\(\int \frac {e^{\text {arctanh}(a x)} (c-a^2 c x^2)^p}{x^2} \, dx\) [1039]

Optimal result

Integrand size = 23, antiderivative size = 113 \[ \int \frac {e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx=-\frac {\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2}-p,\frac {1}{2},a^2 x^2\right )}{x}-\frac {a \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+p,\frac {3}{2}+p,1-a^2 x^2\right )}{1+2 p} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx=\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2}-p,\frac {1}{2},a^2 x^2\right )}{x}-\frac {a \left (1-a^2 x^2\right )^{\frac {1}{2}+p} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+p,\frac {3}{2}+p,1-a^2 x^2\right )}{2 \left (\frac {1}{2}+p\right )}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6703, 6698, 542, 243, 75, 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.

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

\(\Big \downarrow \) 6703

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

\(\Big \downarrow \) 6698

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

\(\Big \downarrow \) 542

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 75

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 75

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 7.70 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.64 \[ \int \frac {e^{\text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx=- \frac {a a^{2 p} c^{p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, p \\ p + 1, p + 1 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (1 - p\right ) \Gamma \left (p + 1\right )} - \frac {a a^{2 p} c^{p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, - p \\ \frac {1}{2}, 1 - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (1 - p\right ) \Gamma \left (p + 1\right )} - \frac {a a^{2 p - 1} c^{p} x^{2 p - 1} e^{i \pi p} \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, p - \frac {1}{2} \\ p + \frac {1}{2}, p + 1 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (\frac {3}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} c^{p} x^{2 p - 1} e^{i \pi p} \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, \frac {1}{2} - p \\ \frac {1}{2}, \frac {3}{2} - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (\frac {3}{2} - p\right ) \Gamma \left (p + 1\right )} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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