\(\int e^{3 \coth ^{-1}(a x)} (c-a c x)^p \, dx\) [279]

Optimal result

Integrand size = 18, antiderivative size = 94 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {\left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {3}{2}-p} \left (1+\frac {1}{a x}\right )^{5/2} x (c-a c x)^p \operatorname {Hypergeometric2F1}\left (-1-p,\frac {3}{2}-p,-p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{(1+p) \left (1-\frac {1}{a x}\right )^{3/2}} \] Output:

((a-1/x)/(a+1/x))^(3/2-p)*(1+1/a/x)^(5/2)*x*(-a*c*x+c)^p*hypergeom([-1-p, 
3/2-p],[-p],2/(a+1/x)/x)/(p+1)/(1-1/a/x)^(3/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.16 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.47 \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\frac {\left (\frac {-1+a x}{1+a x}\right )^{-p} (c-a c x)^p \left ((-1+p) \left (\frac {-1+a x}{1+a x}\right )^p (1+a x) (3+p+a p x)+3 \sqrt {\frac {-1+a x}{1+a x}} \operatorname {Hypergeometric2F1}\left (1-p,\frac {3}{2}-p,2-p,\frac {2}{1+a x}\right )\right )}{a^2 (-1+p) p (1+p) \sqrt {1-\frac {1}{a^2 x^2}} x} \] Input:

Integrate[E^(3*ArcCoth[a*x])*(c - a*c*x)^p,x]
 

Output:

((c - a*c*x)^p*((-1 + p)*((-1 + a*x)/(1 + a*x))^p*(1 + a*x)*(3 + p + a*p*x 
) + 3*Sqrt[(-1 + a*x)/(1 + a*x)]*Hypergeometric2F1[1 - p, 3/2 - p, 2 - p, 
2/(1 + a*x)]))/(a^2*(-1 + p)*p*(1 + p)*Sqrt[1 - 1/(a^2*x^2)]*x*((-1 + a*x) 
/(1 + a*x))^p)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(233\) vs. \(2(94)=188\).

Time = 0.58 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.48, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6727, 105, 105, 142}

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.

\(\displaystyle \int e^{3 \coth ^{-1}(a x)} (c-a c x)^p \, dx\)

\(\Big \downarrow \) 6727

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 142

\(\displaystyle \left (\frac {1}{x}\right )^p \left (-\left (1-\frac {1}{a x}\right )^{-p}\right ) (c-a c x)^p \left (\frac {3 \left (\frac {\sqrt {\frac {1}{a x}+1} \left (\frac {1}{x}\right )^{1-p} \left (\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )^{\frac {3}{2}-p} \left (1-\frac {1}{a x}\right )^{p-\frac {3}{2}} \operatorname {Hypergeometric2F1}\left (1-p,\frac {3}{2}-p,2-p,\frac {2}{\left (a+\frac {1}{x}\right ) x}\right )}{a (1-p) p}-\frac {\sqrt {\frac {1}{a x}+1} \left (\frac {1}{x}\right )^{-p} \left (1-\frac {1}{a x}\right )^{p-\frac {1}{2}}}{p}\right )}{a (p+1)}-\frac {\left (\frac {1}{a x}+1\right )^{3/2} \left (\frac {1}{x}\right )^{-p-1} \left (1-\frac {1}{a x}\right )^{p-\frac {1}{2}}}{p+1}\right )\)

Input:

Int[E^(3*ArcCoth[a*x])*(c - a*c*x)^p,x]
 

Output:

-(((x^(-1))^p*(c - a*c*x)^p*(-(((1 - 1/(a*x))^(-1/2 + p)*(1 + 1/(a*x))^(3/ 
2)*(x^(-1))^(-1 - p))/(1 + p)) + (3*(-(((1 - 1/(a*x))^(-1/2 + p)*Sqrt[1 + 
1/(a*x)])/(p*(x^(-1))^p)) + (((a - x^(-1))/(a + x^(-1)))^(3/2 - p)*(1 - 1/ 
(a*x))^(-3/2 + p)*Sqrt[1 + 1/(a*x)]*(x^(-1))^(1 - p)*Hypergeometric2F1[1 - 
 p, 3/2 - p, 2 - p, 2/((a + x^(-1))*x)])/(a*(1 - p)*p)))/(a*(1 + p))))/(1 
- 1/(a*x))^p)
 

Defintions of rubi rules used

rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f)))   Int[(a 
+ b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, 
e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] 
 ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]
 

rule 142

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((b*e 
 - a*f)*(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f))*((a + 
b*x)/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*((c + d*x)/((b*c - a*d)*(e + f 
*x))))^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 
 0] &&  !IntegerQ[n]
 

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 [F]

\[\int \frac {\left (-a c x +c \right )^{p}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}d x\]

Input:

int(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^p,x)
 

Output:

int(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^p,x)
 

Fricas [F]

\[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\int { \frac {{\left (-a c x + c\right )}^{p}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^p,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\int \frac {\left (- c \left (a x - 1\right )\right )^{p}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/((a*x-1)/(a*x+1))**(3/2)*(-a*c*x+c)**p,x)
 

Output:

Integral((-c*(a*x - 1))**p/((a*x - 1)/(a*x + 1))**(3/2), x)
 

Maxima [F]

\[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\int { \frac {{\left (-a c x + c\right )}^{p}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^p,x, algorithm="maxima")
 

Output:

integrate((-a*c*x + c)^p/((a*x - 1)/(a*x + 1))^(3/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^p,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\int \frac {{\left (c-a\,c\,x\right )}^p}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \] Input:

int((c - a*c*x)^p/((a*x - 1)/(a*x + 1))^(3/2),x)
 

Output:

int((c - a*c*x)^p/((a*x - 1)/(a*x + 1))^(3/2), x)
 

Reduce [F]

\[ \int e^{3 \coth ^{-1}(a x)} (c-a c x)^p \, dx=\left (\int \frac {\sqrt {a x +1}\, \left (-a c x +c \right )^{p} x}{\sqrt {a x -1}\, a x -\sqrt {a x -1}}d x \right ) a +\int \frac {\sqrt {a x +1}\, \left (-a c x +c \right )^{p}}{\sqrt {a x -1}\, a x -\sqrt {a x -1}}d x \] Input:

int(1/((a*x-1)/(a*x+1))^(3/2)*(-a*c*x+c)^p,x)
 

Output:

int((sqrt(a*x + 1)*( - a*c*x + c)**p*x)/(sqrt(a*x - 1)*a*x - sqrt(a*x - 1) 
),x)*a + int((sqrt(a*x + 1)*( - a*c*x + c)**p)/(sqrt(a*x - 1)*a*x - sqrt(a 
*x - 1)),x)