\(\int e^{\text {arctanh}(a x)} x^m (c-a^2 c x^2) \, dx\) [1010]

Optimal result

Integrand size = 21, antiderivative size = 76 \[ \int e^{\text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right ) \, dx=\frac {c x^{1+m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{1+m}+\frac {a c x^{2+m} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},a^2 x^2\right )}{2+m} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.95 \[ \int e^{\text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right ) \, dx=c x^{1+m} \left (\frac {a x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1+\frac {m}{2},2+\frac {m}{2},a^2 x^2\right )}{2+m}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1+m}{2},\frac {3+m}{2},a^2 x^2\right )}{1+m}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {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.

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

\(\Big \downarrow \) 6698

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

\(\Big \downarrow \) 557

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

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

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(142\) vs. \(2(68)=136\).

Time = 0.19 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.88

method	result	size
meijerg	\(-\frac {a^{3} c \,x^{4+m} \operatorname {hypergeom}\left (\left [\frac {1}{2}, 2+\frac {m}{2}\right ], \left [\frac {m}{2}+3\right ], a^{2} x^{2}\right )}{4+m}+\frac {a c \,x^{2+m} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {m}{2}+1\right ], \left [2+\frac {m}{2}\right ], a^{2} x^{2}\right )}{2+m}-\frac {c \,a^{2} x^{3+m} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {3}{2}+\frac {m}{2}\right ], \left [\frac {5}{2}+\frac {m}{2}\right ], a^{2} x^{2}\right )}{3+m}+\frac {c \,x^{1+m} \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {m}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}+\frac {m}{2}\right ], a^{2} x^{2}\right )}{1+m}\)	\(143\)

Input:

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

Output:

-a^3*c/(4+m)*x^(4+m)*hypergeom([1/2,2+1/2*m],[1/2*m+3],a^2*x^2)+a*c/(2+m)* 
x^(2+m)*hypergeom([1/2,1/2*m+1],[2+1/2*m],a^2*x^2)-c*a^2/(3+m)*x^(3+m)*hyp 
ergeom([1/2,3/2+1/2*m],[5/2+1/2*m],a^2*x^2)+c/(1+m)*x^(1+m)*hypergeom([1/2 
,1/2*m+1/2],[3/2+1/2*m],a^2*x^2)
 

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.37 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.68 \[ \int e^{\text {arctanh}(a x)} x^m \left (c-a^2 c x^2\right ) \, dx=- \frac {a^{3} c x^{m + 4} \Gamma \left (\frac {m}{2} + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{2} + 2 \\ \frac {m}{2} + 3 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {m}{2} + 3\right )} - \frac {a^{2} c x^{m + 3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {a c x^{m + 2} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {m}{2} + 2\right )} + \frac {c x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} \] Input:

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

Output:

-a**3*c*x**(m + 4)*gamma(m/2 + 2)*hyper((1/2, m/2 + 2), (m/2 + 3,), a**2*x 
**2*exp_polar(2*I*pi))/(2*gamma(m/2 + 3)) - a**2*c*x**(m + 3)*gamma(m/2 + 
3/2)*hyper((1/2, m/2 + 3/2), (m/2 + 5/2,), a**2*x**2*exp_polar(2*I*pi))/(2 
*gamma(m/2 + 5/2)) + a*c*x**(m + 2)*gamma(m/2 + 1)*hyper((1/2, m/2 + 1), ( 
m/2 + 2,), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(m/2 + 2)) + c*x**(m + 1)* 
gamma(m/2 + 1/2)*hyper((1/2, m/2 + 1/2), (m/2 + 3/2,), a**2*x**2*exp_polar 
(2*I*pi))/(2*gamma(m/2 + 3/2))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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