\(\int e^{2 \coth ^{-1}(a x)} (c-a^2 c x^2)^p \, dx\) [740]

Optimal result

Integrand size = 22, antiderivative size = 54 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {2^{1+p} (1+a x)^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-p,p,1+p,\frac {1}{2} (1-a x)\right )}{a p} \] Output:

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.24 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {2^{1+p} (1-a x)^p \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-p,p,1+p,\frac {1}{2} (1-a x)\right )}{a p} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6717, 6691, 473, 79}

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.

\(\displaystyle \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx\)

\(\Big \downarrow \) 6717

\(\displaystyle -\int e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^pdx\)

\(\Big \downarrow \) 6691

\(\displaystyle -c \int (a x+1)^2 \left (c-a^2 c x^2\right )^{p-1}dx\)

\(\Big \downarrow \) 473

\(\displaystyle -c (a x+1)^{-p} (c-a c x)^{-p} \left (c-a^2 c x^2\right )^p \int (a x+1)^{p+1} (c-a c x)^{p-1}dx\)

\(\Big \downarrow \) 79

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 473

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
c^(n - 1)*((a + b*x^2)^(p + 1)/((1 + d*(x/c))^(p + 1)*(a/c + (b*x)/d)^(p + 
1)))   Int[(1 + d*(x/c))^(n + p)*(a/c + (b/d)*x)^p, x], x] /; FreeQ[{a, b, 
c, d, n}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[n] || GtQ[c, 0]) &&  !Gt 
Q[a, 0] &&  !(IntegerQ[n] && (IntegerQ[3*p] || IntegerQ[4*p]))
 

rule 6691

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

rule 6717

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2)   Int[ 
u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 13.40 (sec) , antiderivative size = 648, normalized size of antiderivative = 12.00 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\text {Too large to display} \] Input:

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

Output:

a*Piecewise((0**p*x/a - 0**p*log(1/(a**2*x**2))/(2*a**2) + 0**p*log(-1 + 1 
/(a**2*x**2))/(2*a**2) - 0**p*acoth(1/(a*x))/a**2 - a**(2*p - 1)*c**p*p*x* 
*(2*p + 1)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((1 - p, -p - 1/2), ( 
1/2 - p,), 1/(a**2*x**2))/(2*gamma(1/2 - p)*gamma(p + 1)) + c**p*x**2*gamm 
a(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), a**2*x**2*exp_polar(2*I*pi) 
)/(2*gamma(-p)*gamma(p + 1)), 1/Abs(a**2*x**2) > 1), (0**p*x/a - 0**p*log( 
1/(a**2*x**2))/(2*a**2) + 0**p*log(1 - 1/(a**2*x**2))/(2*a**2) - 0**p*atan 
h(1/(a*x))/a**2 - a**(2*p - 1)*c**p*p*x**(2*p + 1)*exp(I*pi*p)*gamma(p)*ga 
mma(-p - 1/2)*hyper((1 - p, -p - 1/2), (1/2 - p,), 1/(a**2*x**2))/(2*gamma 
(1/2 - p)*gamma(p + 1)) + c**p*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - 
 p), (2, 2), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(-p)*gamma(p + 1)), True 
)) + Piecewise((0**p*log(a**2*x**2 - 1)/(2*a) - 0**p*acoth(a*x)/a + a*c**p 
*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), a**2*x**2*exp_pol 
ar(2*I*pi))/(2*gamma(-p)*gamma(p + 1)) - a**(2*p - 2)*c**p*p*x**(2*p - 1)* 
exp(I*pi*p)*gamma(p)*gamma(1/2 - p)*hyper((1 - p, 1/2 - p), (3/2 - p,), 1/ 
(a**2*x**2))/(2*gamma(3/2 - p)*gamma(p + 1)), Abs(a**2*x**2) > 1), (0**p*l 
og(-a**2*x**2 + 1)/(2*a) - 0**p*atanh(a*x)/a + a*c**p*x**2*gamma(p)*gamma( 
1 - p)*hyper((2, 1, 1 - p), (2, 2), a**2*x**2*exp_polar(2*I*pi))/(2*gamma( 
-p)*gamma(p + 1)) - a**(2*p - 2)*c**p*p*x**(2*p - 1)*exp(I*pi*p)*gamma(p)* 
gamma(1/2 - p)*hyper((1 - p, 1/2 - p), (3/2 - p,), 1/(a**2*x**2))/(2*ga...
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {\left (-a^{2} c \,x^{2}+c \right )^{p} a p x +2 \left (-a^{2} c \,x^{2}+c \right )^{p} p +\left (-a^{2} c \,x^{2}+c \right )^{p}+4 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p}}{2 a^{2} p \,x^{2}+a^{2} x^{2}-2 p -1}d x \right ) a \,p^{3}+6 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p}}{2 a^{2} p \,x^{2}+a^{2} x^{2}-2 p -1}d x \right ) a \,p^{2}+2 \left (\int \frac {\left (-a^{2} c \,x^{2}+c \right )^{p}}{2 a^{2} p \,x^{2}+a^{2} x^{2}-2 p -1}d x \right ) a p}{a p \left (2 p +1\right )} \] Input:

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

Output:

(( - a**2*c*x**2 + c)**p*a*p*x + 2*( - a**2*c*x**2 + c)**p*p + ( - a**2*c* 
x**2 + c)**p + 4*int(( - a**2*c*x**2 + c)**p/(2*a**2*p*x**2 + a**2*x**2 - 
2*p - 1),x)*a*p**3 + 6*int(( - a**2*c*x**2 + c)**p/(2*a**2*p*x**2 + a**2*x 
**2 - 2*p - 1),x)*a*p**2 + 2*int(( - a**2*c*x**2 + c)**p/(2*a**2*p*x**2 + 
a**2*x**2 - 2*p - 1),x)*a*p)/(a*p*(2*p + 1))