3.3.37 \(\int e^{n \arctan (a+b x)} x^3 \, dx\) [237]

3.3.37.1 Optimal result

Integrand size = 14, antiderivative size = 260 \[ \int e^{n \arctan (a+b x)} x^3 \, dx=\frac {x^2 (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{4 b^2}-\frac {(1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}} \left (6-18 a^2-10 a n-n^2+2 b (6 a+n) x\right )}{24 b^4}+\frac {2^{-2-\frac {i n}{2}} \left (24 a^3+36 a^2 n-12 a \left (2-n^2\right )-n \left (8-n^2\right )\right ) (1-i a-i b x)^{1+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (1+\frac {i n}{2},\frac {i n}{2},2+\frac {i n}{2},\frac {1}{2} (1-i a-i b x)\right )}{3 b^4 (2 i-n)} \]

output

1/4*x^2*(1-I*a-I*b*x)^(1+1/2*I*n)*(1+I*a+I*b*x)^(1-1/2*I*n)/b^2-1/24*(1-I* 
a-I*b*x)^(1+1/2*I*n)*(1+I*a+I*b*x)^(1-1/2*I*n)*(6-18*a^2-10*a*n-n^2+2*b*(6 
*a+n)*x)/b^4+1/3*2^(-2-1/2*I*n)*(24*a^3+36*a^2*n-12*a*(-n^2+2)-n*(-n^2+8)) 
*(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)/b^4/(2*I-n)
 

3.3.37.2 Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.05 \[ \int e^{n \arctan (a+b x)} x^3 \, dx=\frac {(-i (i+a+b x))^{1+\frac {i n}{2}} \left (b^2 (2 i-n) x^2 (1+i a+i b x)^{1-\frac {i n}{2}}-2^{3-\frac {i n}{2}} (6 a+n) \operatorname {Hypergeometric2F1}\left (-2+\frac {i n}{2},1+\frac {i n}{2},2+\frac {i n}{2},-\frac {1}{2} i (i+a+b x)\right )+2^{3-\frac {i n}{2}} (1+i a) (-i+5 a+n) \operatorname {Hypergeometric2F1}\left (-1+\frac {i n}{2},1+\frac {i n}{2},2+\frac {i n}{2},-\frac {1}{2} i (i+a+b x)\right )+2^{1-\frac {i n}{2}} (-i+a)^2 (-2 i+4 a+n) \operatorname {Hypergeometric2F1}\left (1+\frac {i n}{2},\frac {i n}{2},2+\frac {i n}{2},-\frac {1}{2} i (i+a+b x)\right )\right )}{4 b^4 (2 i-n)} \]

input

Integrate[E^(n*ArcTan[a + b*x])*x^3,x]
 

output

(((-I)*(I + a + b*x))^(1 + (I/2)*n)*(b^2*(2*I - n)*x^2*(1 + I*a + I*b*x)^( 
1 - (I/2)*n) - 2^(3 - (I/2)*n)*(6*a + n)*Hypergeometric2F1[-2 + (I/2)*n, 1 
 + (I/2)*n, 2 + (I/2)*n, (-1/2*I)*(I + a + b*x)] + 2^(3 - (I/2)*n)*(1 + I* 
a)*(-I + 5*a + n)*Hypergeometric2F1[-1 + (I/2)*n, 1 + (I/2)*n, 2 + (I/2)*n 
, (-1/2*I)*(I + a + b*x)] + 2^(1 - (I/2)*n)*(-I + a)^2*(-2*I + 4*a + n)*Hy 
pergeometric2F1[1 + (I/2)*n, (I/2)*n, 2 + (I/2)*n, (-1/2*I)*(I + a + b*x)] 
))/(4*b^4*(2*I - n))
 

3.3.37.3 Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5618, 111, 25, 164, 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.

