\(\int e^{\text {arctanh}(a x)} x (c-a^2 c x^2)^p \, dx\) [1051]

Optimal result

Integrand size = 21, antiderivative size = 96 \[ \int e^{\text {arctanh}(a x)} x \left (c-a^2 c x^2\right )^p \, dx=-\frac {\sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p}{a^2 (1+2 p)}+\frac {1}{3} a x^3 \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{2}-p,\frac {5}{2},a^2 x^2\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.92 \[ \int e^{\text {arctanh}(a x)} x \left (c-a^2 c x^2\right )^p \, dx=\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (-\frac {\left (1-a^2 x^2\right )^{\frac {1}{2}+p}}{2 a^2 \left (\frac {1}{2}+p\right )}+\frac {1}{3} a x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{2}-p,\frac {5}{2},a^2 x^2\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6703, 6698, 542, 241, 278}

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.

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

\(\Big \downarrow \) 6703

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

\(\Big \downarrow \) 6698

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

\(\Big \downarrow \) 542

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

((c - a^2*c*x^2)^p*(-((1 - a^2*x^2)^(1/2 + p)/(a^2*(1 + 2*p))) + (a*x^3*Hy 
pergeometric2F1[3/2, 1/2 - p, 5/2, a^2*x^2])/3))/(1 - a^2*x^2)^p
 

Defintions of rubi rules used

rule 241

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ 
(2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 542

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Simp[c   Int[x^m*(a + b*x^2)^p, x], x] + Simp[d   Int[x^(m + 1)*(a + b*x^2 
)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && IntegerQ[m] &&  !IntegerQ[2*p]
 

rule 6698

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

rule 6703

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 7.40 (sec) , antiderivative size = 311, normalized size of antiderivative = 3.24 \[ \int e^{\text {arctanh}(a x)} x \left (c-a^2 c x^2\right )^p \, dx=- \frac {a a^{2 p} c^{p} x^{2 p + 3} e^{i \pi p} \Gamma \left (- p - \frac {3}{2}\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, - p - \frac {3}{2} \\ \frac {1}{2}, - p - \frac {1}{2} \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (- p - \frac {1}{2}\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} c^{p} x^{2 p + 2} e^{i \pi p} \Gamma \left (- p - 1\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, 1 \\ p + 2 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} c^{p} x^{2 p + 2} e^{i \pi p} \Gamma \left (- p - 1\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} 1, - p - 1 \\ \frac {1}{2} \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p + 3} c^{p} x^{2 p + 3} e^{i \pi p} \Gamma \left (- p - \frac {3}{2}\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, p + \frac {3}{2} \\ p + 1, p + \frac {5}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } a^{2} \Gamma \left (- p - \frac {1}{2}\right ) \Gamma \left (p + 1\right )} \] Input:

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

Output:

-a*a**(2*p)*c**p*x**(2*p + 3)*exp(I*pi*p)*gamma(-p - 3/2)*gamma(p + 1/2)*h 
yper((1, -p, -p - 3/2), (1/2, -p - 1/2), 1/(a**2*x**2))/(2*sqrt(pi)*gamma( 
-p - 1/2)*gamma(p + 1)) - a**(2*p)*c**p*x**(2*p + 2)*exp(I*pi*p)*gamma(-p 
- 1)*gamma(p + 1/2)*hyper((1/2, 1), (p + 2,), a**2*x**2*exp_polar(2*I*pi)) 
/(2*sqrt(pi)*gamma(-p)*gamma(p + 1)) - a**(2*p)*c**p*x**(2*p + 2)*exp(I*pi 
*p)*gamma(-p - 1)*gamma(p + 1/2)*hyper((1, -p - 1), (1/2,), 1/(a**2*x**2)) 
/(2*sqrt(pi)*gamma(-p)*gamma(p + 1)) - a**(2*p + 3)*c**p*x**(2*p + 3)*exp( 
I*pi*p)*gamma(-p - 3/2)*gamma(p + 1/2)*hyper((1/2, 1, p + 3/2), (p + 1, p 
+ 5/2), a**2*x**2*exp_polar(2*I*pi))/(2*sqrt(pi)*a**2*gamma(-p - 1/2)*gamm 
a(p + 1))
 

Maxima [F]

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

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

Output:

a*c^p*integrate(x^2*e^(p*log(a*x + 1) + p*log(-a*x + 1))/(sqrt(a*x + 1)*sq 
rt(-a*x + 1)), x) + (a^2*c^p*x^2 - c^p)*(-a^2*x^2 + 1)^p/(sqrt(-a^2*x^2 + 
1)*a^2*(2*p + 1))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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