\(\int e^{n \arctan (a+b x)} x^2 \, dx\) [252]

Optimal result

Integrand size = 14, antiderivative size = 220 \[ \int e^{n \arctan (a+b x)} x^2 \, dx=-\frac {(4 a+n) (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{6 b^3}+\frac {x (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{3 b^2}+\frac {2^{-\frac {i n}{2}} \left (2-6 a^2-6 a n-n^2\right ) (1-i a-i b x)^{1+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (1+\frac {i n}{2},\frac {i n}{2},2+\frac {i n}{2},\frac {1}{2} (1-i a-i b x)\right )}{3 b^3 (2 i-n)} \] Output:

-1/6*(4*a+n)*(1-I*a-I*b*x)^(1+1/2*I*n)*(1+I*a+I*b*x)^(1-1/2*I*n)/b^3+1/3*x 
*(1-I*a-I*b*x)^(1+1/2*I*n)*(1+I*a+I*b*x)^(1-1/2*I*n)/b^2+1/3*(-6*a^2-6*a*n 
-n^2+2)*(1-I*a-I*b*x)^(1+1/2*I*n)*hypergeom([1/2*I*n, 1+1/2*I*n],[2+1/2*I* 
n],1/2-1/2*I*a-1/2*I*b*x)/(2^(1/2*I*n))/b^3/(2*I-n)
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.73 \[ \int e^{n \arctan (a+b x)} x^2 \, dx=\frac {(-i (i+a+b x))^{1+\frac {i n}{2}} \left (-\left ((4 a+n) (1+i a+i b x)^{1-\frac {i n}{2}}\right )+2 b x (1+i a+i b x)^{1-\frac {i n}{2}}+\frac {2^{1-\frac {i n}{2}} \left (-2+6 a^2+6 a n+n^2\right ) \operatorname {Hypergeometric2F1}\left (1+\frac {i n}{2},\frac {i n}{2},2+\frac {i n}{2},-\frac {1}{2} i (i+a+b x)\right )}{-2 i+n}\right )}{6 b^3} \] Input:

Integrate[E^(n*ArcTan[a + b*x])*x^2,x]
 

Output:

(((-I)*(I + a + b*x))^(1 + (I/2)*n)*(-((4*a + n)*(1 + I*a + I*b*x)^(1 - (I 
/2)*n)) + 2*b*x*(1 + I*a + I*b*x)^(1 - (I/2)*n) + (2^(1 - (I/2)*n)*(-2 + 6 
*a^2 + 6*a*n + n^2)*Hypergeometric2F1[1 + (I/2)*n, (I/2)*n, 2 + (I/2)*n, ( 
-1/2*I)*(I + a + b*x)])/(-2*I + n)))/(6*b^3)
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5618, 101, 25, 90, 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.

\(\displaystyle \int x^2 e^{n \arctan (a+b x)} \, dx\)

\(\Big \downarrow \) 5618

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

\(\Big \downarrow \) 101

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 90

\(\displaystyle \frac {x (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{3 b^2}-\frac {\frac {1}{2} \left (-6 a^2-6 a n-n^2+2\right ) \int (-i a-i b x+1)^{\frac {i n}{2}} (i a+i b x+1)^{-\frac {i n}{2}}dx+\frac {(4 a+n) (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{2 b}}{3 b^2}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {x (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{3 b^2}-\frac {\frac {(4 a+n) (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{2 b}-\frac {2^{-\frac {i n}{2}} \left (-6 a^2-6 a n-n^2+2\right ) (-i a-i b x+1)^{1+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2}+1,\frac {i n}{2},\frac {i n}{2}+2,\frac {1}{2} (-i a-i b x+1)\right )}{b (-n+2 i)}}{3 b^2}\)

Input:

Int[E^(n*ArcTan[a + b*x])*x^2,x]
 

Output:

(x*(1 - I*a - I*b*x)^(1 + (I/2)*n)*(1 + I*a + I*b*x)^(1 - (I/2)*n))/(3*b^2 
) - (((4*a + n)*(1 - I*a - I*b*x)^(1 + (I/2)*n)*(1 + I*a + I*b*x)^(1 - (I/ 
2)*n))/(2*b) - ((2 - 6*a^2 - 6*a*n - n^2)*(1 - I*a - I*b*x)^(1 + (I/2)*n)* 
Hypergeometric2F1[1 + (I/2)*n, (I/2)*n, 2 + (I/2)*n, (1 - I*a - I*b*x)/2]) 
/(2^((I/2)*n)*b*(2*I - n)))/(3*b^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 90

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

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 5618

Int[E^(ArcTan[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), 
 x_Symbol] :> Int[(d + e*x)^m*((1 - I*a*c - I*b*c*x)^(I*(n/2))/(1 + I*a*c + 
 I*b*c*x)^(I*(n/2))), x] /; FreeQ[{a, b, c, d, e, m, n}, x]
 

Maple [F]

\[\int {\mathrm e}^{n \arctan \left (b x +a \right )} x^{2}d x\]

Input:

int(exp(n*arctan(b*x+a))*x^2,x)
 

Output:

int(exp(n*arctan(b*x+a))*x^2,x)
 

Fricas [F]

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

integrate(exp(n*arctan(b*x+a))*x^2,x, algorithm="fricas")
 

Output:

integral(x^2*e^(n*arctan(b*x + a)), x)
 

Sympy [F]

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

integrate(exp(n*atan(b*x+a))*x**2,x)
 

Output:

Integral(x**2*exp(n*atan(a + b*x)), x)
 

Maxima [F]

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

integrate(exp(n*arctan(b*x+a))*x^2,x, algorithm="maxima")
 

Output:

integrate(x^2*e^(n*arctan(b*x + a)), x)
 

Giac [F(-1)]

Timed out. \[ \int e^{n \arctan (a+b x)} x^2 \, dx=\text {Timed out} \] Input:

integrate(exp(n*arctan(b*x+a))*x^2,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

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

int(x^2*exp(n*atan(a + b*x)),x)
 

Output:

int(x^2*exp(n*atan(a + b*x)), x)
 

Reduce [F]

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

int(exp(n*atan(b*x+a))*x^2,x)
 

Output:

int(e**(atan(a + b*x)*n)*x**2,x)