3.2.89 \(\int e^{-i \text {ArcTan}(a+b x)} x^4 \, dx\) [189]

Optimal. Leaf size=276 \[ -\frac {\left (3 i-12 a-24 i a^2+16 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}+\frac {(i-8 a) x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}+\frac {x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (19 i-114 a-86 i a^2+96 a^3+2 \left (13+14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac {\left (3+12 i a-24 a^2-16 i a^3+8 a^4\right ) \sinh ^{-1}(a+b x)}{8 b^5} \]

[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

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Rubi [A]
time = 0.16, antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5203, 102, 158, 152, 52, 55, 633, 221} \begin {gather*} -\frac {(-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1} \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 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 {\left (8 a^4-16 i a^3-24 a^2+12 i a+3\right ) \sinh ^{-1}(a+b x)}{8 b^5}+\frac {(-8 a+i) x^2 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}{20 b^3}+\frac {x^3 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}{5 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/E^(I*ArcTan[a + b*x]),x]

[Out]

-1/8*((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])/b^5 + ((I -
8*a)*x^2*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x])/(20*b^3) + (x^3*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a +
I*b*x])/(5*b^2) - ((1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x]*(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*} \int e^{-i \tan ^{-1}(a+b x)} x^4 \, dx &=\int \frac {x^4 \sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}} \, dx\\ &=\frac {x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{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 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}+\frac {x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}+\frac {\int \frac {x \sqrt {1-i a-i b x} \left (2 (i-8 a) (i-a) (i+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 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}+\frac {x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \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 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}+\frac {x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \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 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}+\frac {x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \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 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}+\frac {x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \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 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}+\frac {x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \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 ) \sinh ^{-1}(a+b x)}{8 b^5}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 158, normalized size = 0.57 \begin {gather*} \frac {-\sqrt {1+a^2+2 a b x+b^2 x^2} \left (64 i-275 a-332 i a^2+250 a^3+24 i a^4+\left (45+116 i a-130 a^2-24 i a^3\right ) b x+2 \left (-16 i+35 a+12 i a^2\right ) b^2 x^2-6 (5+4 i a) b^3 x^3+24 i b^4 x^4\right )+15 \left (3+12 i a-24 a^2-16 i a^3+8 a^4\right ) \sinh ^{-1}(a+b x)}{120 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(64*I - 275*a - (332*I)*a^2 + 250*a^3 + (24*I)*a^4 + (45 + (116*I)*a - 13
0*a^2 - (24*I)*a^3)*b*x + 2*(-16*I + 35*a + (12*I)*a^2)*b^2*x^2 - 6*(5 + (4*I)*a)*b^3*x^3 + (24*I)*b^4*x^4)) +
15*(3 + (12*I)*a - 24*a^2 - (16*I)*a^3 + 8*a^4)*ArcSinh[a + b*x])/(120*b^5)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1151 vs. \(2 (225 ) = 450\).
time = 0.17, size = 1152, normalized size = 4.17

method	result	size
risch	\(-\frac {i \left (24 x^{4} b^{4}-24 a \,b^{3} x^{3}+30 i b^{3} x^{3}+24 a^{2} b^{2} x^{2}-70 i a \,b^{2} x^{2}-24 a^{3} b x +130 i x \,a^{2} b +24 a^{4}-250 i a^{3}-32 b^{2} x^{2}+116 a b x -45 i b x -332 a^{2}+275 i a +64\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{120 b^{5}}-\frac {2 i \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right ) a^{3}}{b^{4} \sqrt {b^{2}}}+\frac {\ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right ) a^{4}}{b^{4} \sqrt {b^{2}}}+\frac {3 i \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right ) a}{2 b^{4} \sqrt {b^{2}}}-\frac {3 \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right ) a^{2}}{b^{4} \sqrt {b^{2}}}+\frac {3 \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{4} \sqrt {b^{2}}}\)	\(370\)
default	\(-\frac {i \left (\frac {x^{2} \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{5 b^{2}}-\frac {7 a \left (\frac {x \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{4 b^{2}}-\frac {5 a \left (\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{3 b^{2}}-\frac {a \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\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^{2} \sqrt {b^{2}}}\right )}{b}\right )}{4 b}-\frac {\left (a^{2}+1\right ) \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\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^{2} \sqrt {b^{2}}}\right )}{4 b^{2}}\right )}{5 b}-\frac {2 \left (a^{2}+1\right ) \left (\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{3 b^{2}}-\frac {a \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\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^{2} \sqrt {b^{2}}}\right )}{b}\right )}{5 b^{2}}\right )}{b}-\frac {i \left (i-a \right ) \left (\frac {x \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{4 b^{2}}-\frac {5 a \left (\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{3 b^{2}}-\frac {a \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\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^{2} \sqrt {b^{2}}}\right )}{b}\right )}{4 b}-\frac {\left (a^{2}+1\right ) \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\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^{2} \sqrt {b^{2}}}\right )}{4 b^{2}}\right )}{b^{2}}-\frac {i \left (i-a \right )^{2} \left (\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{3 b^{2}}-\frac {a \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\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^{2} \sqrt {b^{2}}}\right )}{b}\right )}{b^{3}}-\frac {i \left (i-a \right )^{3} \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\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^{2} \sqrt {b^{2}}}\right )}{b^{4}}+\frac {\left (-i a^{4}-4 a^{3}+6 i a^{2}+4 a -i\right ) \left (\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}+\frac {i b \ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right )}{\sqrt {b^{2}}}\right )}{b^{5}}\)	\(1152\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I/b*(1/5*x^2*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)/b^2-7/5*a/b*(1/4*x*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)/b^2-5/4*a/b*(1/3*
(b^2*x^2+2*a*b*x+a^2+1)^(3/2)/b^2-a/b*(1/4*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/8*(4*b^2*(a^2+1
)-4*a^2*b^2)/b^2*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)))-1/4*(a^2+1)/b^2*(1/4*
(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/8*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*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)))-2/5*(a^2+1)/b^2*(1/3*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)/b^2-a/b*(1/4*
(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/8*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*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))))-I*(I-a)/b^2*(1/4*x*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)/b^2-5/4*a/b*(1
/3*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)/b^2-a/b*(1/4*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/8*(4*b^2*(a^
2+1)-4*a^2*b^2)/b^2*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)))-1/4*(a^2+1)/b^2*(1
/4*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/8*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*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)))-I*(I-a)^2/b^3*(1/3*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)/b^2-a/b*(1/4
*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/8*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*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)))-I*(I-a)^3/b^4*(1/4*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^
(1/2)+1/8*(4*b^2*(a^2+1)-4*a^2*b^2)/b^2*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))
+(-I*a^4+6*I*a^2-4*a^3-I+4*a)/b^5*(((x-(I-a)/b)^2*b^2+2*I*b*(x-(I-a)/b))^(1/2)+I*b*ln((I*b+(x-(I-a)/b)*b^2)/(b
^2)^(1/2)+((x-(I-a)/b)^2*b^2+2*I*b*(x-(I-a)/b))^(1/2))/(b^2)^(1/2))

