\(\int e^{i \arctan (a+b x)} x^4 \, dx\) [162]

Optimal result

Integrand size = 16, antiderivative size = 276 \[ \int e^{i \arctan (a+b x)} x^4 \, dx=\frac {\left (3 i+12 a-24 i a^2-16 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}-\frac {(i+8 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}+\frac {x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (19 i+114 a-86 i a^2-96 a^3-2 \left (13-14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac {\left (3-12 i a-24 a^2+16 i a^3+8 a^4\right ) \text {arcsinh}(a+b x)}{8 b^5} \]

[Out]

1/8*(3-12*I*a-24*a^2+16*I*a^3+8*a^4)*arcsinh(b*x+a)/b^5-1/20*(I+8*a)*x^2*(1+I*a+I*b*x)^(3/2)*(1-I*a-I*b*x)^(1/
2)/b^3+1/5*x^3*(1+I*a+I*b*x)^(3/2)*(1-I*a-I*b*x)^(1/2)/b^2+1/120*(1+I*a+I*b*x)^(3/2)*(19*I+114*a-86*I*a^2-96*a
^3-2*(13-14*I*a-36*a^2)*b*x)*(1-I*a-I*b*x)^(1/2)/b^5+1/8*(3*I+12*a-24*I*a^2-16*a^3+8*I*a^4)*(1-I*a-I*b*x)^(1/2
)*(1+I*a+I*b*x)^(1/2)/b^5

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5203, 102, 158, 152, 52, 55, 633, 221} \[ \int e^{i \arctan (a+b x)} x^4 \, dx=\frac {\sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2} \left (-96 a^3-2 \left (-36 a^2-14 i a+13\right ) b x-86 i a^2+114 a+19 i\right )}{120 b^5}+\frac {\left (8 a^4+16 i a^3-24 a^2-12 i a+3\right ) \text {arcsinh}(a+b x)}{8 b^5}+\frac {\left (8 i a^4-16 a^3-24 i a^2+12 a+3 i\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{8 b^5}-\frac {(8 a+i) x^2 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{20 b^3}+\frac {x^3 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{5 b^2} \]

[In]

Int[E^(I*ArcTan[a + b*x])*x^4,x]

[Out]

((3*I + 12*a - (24*I)*a^2 - 16*a^3 + (8*I)*a^4)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(8*b^5) - ((I + 8
*a)*x^2*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/(20*b^3) + (x^3*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)
^(3/2))/(5*b^2) + (Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2)*(19*I + 114*a - (86*I)*a^2 - 96*a^3 - 2*(13 -
(14*I)*a - 36*a^2)*b*x))/(120*b^5) + ((3 - (12*I)*a - 24*a^2 + (16*I)*a^3 + 8*a^4)*ArcSinh[a + b*x])/(8*b^5)

Rule 52

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

Rule 55

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Int[1/Sqrt[a*c - b*(a - c)*x - b^2*x^2]
, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + c, 0]

Rule 102

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 1))), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(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 152

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> 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] + Dist[(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 158