\(\displaystyle \int x^3 e^{n \arctan (a+b x)} \, dx\)

\(\Big \downarrow \) 5618

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

\(\Big \downarrow \) 111

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 164

\(\displaystyle \frac {x^2 (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{4 b^2}-\frac {\frac {\left (24 a^3+36 a^2 n-12 a \left (2-n^2\right )-n \left (8-n^2\right )\right ) \int (-i a-i b x+1)^{\frac {i n}{2}} (i a+i b x+1)^{-\frac {i n}{2}}dx}{6 b}+\frac {(-i a-i b x+1)^{1+\frac {i n}{2}} \left (-18 a^2+2 b x (6 a+n)-10 a n-n^2+6\right ) (i a+i b x+1)^{1-\frac {i n}{2}}}{6 b^2}}{4 b^2}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {x^2 (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{4 b^2}-\frac {\frac {(-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}} \left (-18 a^2+2 b x (6 a+n)-10 a n-n^2+6\right )}{6 b^2}-\frac {2^{-\frac {i n}{2}} \left (24 a^3+36 a^2 n-12 a \left (2-n^2\right )-n \left (8-n^2\right )\right ) (-i a-i b x+1)^{1+\frac {i n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {i n}{2}+1,\frac {i n}{2},\frac {i n}{2}+2,\frac {1}{2} (-i a-i b x+1)\right )}{3 b^2 (-n+2 i)}}{4 b^2}\)

input

Int[E^(n*ArcTan[a + b*x])*x^3,x]
 

output

(x^2*(1 - I*a - I*b*x)^(1 + (I/2)*n)*(1 + I*a + I*b*x)^(1 - (I/2)*n))/(4*b 
^2) - (((1 - I*a - I*b*x)^(1 + (I/2)*n)*(1 + I*a + I*b*x)^(1 - (I/2)*n)*(6 
 - 18*a^2 - 10*a*n - n^2 + 2*b*(6*a + n)*x))/(6*b^2) - ((24*a^3 + 36*a^2*n 
 - 12*a*(2 - n^2) - n*(8 - n^2))*(1 - I*a - I*b*x)^(1 + (I/2)*n)*Hypergeom 
etric2F1[1 + (I/2)*n, (I/2)*n, 2 + (I/2)*n, (1 - I*a - I*b*x)/2])/(3*2^((I 
/2)*n)*b^2*(2*I - n)))/(4*b^2)
 

3.3.37.3.1 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 164

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

3.3.37.4 Maple [F]

\[\int {\mathrm e}^{n \arctan \left (b x +a \right )} x^{3}d x\]

input

int(exp(n*arctan(b*x+a))*x^3,x)
 

output

int(exp(n*arctan(b*x+a))*x^3,x)
 

3.3.37.5 Fricas [F]

\[ \int e^{n \arctan (a+b x)} x^3 \, dx=\int { x^{3} e^{\left (n \arctan \left (b x + a\right )\right )} \,d x } \]

input

integrate(exp(n*arctan(b*x+a))*x^3,x, algorithm="fricas")
 

output

integral(x^3*e^(n*arctan(b*x + a)), x)
 

3.3.37.6 Sympy [F]

\[ \int e^{n \arctan (a+b x)} x^3 \, dx=\int x^{3} e^{n \operatorname {atan}{\left (a + b x \right )}}\, dx \]

input

integrate(exp(n*atan(b*x+a))*x**3,x)
 

output

Integral(x**3*exp(n*atan(a + b*x)), x)
 

3.3.37.7 Maxima [F]

\[ \int e^{n \arctan (a+b x)} x^3 \, dx=\int { x^{3} e^{\left (n \arctan \left (b x + a\right )\right )} \,d x } \]

input

integrate(exp(n*arctan(b*x+a))*x^3,x, algorithm="maxima")
 

output

integrate(x^3*e^(n*arctan(b*x + a)), x)
 

3.3.37.8 Giac [F(-1)]

Timed out. \[ \int e^{n \arctan (a+b x)} x^3 \, dx=\text {Timed out} \]

input

integrate(exp(n*arctan(b*x+a))*x^3,x, algorithm="giac")
 

output

Timed out
 

3.3.37.9 Mupad [F(-1)]

Timed out. \[ \int e^{n \arctan (a+b x)} x^3 \, dx=\int x^3\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a+b\,x\right )} \,d x \]

input

int(x^3*exp(n*atan(a + b*x)),x)
 

output

int(x^3*exp(n*atan(a + b*x)), x)