\(\int e^{\text {arctanh}(a x)} (c-\frac {c}{a^2 x^2})^p \, dx\) [777]

Optimal result

Integrand size = 20, antiderivative size = 137 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x \left (1-a^2 x^2\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 p),\frac {1}{2}-p,\frac {1}{2} (3-2 p),a^2 x^2\right )}{1-2 p}+\frac {a \left (c-\frac {c}{a^2 x^2}\right )^p x^2 \left (1-a^2 x^2\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,1-p,2-p,a^2 x^2\right )}{2 (1-p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x \left (1-a^2 x^2\right )^{-p} \left (2 (-1+p) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,\frac {1}{2}-p,\frac {3}{2}-p,a^2 x^2\right )+a (-1+2 p) x \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,1-p,2-p,a^2 x^2\right )\right )}{2 (-1+p) (-1+2 p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6710, 6698, 557, 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.

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

\(\Big \downarrow \) 6710

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

\(\Big \downarrow \) 6698

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

\(\Big \downarrow \) 557

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

((c - c/(a^2*x^2))^p*x^(2*p)*((x^(1 - 2*p)*Hypergeometric2F1[(1 - 2*p)/2, 
1/2 - p, (3 - 2*p)/2, a^2*x^2])/(1 - 2*p) + (a*x^(2 - 2*p)*Hypergeometric2 
F1[1/2 - p, 1 - p, 2 - p, a^2*x^2])/(2*(1 - p))))/(1 - a^2*x^2)^p
 

Defintions of rubi rules used

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 557

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

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 6710

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 17.18 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.30 \[ \int e^{\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {a c^{p} x^{2} \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, 1 \\ 2, p + 1 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (p + 1\right )} + \frac {c^{p} x \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {1}{2}, 1, - p \\ \frac {1}{2}, \frac {1}{2} \end {matrix}\middle | {\frac {e^{2 i \pi }}{a^{2} x^{2}}} \right )}}{\sqrt {\pi } \Gamma \left (p + 1\right )} + \frac {c^{p} x \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, \frac {1}{2}, 1 \\ \frac {3}{2}, p + 1 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{\sqrt {\pi } \Gamma \left (p + 1\right )} - \frac {c^{p} {G_{3, 3}^{2, 2}\left (\begin {matrix} -1, p & 1 \\-1, 0 & - \frac {1}{2} \end {matrix} \middle | {\frac {e^{i \pi }}{a^{2} x^{2}}} \right )} \Gamma \left (p + \frac {1}{2}\right )}{2 a \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} \] Input:

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

Output:

a*c**p*x**2*gamma(p + 1/2)*hyper((1/2, 1, 1), (2, p + 1), a**2*x**2*exp_po 
lar(2*I*pi))/(2*sqrt(pi)*gamma(p + 1)) + c**p*x*gamma(p + 1/2)*hyper((-1/2 
, 1, -p), (1/2, 1/2), exp_polar(2*I*pi)/(a**2*x**2))/(sqrt(pi)*gamma(p + 1 
)) + c**p*x*gamma(p + 1/2)*hyper((1/2, 1/2, 1), (3/2, p + 1), a**2*x**2*ex 
p_polar(2*I*pi))/(sqrt(pi)*gamma(p + 1)) - c**p*meijerg(((-1, p), (1,)), ( 
(-1, 0), (-1/2,)), exp_polar(I*pi)/(a**2*x**2))*gamma(p + 1/2)/(2*a*gamma( 
-p)*gamma(p + 1))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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