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

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 5203

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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^4 \sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}} \, dx \\ & = \frac {x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac {\int \frac {x^2 \sqrt {1+i a+i b x} \left (-3 \left (1+a^2\right )-(i+8 a) b x\right )}{\sqrt {1-i a-i b x}} \, dx}{5 b^2} \\ & = -\frac {(i+8 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}+\frac {x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac {\int \frac {x \sqrt {1+i a+i b x} \left (-2 (i-a) (i+a) (i+8 a) b-\left (13-14 i a-36 a^2\right ) b^2 x\right )}{\sqrt {1-i a-i b x}} \, dx}{20 b^4} \\ & = -\frac {(i+8 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}+\frac {x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (19 i+114 a-86 i a^2-96 a^3-2 \left (13-14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac {\left (3-12 i a-24 a^2+16 i a^3+8 a^4\right ) \int \frac {\sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}} \, dx}{8 b^4} \\ & = \frac {\left (3 i+12 a-24 i a^2-16 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}-\frac {(i+8 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}+\frac {x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (19 i+114 a-86 i a^2-96 a^3-2 \left (13-14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac {\left (3-12 i a-24 a^2+16 i a^3+8 a^4\right ) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{8 b^4} \\ & = \frac {\left (3 i+12 a-24 i a^2-16 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}-\frac {(i+8 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}+\frac {x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (19 i+114 a-86 i a^2-96 a^3-2 \left (13-14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac {\left (3-12 i a-24 a^2+16 i a^3+8 a^4\right ) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{8 b^4} \\ & = \frac {\left (3 i+12 a-24 i a^2-16 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}-\frac {(i+8 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}+\frac {x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (19 i+114 a-86 i a^2-96 a^3-2 \left (13-14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac {\left (3-12 i a-24 a^2+16 i a^3+8 a^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4 b^2}}} \, dx,x,2 a b+2 b^2 x\right )}{16 b^6} \\ & = \frac {\left (3 i+12 a-24 i a^2-16 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}-\frac {(i+8 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}+\frac {x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (19 i+114 a-86 i a^2-96 a^3-2 \left (13-14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac {\left (3-12 i a-24 a^2+16 i a^3+8 a^4\right ) \text {arcsinh}(a+b x)}{8 b^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.79 \[ \int e^{i \arctan (a+b x)} x^4 \, dx=\frac {i \sqrt {1+a^2+2 a b x+b^2 x^2} \left (64+24 a^4+45 i b x-32 b^2 x^2-30 i b^3 x^3+24 b^4 x^4+a^3 (250 i-24 b x)+2 a^2 \left (-166-65 i b x+12 b^2 x^2\right )+a \left (-275 i+116 b x+70 i b^2 x^2-24 b^3 x^3\right )\right )}{120 b^5}+\frac {\sqrt [4]{-1} \left (3-12 i a-24 a^2+16 i a^3+8 a^4\right ) \sqrt {-i b} \text {arcsinh}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (i+a+b x)}}{\sqrt {-i b}}\right )}{4 b^{11/2}} \]

[In]

Integrate[E^(I*ArcTan[a + b*x])*x^4,x]

[Out]

((I/120)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(64 + 24*a^4 + (45*I)*b*x - 32*b^2*x^2 - (30*I)*b^3*x^3 + 24*b^4*x^
4 + a^3*(250*I - 24*b*x) + 2*a^2*(-166 - (65*I)*b*x + 12*b^2*x^2) + a*(-275*I + 116*b*x + (70*I)*b^2*x^2 - 24*
b^3*x^3)))/b^5 + ((-1)^(1/4)*(3 - (12*I)*a - 24*a^2 + (16*I)*a^3 + 8*a^4)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sq
rt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/(4*b^(11/2))

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.71

method	result	size
risch	\(\frac {i \left (24 x^{4} b^{4}-24 a \,b^{3} x^{3}-30 i b^{3} x^{3}+24 a^{2} b^{2} x^{2}+70 i a \,b^{2} x^{2}-24 a^{3} b x -130 i a^{2} b x +24 a^{4}+250 i a^{3}-32 b^{2} x^{2}+116 a b x +45 b x i-332 a^{2}-275 i a +64\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{120 b^{5}}+\frac {\left (8 a^{4}+16 i a^{3}-24 a^{2}-12 i a +3\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{4} \sqrt {b^{2}}}\)	\(197\)
default	\(\text {Expression too large to display}\)	\(1260\)

[In]

int((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2)*x^4,x,method=_RETURNVERBOSE)

[Out]

1/120*I*(24*x^4*b^4-30*I*b^3*x^3-24*a*b^3*x^3+70*I*a*b^2*x^2+24*a^2*b^2*x^2-130*I*a^2*b*x-24*a^3*b*x+250*I*a^3
+24*a^4-32*b^2*x^2+45*I*b*x+116*a*b*x-275*I*a-332*a^2+64)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)/b^5+1/8*(3-12*I*a-24*a
^2+16*I*a^3+8*a^4)/b^4*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.64 \[ \int e^{i \arctan (a+b x)} x^4 \, dx=\frac {186 i \, a^{5} - 1345 \, a^{4} - 1730 i \, a^{3} + 1320 \, a^{2} - 120 \, {\left (8 \, a^{4} + 16 i \, a^{3} - 24 \, a^{2} - 12 i \, a + 3\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 8 \, {\left (-24 i \, b^{4} x^{4} + 6 \, {\left (4 i \, a - 5\right )} b^{3} x^{3} + 2 \, {\left (-12 i \, a^{2} + 35 \, a + 16 i\right )} b^{2} x^{2} - 24 i \, a^{4} + 250 \, a^{3} + {\left (24 i \, a^{3} - 130 \, a^{2} - 116 i \, a + 45\right )} b x + 332 i \, a^{2} - 275 \, a - 64 i\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 300 i \, a}{960 \, b^{5}} \]

