\(\int e^{3 \text {arctanh}(a x)} (c-a^2 c x^2)^p \, dx\) [1200]

Optimal result

Integrand size = 22, antiderivative size = 85 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {2^{\frac {5}{2}+p} (1-a x)^{-\frac {1}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}-p,-\frac {1}{2}+p,\frac {1}{2}+p,\frac {1}{2} (1-a x)\right )}{a (1-2 p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.98 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {2^{\frac {5}{2}+p} (1-a x)^{-\frac {1}{2}+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}-p,-\frac {1}{2}+p,\frac {1}{2}+p,\frac {1}{2} (1-a x)\right )}{a-2 a p} \] Input:

Integrate[E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^p,x]
 

Output:

(2^(5/2 + p)*(1 - a*x)^(-1/2 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-3/2 
 - p, -1/2 + p, 1/2 + p, (1 - a*x)/2])/((a - 2*a*p)*(1 - a^2*x^2)^p)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6693, 6690, 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.

\(\displaystyle \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx\)

\(\Big \downarrow \) 6693

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int e^{3 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^pdx\)

\(\Big \downarrow \) 6690

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {2^{p+\frac {5}{2}} (1-a x)^{p-\frac {1}{2}} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-p-\frac {3}{2},p-\frac {1}{2},p+\frac {1}{2},\frac {1}{2} (1-a x)\right )}{a (1-2 p)}\)

Input:

Int[E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^p,x]
 

Output:

(2^(5/2 + p)*(1 - a*x)^(-1/2 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-3/2 
 - p, -1/2 + p, 1/2 + p, (1 - a*x)/2])/(a*(1 - 2*p)*(1 - a^2*x^2)^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 6690

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

rule 6693

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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