\(\int \frac {e^{n \arctan (a x)} x^3}{\sqrt {c+a^2 c x^2}} \, dx\) [367]

Optimal result

Integrand size = 26, antiderivative size = 390 \[ \int \frac {e^{n \arctan (a x)} x^3}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {(2 i+n) (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \sqrt {1+a^2 x^2}}{3 a^4 (i-n) \sqrt {c+a^2 c x^2}}+\frac {x^2 (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \sqrt {1+a^2 x^2}}{3 a^2 \sqrt {c+a^2 c x^2}}-\frac {i n (1-i a x)^{\frac {1}{2} (3+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \sqrt {1+a^2 x^2}}{a^4 \left (1+n^2\right ) \sqrt {c+a^2 c x^2}}-\frac {2^{\frac {3}{2}-\frac {i n}{2}} n \left (5-n^2\right ) (1-i a x)^{\frac {1}{2} (3+i n)} \sqrt {1+a^2 x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-1+i n),\frac {1}{2} (3+i n),\frac {1}{2} (5+i n),\frac {1}{2} (1-i a x)\right )}{3 a^4 \left (3 i-n+3 i n^2-n^3\right ) \sqrt {c+a^2 c x^2}} \] Output:

-1/3*(2*I+n)*(1-I*a*x)^(1/2+1/2*I*n)*(1+I*a*x)^(1/2-1/2*I*n)*(a^2*x^2+1)^( 
1/2)/a^4/(I-n)/(a^2*c*x^2+c)^(1/2)+1/3*x^2*(1-I*a*x)^(1/2+1/2*I*n)*(1+I*a* 
x)^(1/2-1/2*I*n)*(a^2*x^2+1)^(1/2)/a^2/(a^2*c*x^2+c)^(1/2)-I*n*(1-I*a*x)^( 
3/2+1/2*I*n)*(1+I*a*x)^(1/2-1/2*I*n)*(a^2*x^2+1)^(1/2)/a^4/(n^2+1)/(a^2*c* 
x^2+c)^(1/2)-1/3*2^(3/2-1/2*I*n)*n*(-n^2+5)*(1-I*a*x)^(3/2+1/2*I*n)*(a^2*x 
^2+1)^(1/2)*hypergeom([3/2+1/2*I*n, -1/2+1/2*I*n],[5/2+1/2*I*n],1/2-1/2*I* 
a*x)/a^4/(3*I-n+3*I*n^2-n^3)/(a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.37 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.64 \[ \int \frac {e^{n \arctan (a x)} x^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {2^{-\frac {3}{2}-\frac {i n}{2}} (1-i a x)^{\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \sqrt {1+a^2 x^2} \left (2^{\frac {1}{2}+\frac {i n}{2}} (-3 i+n) \sqrt {1+i a x} \left (-n^2 (i+a x)-2 i \left (-2+a^2 x^2\right )+n \left (1+i a x+2 a^2 x^2\right )\right )+2 n \left (-5+n^2\right ) (1+i a x)^{\frac {i n}{2}} (i+a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+\frac {i n}{2},\frac {3}{2}+\frac {i n}{2},\frac {5}{2}+\frac {i n}{2},\frac {1}{2}-\frac {i a x}{2}\right )\right )}{3 a^4 \left (-3-4 i n+n^2\right ) \sqrt {c+a^2 c x^2}} \] Input:

Integrate[(E^(n*ArcTan[a*x])*x^3)/Sqrt[c + a^2*c*x^2],x]
 

Output:

(2^(-3/2 - (I/2)*n)*(1 - I*a*x)^(1/2 + (I/2)*n)*Sqrt[1 + a^2*x^2]*(2^(1/2 
+ (I/2)*n)*(-3*I + n)*Sqrt[1 + I*a*x]*(-(n^2*(I + a*x)) - (2*I)*(-2 + a^2* 
x^2) + n*(1 + I*a*x + 2*a^2*x^2)) + 2*n*(-5 + n^2)*(1 + I*a*x)^((I/2)*n)*( 
I + a*x)*Hypergeometric2F1[1/2 + (I/2)*n, 3/2 + (I/2)*n, 5/2 + (I/2)*n, 1/ 
2 - (I/2)*a*x]))/(3*a^4*(-3 - (4*I)*n + n^2)*(1 + I*a*x)^((I/2)*n)*Sqrt[c 
+ a^2*c*x^2])
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.71, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5608, 5605, 111, 25, 163, 79}

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.

