\(\int \frac {e^{n \arctan (a x)} x^m}{\sqrt {c+a^2 c x^2}} \, dx\) [382]

Optimal result

Integrand size = 26, antiderivative size = 79 \[ \int \frac {e^{n \arctan (a x)} x^m}{\sqrt {c+a^2 c x^2}} \, dx=\frac {x^{1+m} \sqrt {1+a^2 x^2} \operatorname {AppellF1}\left (1+m,\frac {1}{2} (1-i n),\frac {1}{2} (1+i n),2+m,i a x,-i a x\right )}{(1+m) \sqrt {c+a^2 c x^2}} \] Output:

x^(1+m)*(a^2*x^2+1)^(1/2)*AppellF1(1+m,1/2+1/2*I*n,1/2-1/2*I*n,2+m,-I*a*x, 
I*a*x)/(1+m)/(a^2*c*x^2+c)^(1/2)
 

Mathematica [F]

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

Integrate[(E^(n*ArcTan[a*x])*x^m)/Sqrt[c + a^2*c*x^2],x]
 

Output:

Integrate[(E^(n*ArcTan[a*x])*x^m)/Sqrt[c + a^2*c*x^2], x]
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {5608, 5605, 150}

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.

\(\displaystyle \int \frac {x^m e^{n \arctan (a x)}}{\sqrt {a^2 c x^2+c}} \, dx\)

\(\Big \downarrow \) 5608

\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {e^{n \arctan (a x)} x^m}{\sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 5605

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

\(\Big \downarrow \) 150

\(\displaystyle \frac {\sqrt {a^2 x^2+1} x^{m+1} \operatorname {AppellF1}\left (m+1,\frac {1}{2} (1-i n),\frac {1}{2} (i n+1),m+2,i a x,-i a x\right )}{(m+1) \sqrt {a^2 c x^2+c}}\)

Input:

Int[(E^(n*ArcTan[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, (1 - I*n)/2, (1 + I*n)/2, 2 + 
 m, I*a*x, (-I)*a*x])/((1 + m)*Sqrt[c + a^2*c*x^2])
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 5605

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Simp[c^p   Int[x^m*(1 - I*a*x)^(p + I*(n/2))*(1 + I*a*x)^(p - I* 
(n/2)), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && (Integer 
Q[p] || GtQ[c, 0])
 

rule 5608

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 + a^2*x^2)^FracPart 
[p])   Int[x^m*(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c, d, 
 m, n, p}, x] && EqQ[d, a^2*c] &&  !(IntegerQ[p] || GtQ[c, 0])
 

Maple [F]

\[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x^{m}}{\sqrt {a^{2} c \,x^{2}+c}}d x\]

Input:

int(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^(1/2),x)
 

Output:

int(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^(1/2),x)
 

Fricas [F]

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

integrate(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas" 
)
 

Output:

integral(x^m*e^(n*arctan(a*x))/sqrt(a^2*c*x^2 + c), x)
 

Sympy [F]

\[ \int \frac {e^{n \arctan (a x)} x^m}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^{m} e^{n \operatorname {atan}{\left (a x \right )}}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \] Input:

integrate(exp(n*atan(a*x))*x**m/(a**2*c*x**2+c)**(1/2),x)
 

Output:

Integral(x**m*exp(n*atan(a*x))/sqrt(c*(a**2*x**2 + 1)), x)
 

Maxima [F]

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

integrate(exp(n*arctan(a*x))*x^m/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima" 
)
 

Output:

integrate(x^m*e^(n*arctan(a*x))/sqrt(a^2*c*x^2 + c), x)
 

Giac [F]

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

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

Output:

integrate(x^m*e^(n*arctan(a*x))/sqrt(a^2*c*x^2 + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \arctan (a x)} x^m}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^m\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{\sqrt {c\,a^2\,x^2+c}} \,d x \] Input:

int((x^m*exp(n*atan(a*x)))/(c + a^2*c*x^2)^(1/2),x)
 

Output:

int((x^m*exp(n*atan(a*x)))/(c + a^2*c*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {e^{n \arctan (a x)} x^m}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\int \frac {x^{m} e^{\mathit {atan} \left (a x \right ) n}}{\sqrt {a^{2} x^{2}+1}}d x}{\sqrt {c}} \] Input:

int(exp(n*atan(a*x))*x^m/(a^2*c*x^2+c)^(1/2),x)
 

Output:

int((x**m*e**(atan(a*x)*n))/sqrt(a**2*x**2 + 1),x)/sqrt(c)