\(\int e^{\text {arctanh}(a x)} x^m (c-a^2 c x^2)^p \, dx\) [1026]

Optimal result

Integrand size = 23, antiderivative size = 136 \[ \int e^{\text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx=\frac {x^{1+m} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {1}{2}-p,\frac {3+m}{2},a^2 x^2\right )}{1+m}+\frac {a x^{2+m} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},\frac {1}{2}-p,\frac {4+m}{2},a^2 x^2\right )}{2+m} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.84 \[ \int e^{\text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx=\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},\frac {1}{2}-p,1+\frac {1+m}{2},a^2 x^2\right )}{1+m}+\frac {a x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},\frac {1}{2}-p,1+\frac {2+m}{2},a^2 x^2\right )}{2+m}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6703, 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.

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

\(\Big \downarrow \) 6703

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

\(\Big \downarrow \) 6698

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

\(\Big \downarrow \) 557

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

\(\Big \downarrow \) 278

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},\frac {1}{2}-p,\frac {m+3}{2},a^2 x^2\right )}{m+1}+\frac {a x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},\frac {1}{2}-p,\frac {m+4}{2},a^2 x^2\right )}{m+2}\right )\)

Input:

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

Output:

((c - a^2*c*x^2)^p*((x^(1 + m)*Hypergeometric2F1[(1 + m)/2, 1/2 - p, (3 + 
m)/2, a^2*x^2])/(1 + m) + (a*x^(2 + m)*Hypergeometric2F1[(2 + m)/2, 1/2 - 
p, (4 + m)/2, a^2*x^2])/(2 + m)))/(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 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 41.73 (sec) , antiderivative size = 391, normalized size of antiderivative = 2.88 \[ \int e^{\text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right )^p \, dx=- \frac {a a^{2 p} c^{p} x^{m + 2 p + 2} e^{i \pi p} \Gamma \left (p + \frac {1}{2}\right ) \Gamma \left (- \frac {m}{2} - p - 1\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, - \frac {m}{2} - p - 1 \\ \frac {1}{2}, - \frac {m}{2} - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (- \frac {m}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {a a^{- m - 2} a^{m + 2 p + 2} c^{p} x^{m + 2 p + 2} e^{i \pi p} \Gamma \left (p + \frac {1}{2}\right ) \Gamma \left (- \frac {m}{2} - p - 1\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, \frac {m}{2} + p + 1 \\ p + 1, \frac {m}{2} + p + 2 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (- \frac {m}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} c^{p} x^{m + 2 p + 1} e^{i \pi p} \Gamma \left (p + \frac {1}{2}\right ) \Gamma \left (- \frac {m}{2} - p - \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, - \frac {m}{2} - p - \frac {1}{2} \\ \frac {1}{2}, - \frac {m}{2} - p + \frac {1}{2} \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (p + 1\right ) \Gamma \left (- \frac {m}{2} - p + \frac {1}{2}\right )} - \frac {a^{- m - 1} a^{m + 2 p + 1} c^{p} x^{m + 2 p + 1} e^{i \pi p} \Gamma \left (p + \frac {1}{2}\right ) \Gamma \left (- \frac {m}{2} - p - \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, \frac {m}{2} + p + \frac {1}{2} \\ p + 1, \frac {m}{2} + p + \frac {3}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (p + 1\right ) \Gamma \left (- \frac {m}{2} - p + \frac {1}{2}\right )} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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