\(\displaystyle \int \frac {x^3 e^{n \arctan (a x)}}{\sqrt {a^2 c x^2+c}} \, dx\)

\(\Big \downarrow \) 5608

\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {e^{n \arctan (a x)} x^3}{\sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 5605

\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int x^3 (1-i a x)^{\frac {1}{2} (i n-1)} (i a x+1)^{\frac {1}{2} (-i n-1)}dx}{\sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 163

\(\displaystyle \frac {\sqrt {a^2 x^2+1} \left (\frac {x^2 (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)}}{3 a^2}-\frac {\frac {(1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \left (a (1+i n) n x-n^2-i n+4\right )}{2 a^2 (1+i n)}-\frac {n \left (5-n^2\right ) \int (1-i a x)^{\frac {1}{2} (i n+1)} (i a x+1)^{\frac {1}{2} (-i n-1)}dx}{2 a (1+i n)}}{3 a^2}\right )}{\sqrt {a^2 c x^2+c}}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {\sqrt {a^2 x^2+1} \left (\frac {x^2 (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)}}{3 a^2}-\frac {\frac {2^{-\frac {1}{2}-\frac {i n}{2}} n \left (5-n^2\right ) (1-i a x)^{\frac {1}{2} (3+i n)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (i n+1),\frac {1}{2} (i n+3),\frac {1}{2} (i n+5),\frac {1}{2} (1-i a x)\right )}{a^2 (-n+3 i) (1+i n)}+\frac {(1-i a x)^{\frac {1}{2} (1+i n)} \left (a (1+i n) n x-n^2-i n+4\right ) (1+i a x)^{\frac {1}{2} (1-i n)}}{2 a^2 (1+i n)}}{3 a^2}\right )}{\sqrt {a^2 c x^2+c}}\)

Input:

Int[(E^(n*ArcTan[a*x])*x^3)/Sqrt[c + a^2*c*x^2],x]
 

Output:

(Sqrt[1 + a^2*x^2]*((x^2*(1 - I*a*x)^((1 + I*n)/2)*(1 + I*a*x)^((1 - I*n)/ 
2))/(3*a^2) - (((1 - I*a*x)^((1 + I*n)/2)*(1 + I*a*x)^((1 - I*n)/2)*(4 - I 
*n - n^2 + a*(1 + I*n)*n*x))/(2*a^2*(1 + I*n)) + (2^(-1/2 - (I/2)*n)*n*(5 
- n^2)*(1 - I*a*x)^((3 + I*n)/2)*Hypergeometric2F1[(1 + I*n)/2, (3 + I*n)/ 
2, (5 + I*n)/2, (1 - I*a*x)/2])/(a^2*(3*I - n)*(1 + I*n)))/(3*a^2)))/Sqrt[ 
c + a^2*c*x^2]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 79

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

rule 111

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

rule 163

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n 
+ 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(b*c - a*d)* 
(m + 1)*x)/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)))*(a + b*x)^(m + 1)*(c + 
d*x)^(n + 1), x] - Simp[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f 
*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c* 
d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2* 
d*(b*c - a*d)*(m + 1)*(m + n + 3))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((GeQ[m, -2] && LtQ[m, - 
1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 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]

\[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x^{3}}{\sqrt {a^{2} c \,x^{2}+c}}d x\]

Input:

int(exp(n*arctan(a*x))*x^3/(a^2*c*x^2+c)^(1/2),x)
 

Output:

int(exp(n*arctan(a*x))*x^3/(a^2*c*x^2+c)^(1/2),x)
 

Fricas [F]

\[ \int \frac {e^{n \arctan (a x)} x^3}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x^{3} e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \] Input:

integrate(exp(n*arctan(a*x))*x^3/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas" 
)
 

Output:

integral(x^3*e^(n*arctan(a*x))/sqrt(a^2*c*x^2 + c), x)
 

Sympy [F]

\[ \int \frac {e^{n \arctan (a x)} x^3}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^{3} e^{n \operatorname {atan}{\left (a x \right )}}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \] Input:

integrate(exp(n*atan(a*x))*x**3/(a**2*c*x**2+c)**(1/2),x)
 

Output:

Integral(x**3*exp(n*atan(a*x))/sqrt(c*(a**2*x**2 + 1)), x)
 

Maxima [F]

\[ \int \frac {e^{n \arctan (a x)} x^3}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x^{3} e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \] Input:

integrate(exp(n*arctan(a*x))*x^3/(a^2*c*x^2+c)^(1/2),x, algorithm="maxima" 
)
 

Output:

integrate(x^3*e^(n*arctan(a*x))/sqrt(a^2*c*x^2 + c), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{n \arctan (a x)} x^3}{\sqrt {c+a^2 c x^2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(exp(n*arctan(a*x))*x^3/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{n \arctan (a x)} x^3}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^3\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{\sqrt {c\,a^2\,x^2+c}} \,d x \] Input:

int((x^3*exp(n*atan(a*x)))/(c + a^2*c*x^2)^(1/2),x)
 

Output:

int((x^3*exp(n*atan(a*x)))/(c + a^2*c*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {e^{n \arctan (a x)} x^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\int \frac {e^{\mathit {atan} \left (a x \right ) n} x^{3}}{\sqrt {a^{2} x^{2}+1}}d x}{\sqrt {c}} \] Input:

int(exp(n*atan(a*x))*x^3/(a^2*c*x^2+c)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

int((e**(atan(a*x)*n)*x**3)/sqrt(a**2*x**2 + 1),x)/sqrt(c)