a^{2} n^{2} + 44 \, a^{2} n + 48 \, a^{2}\right )} x + 48 \, a} \] Input:

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

Output:

2*((a^3*n^2 + 6*a^3*n + 8*a^3)*x^3 - (a^2*n^2 + 14*a^2*n + 24*a^2)*x^2 + n 
^2 - (a*n^2 + 6*a*n - 24*a)*x + 14*n + 56)*(-a*c*x + c)^(1/2*n + 2)*((a*x 
+ 1)/(a*x - 1))^(1/2*n)/(a*n^3 + 12*a*n^2 + (a^3*n^3 + 12*a^3*n^2 + 44*a^3 
*n + 48*a^3)*x^2 + 44*a*n - 2*(a^2*n^3 + 12*a^2*n^2 + 44*a^2*n + 48*a^2)*x 
 + 48*a)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.44 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac {n}{2}} \, dx=\frac {2 \, {\left ({\left (n^{2} + 6 \, n + 8\right )} a^{3} \left (-c\right )^{\frac {1}{2} \, n} c^{2} x^{3} - {\left (n^{2} + 14 \, n + 24\right )} a^{2} \left (-c\right )^{\frac {1}{2} \, n} c^{2} x^{2} - {\left (n^{2} + 6 \, n - 24\right )} a \left (-c\right )^{\frac {1}{2} \, n} c^{2} x + {\left (n^{2} + 14 \, n + 56\right )} \left (-c\right )^{\frac {1}{2} \, n} c^{2}\right )} {\left (a x + 1\right )}^{\frac {1}{2} \, n}}{{\left (n^{3} + 12 \, n^{2} + 44 \, n + 48\right )} a} \] Input:

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

Output:

2*((n^2 + 6*n + 8)*a^3*(-c)^(1/2*n)*c^2*x^3 - (n^2 + 14*n + 24)*a^2*(-c)^( 
1/2*n)*c^2*x^2 - (n^2 + 6*n - 24)*a*(-c)^(1/2*n)*c^2*x + (n^2 + 14*n + 56) 
*(-c)^(1/2*n)*c^2)*(a*x + 1)^(1/2*n)/((n^3 + 12*n^2 + 44*n + 48)*a)
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 13.76 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.80 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{2+\frac {n}{2}} \, dx=\frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {x^3\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+2}\,\left (2\,n^2+12\,n+16\right )}{n^3+12\,n^2+44\,n+48}+\frac {{\left (c-a\,c\,x\right )}^{\frac {n}{2}+2}\,\left (2\,n^2+28\,n+112\right )}{a^3\,\left (n^3+12\,n^2+44\,n+48\right )}-\frac {2\,x\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+2}\,\left (n^2+6\,n-24\right )}{a^2\,\left (n^3+12\,n^2+44\,n+48\right )}-\frac {x^2\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+2}\,\left (2\,n^2+28\,n+48\right )}{a\,\left (n^3+12\,n^2+44\,n+48\right )}\right )}{{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}\,\left (\frac {1}{a^2}-\frac {2\,x}{a}+x^2\right )} \] Input:

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

Output:

(((a*x + 1)/(a*x))^(n/2)*((x^3*(c - a*c*x)^(n/2 + 2)*(12*n + 2*n^2 + 16))/ 
(44*n + 12*n^2 + n^3 + 48) + ((c - a*c*x)^(n/2 + 2)*(28*n + 2*n^2 + 112))/ 
(a^3*(44*n + 12*n^2 + n^3 + 48)) - (2*x*(c - a*c*x)^(n/2 + 2)*(6*n + n^2 - 
 24))/(a^2*(44*n + 12*n^2 + n^3 + 48)) - (x^2*(c - a*c*x)^(n/2 + 2)*(28*n 
+ 2*n^2 + 48))/(a*(44*n + 12*n^2 + n^3 + 48))))/(((a*x - 1)/(a*x))^(n/2)*( 
1/a^2 - (2*x)/a + x^2))
 

Reduce [F]

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

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

Output:

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