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (200) = 400\).
time = 0.51, size = 456, normalized size = 1.65 \begin {gather*} \frac {2 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3} x}{b^{4}} - \frac {i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{5 \, b^{3}} + \frac {a^{4} \operatorname {arsinh}\left (b x + a\right )}{b^{5}} + \frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{4}}{b^{5}} + \frac {3 i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a x}{5 \, b^{4}} + \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} x}{b^{4}} - \frac {2 i \, a^{3} \operatorname {arsinh}\left (b x + a\right )}{b^{5}} - \frac {6 i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{5 \, b^{5}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3}}{b^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} x}{4 \, b^{4}} - \frac {5 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a x}{2 \, b^{4}} - \frac {3 \, a^{2} \operatorname {arsinh}\left (b x + a\right )}{b^{5}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a}{12 \, b^{5}} + \frac {7 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2}}{2 \, b^{5}} - \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x}{8 \, b^{4}} + \frac {3 i \, a \operatorname {arsinh}\left (b x + a\right )}{2 \, b^{5}} + \frac {7 i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{15 \, b^{5}} + \frac {27 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{8 \, b^{5}} + \frac {3 \, \operatorname {arsinh}\left (b x + a\right )}{8 \, b^{5}} - \frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^3*x/b^4 - 1/5*I*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*x^2/b^3 + a^4*arcs
inh(b*x + a)/b^5 + I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^4/b^5 + 3/5*I*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*a*x
/b^4 + 3*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^2*x/b^4 - 2*I*a^3*arcsinh(b*x + a)/b^5 - 6/5*I*(b^2*x^2 + 2*a*b*x
+ a^2 + 1)^(3/2)*a^2/b^5 - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^3/b^5 + 1/4*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2
)*x/b^4 - 5/2*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a*x/b^4 - 3*a^2*arcsinh(b*x + a)/b^5 - 13/12*(b^2*x^2 + 2*a*
b*x + a^2 + 1)^(3/2)*a/b^5 + 7/2*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^2/b^5 - 5/8*sqrt(b^2*x^2 + 2*a*b*x + a^
2 + 1)*x/b^4 + 3/2*I*a*arcsinh(b*x + a)/b^5 + 7/15*I*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)/b^5 + 27/8*sqrt(b^2*x
^2 + 2*a*b*x + a^2 + 1)*a/b^5 + 3/8*arcsinh(b*x + a)/b^5 - I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/b^5

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Fricas [A]
time = 4.00, size = 177, normalized size = 0.64 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - i \int \frac {x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a + b x - i}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*Integral(x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/(a + b*x - I), x)

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Giac [A]
time = 0.45, size = 205, normalized size = 0.74 \begin {gather*} -\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 |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(1+I*(b*x+a))*(1+(b*x+a)^2)^(1/2),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) - sq
rt((b*x + a)^2 + 1))*abs(b))/(b^4*abs(b))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4\,\sqrt {{\left (a+b\,x\right )}^2+1}}{1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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