[In]

integrate((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2)*x^4,x, algorithm="fricas")

[Out]

1/960*(186*I*a^5 - 1345*a^4 - 1730*I*a^3 + 1320*a^2 - 120*(8*a^4 + 16*I*a^3 - 24*a^2 - 12*I*a + 3)*log(-b*x -
a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 8*(-24*I*b^4*x^4 + 6*(4*I*a - 5)*b^3*x^3 + 2*(-12*I*a^2 + 35*a + 16*I
)*b^2*x^2 - 24*I*a^4 + 250*a^3 + (24*I*a^3 - 130*a^2 - 116*I*a + 45)*b*x + 332*I*a^2 - 275*a - 64*I)*sqrt(b^2*
x^2 + 2*a*b*x + a^2 + 1) + 300*I*a)/b^5

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1222 vs. \(2 (236) = 472\).

Time = 1.86 (sec) , antiderivative size = 1222, normalized size of antiderivative = 4.43 \[ \int e^{i \arctan (a+b x)} x^4 \, dx=\text {Too large to display} \]

[In]

integrate((1+I*(b*x+a))/(1+(b*x+a)**2)**(1/2)*x**4,x)

[Out]

Piecewise(((-a*(-3*a*(-5*a*(-7*a*(-4*I*a/5 + 1)/(4*b) - I*(4*a**2 + 4)/(5*b))/(3*b) - (3*a**2 + 3)*(-4*I*a/5 +
1)/(4*b**2))/(2*b) - (2*a**2 + 2)*(-7*a*(-4*I*a/5 + 1)/(4*b) - I*(4*a**2 + 4)/(5*b))/(3*b**2))/b - (a**2 + 1)
*(-5*a*(-7*a*(-4*I*a/5 + 1)/(4*b) - I*(4*a**2 + 4)/(5*b))/(3*b) - (3*a**2 + 3)*(-4*I*a/5 + 1)/(4*b**2))/(2*b**
2))*log(2*a*b + 2*b**2*x + 2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*sqrt(b**2))/sqrt(b**2) + sqrt(a**2 + 2*a*b*x
+ b**2*x**2 + 1)*(I*x**4/(5*b) + x**3*(-4*I*a/5 + 1)/(4*b**2) + x**2*(-7*a*(-4*I*a/5 + 1)/(4*b) - I*(4*a**2 +
4)/(5*b))/(3*b**2) + x*(-5*a*(-7*a*(-4*I*a/5 + 1)/(4*b) - I*(4*a**2 + 4)/(5*b))/(3*b) - (3*a**2 + 3)*(-4*I*a/
5 + 1)/(4*b**2))/(2*b**2) + (-3*a*(-5*a*(-7*a*(-4*I*a/5 + 1)/(4*b) - I*(4*a**2 + 4)/(5*b))/(3*b) - (3*a**2 + 3
)*(-4*I*a/5 + 1)/(4*b**2))/(2*b) - (2*a**2 + 2)*(-7*a*(-4*I*a/5 + 1)/(4*b) - I*(4*a**2 + 4)/(5*b))/(3*b**2))/b
**2), Ne(b**2, 0)), ((I*(a**8*sqrt(a**2 + 2*a*b*x + 1) + 4*a**6*sqrt(a**2 + 2*a*b*x + 1) + 6*a**4*sqrt(a**2 +
2*a*b*x + 1) + 4*a**2*sqrt(a**2 + 2*a*b*x + 1) + (-4*a**2 - 4)*(a**2 + 2*a*b*x + 1)**(7/2)/7 + (a**2 + 2*a*b*x
+ 1)**(9/2)/9 + (a**2 + 2*a*b*x + 1)**(5/2)*(6*a**4 + 12*a**2 + 6)/5 + (a**2 + 2*a*b*x + 1)**(3/2)*(-4*a**6 -
12*a**4 - 12*a**2 - 4)/3 + sqrt(a**2 + 2*a*b*x + 1))/(8*a**3*b**4) + (a**8*sqrt(a**2 + 2*a*b*x + 1) + 4*a**6*
sqrt(a**2 + 2*a*b*x + 1) + 6*a**4*sqrt(a**2 + 2*a*b*x + 1) + 4*a**2*sqrt(a**2 + 2*a*b*x + 1) + (-4*a**2 - 4)*(
a**2 + 2*a*b*x + 1)**(7/2)/7 + (a**2 + 2*a*b*x + 1)**(9/2)/9 + (a**2 + 2*a*b*x + 1)**(5/2)*(6*a**4 + 12*a**2 +
6)/5 + (a**2 + 2*a*b*x + 1)**(3/2)*(-4*a**6 - 12*a**4 - 12*a**2 - 4)/3 + sqrt(a**2 + 2*a*b*x + 1))/(8*a**4*b*
*4) + I*(-a**10*sqrt(a**2 + 2*a*b*x + 1) - 5*a**8*sqrt(a**2 + 2*a*b*x + 1) - 10*a**6*sqrt(a**2 + 2*a*b*x + 1)
- 10*a**4*sqrt(a**2 + 2*a*b*x + 1) - 5*a**2*sqrt(a**2 + 2*a*b*x + 1) + (-5*a**2 - 5)*(a**2 + 2*a*b*x + 1)**(9/
2)/9 + (a**2 + 2*a*b*x + 1)**(11/2)/11 + (a**2 + 2*a*b*x + 1)**(7/2)*(10*a**4 + 20*a**2 + 10)/7 + (a**2 + 2*a*
b*x + 1)**(5/2)*(-10*a**6 - 30*a**4 - 30*a**2 - 10)/5 + (a**2 + 2*a*b*x + 1)**(3/2)*(5*a**8 + 20*a**6 + 30*a**
4 + 20*a**2 + 5)/3 - sqrt(a**2 + 2*a*b*x + 1))/(16*a**5*b**4))/(2*a*b), Ne(a*b, 0)), ((I*a*x**5/5 + I*b*x**6/6
+ x**5/5)/sqrt(a**2 + 1), True))

