\(\int \frac {e^{2 \text {arctanh}(a x)} x^m}{\sqrt {c-a^2 c x^2}} \, dx\) [1156]

Optimal result

Integrand size = 27, antiderivative size = 166 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^m}{\sqrt {c-a^2 c x^2}} \, dx=-\frac {x^{1+m}}{m \sqrt {c-a^2 c x^2}}+\frac {(1+2 m) x^{1+m} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{m (1+m) \sqrt {c-a^2 c x^2}}+\frac {2 a x^{2+m} \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{(2+m) \sqrt {c-a^2 c x^2}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.40 \[ \int \frac {e^{2 \text {arctanh}(a x)} x^m}{\sqrt {c-a^2 c x^2}} \, dx=\frac {x^{1+m} \sqrt {1-a^2 x^2} \operatorname {AppellF1}\left (1+m,\frac {3}{2},-\frac {1}{2},2+m,a x,-a x\right )}{(1+m) \sqrt {-1+a x} \sqrt {-c (1+a x)}} \] Input:

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

Output:

(x^(1 + m)*Sqrt[1 - a^2*x^2]*AppellF1[1 + m, 3/2, -1/2, 2 + m, a*x, -(a*x) 
])/((1 + m)*Sqrt[-1 + a*x]*Sqrt[-(c*(1 + a*x))])
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6701, 558, 557, 279, 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.

\(\displaystyle \int \frac {x^m e^{2 \text {arctanh}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx\)

\(\Big \downarrow \) 6701

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

\(\Big \downarrow \) 558

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

\(\Big \downarrow \) 557

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

c*((2*x^(1 + m)*(1 + a*x))/(c*Sqrt[c - a^2*c*x^2]) - (((1 + 2*m)*x^(1 + m) 
*Sqrt[1 - a^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2])/ 
((1 + m)*Sqrt[c - a^2*c*x^2]) + (2*a*(1 + m)*x^(2 + m)*Sqrt[1 - a^2*x^2]*H 
ypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^2])/((2 + m)*Sqrt[c - a^ 
2*c*x^2]))/c)
 

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 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], 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 558

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), 
 x_Symbol] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, a + b*x^2, x], f = 
 Coeff[PolynomialRemainder[(c + d*x)^n, a + b*x^2, x], x, 0], g = Coeff[Pol 
ynomialRemainder[(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(-(e*x)^(m + 1))* 
(f + g*x)*((a + b*x^2)^(p + 1)/(2*a*e*(p + 1))), x] + Simp[1/(2*a*(p + 1)) 
  Int[(e*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Qx + f*(m + 2*p + 
 3) + g*(m + 2*p + 4)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, m}, x] && IGt 
Q[n, 1] &&  !IntegerQ[m] && LtQ[p, -1]
 

rule 6701

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[c^(n/2)   Int[x^m*(c + d*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/2, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

-Integral(x**m/(a*x*sqrt(-a**2*c*x**2 + c) - sqrt(-a**2*c*x**2 + c)), x) - 
 Integral(a*x*x**m/(a*x*sqrt(-a**2*c*x**2 + c) - sqrt(-a**2*c*x**2 + c)), 
x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - (int(x**m/(sqrt( - a**2*x**2 + 1)*a*x - sqrt( - a**2*x**2 + 1)),x) + i 
nt((x**m*x)/(sqrt( - a**2*x**2 + 1)*a*x - sqrt( - a**2*x**2 + 1)),x)*a))/s 
qrt(c)