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 749 vs. \(2 (200) = 400\).

Time = 0.21 (sec) , antiderivative size = 749, normalized size of antiderivative = 2.71 \[ \int e^{i \arctan (a+b x)} x^4 \, dx=\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x^{4}}{5 \, b} - \frac {9 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a x^{3}}{20 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (-i \, a - 1\right )} x^{3}}{4 \, b^{2}} + \frac {21 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} x^{2}}{20 \, b^{3}} - \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a {\left (i \, a + 1\right )} x^{2}}{12 \, b^{3}} - \frac {63 i \, a^{5} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{8 \, b^{5}} + \frac {35 \, a^{4} {\left (i \, a + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{8 \, b^{5}} - \frac {21 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3} x}{8 \, b^{4}} + \frac {35 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} {\left (i \, a + 1\right )} x}{24 \, b^{4}} - \frac {4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (i \, a^{2} + i\right )} x^{2}}{15 \, b^{3}} + \frac {35 i \, {\left (a^{2} + 1\right )} a^{3} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{4 \, b^{5}} - \frac {15 \, {\left (a^{2} + 1\right )} a^{2} {\left (i \, a + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{4 \, b^{5}} + \frac {63 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{4}}{8 \, b^{5}} - \frac {35 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3} {\left (i \, a + 1\right )}}{8 \, b^{5}} + \frac {161 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} a x}{120 \, b^{4}} - \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} {\left (i \, a + 1\right )} x}{8 \, b^{4}} - \frac {15 i \, {\left (a^{2} + 1\right )}^{2} a \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{8 \, b^{5}} - \frac {3 \, {\left (a^{2} + 1\right )}^{2} {\left (-i \, a - 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{8 \, b^{5}} - \frac {49 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} a^{2}}{8 \, b^{5}} + \frac {55 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} a {\left (i \, a + 1\right )}}{24 \, b^{5}} + \frac {8 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )}^{2}}{15 \, b^{5}} \]

[In]

integrate((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2)*x^4,x, algorithm="maxima")

[Out]

1/5*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*x^4/b - 9/20*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a*x^3/b^2 - 1/4*sqrt(
b^2*x^2 + 2*a*b*x + a^2 + 1)*(-I*a - 1)*x^3/b^2 + 21/20*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^2*x^2/b^3 - 7/12
*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a*(I*a + 1)*x^2/b^3 - 63/8*I*a^5*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 +
4*(a^2 + 1)*b^2))/b^5 + 35/8*a^4*(I*a + 1)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^5 - 2
1/8*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^3*x/b^4 + 35/24*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^2*(I*a + 1)*x/b^
4 - 4/15*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(I*a^2 + I)*x^2/b^3 + 35/4*I*(a^2 + 1)*a^3*arcsinh(2*(b^2*x + a*b)/
sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^5 - 15/4*(a^2 + 1)*a^2*(I*a + 1)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2
+ 4*(a^2 + 1)*b^2))/b^5 + 63/8*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^4/b^5 - 35/8*sqrt(b^2*x^2 + 2*a*b*x + a^
2 + 1)*a^3*(I*a + 1)/b^5 + 161/120*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)*a*x/b^4 - 3/8*sqrt(b^2*x^2 +
2*a*b*x + a^2 + 1)*(a^2 + 1)*(I*a + 1)*x/b^4 - 15/8*I*(a^2 + 1)^2*a*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 +
4*(a^2 + 1)*b^2))/b^5 - 3/8*(a^2 + 1)^2*(-I*a - 1)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))
/b^5 - 49/8*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)*a^2/b^5 + 55/24*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a
^2 + 1)*a*(I*a + 1)/b^5 + 8/15*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2/b^5

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.74 \[ \int e^{i \arctan (a+b x)} x^4 \, dx=-\frac {1}{120} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left ({\left (2 \, {\left (3 \, x {\left (-\frac {4 i \, x}{b} - \frac {-4 i \, a b^{7} + 5 \, b^{7}}{b^{9}}\right )} - \frac {12 i \, a^{2} b^{6} - 35 \, a b^{6} - 16 i \, b^{6}}{b^{9}}\right )} x - \frac {-24 i \, a^{3} b^{5} + 130 \, a^{2} b^{5} + 116 i \, a b^{5} - 45 \, b^{5}}{b^{9}}\right )} x - \frac {24 i \, a^{4} b^{4} - 250 \, a^{3} b^{4} - 332 i \, a^{2} b^{4} + 275 \, a b^{4} + 64 i \, b^{4}}{b^{9}}\right )} - \frac {{\left (8 \, a^{4} + 16 i \, a^{3} - 24 \, a^{2} - 12 i \, a + 3\right )} \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{8 \, b^{4} {\left | b \right |}} \]

[In]

integrate((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2)*x^4,x, algorithm="giac")

[Out]

-1/120*sqrt((b*x + a)^2 + 1)*((2*(3*x*(-4*I*x/b - (-4*I*a*b^7 + 5*b^7)/b^9) - (12*I*a^2*b^6 - 35*a*b^6 - 16*I*
b^6)/b^9)*x - (-24*I*a^3*b^5 + 130*a^2*b^5 + 116*I*a*b^5 - 45*b^5)/b^9)*x - (24*I*a^4*b^4 - 250*a^3*b^4 - 332*
I*a^2*b^4 + 275*a*b^4 + 64*I*b^4)/b^9) - 1/8*(8*a^4 + 16*I*a^3 - 24*a^2 - 12*I*a + 3)*log(-a*b - (x*abs(b) - s
qrt((b*x + a)^2 + 1))*abs(b))/(b^4*abs(b))

Mupad [F(-1)]

Timed out. \[ \int e^{i \arctan (a+b x)} x^4 \, dx=\int \frac {x^4\,\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}{\sqrt {{\left (a+b\,x\right )}^2+1}} \,d x \]

[In]

int((x^4*(a*1i + b*x*1i + 1))/((a + b*x)^2 + 1)^(1/2),x)

[Out]

int((x^4*(a*1i + b*x*1i + 1))/((a + b*x)^2 + 1)^(1/2), x)