3.3.8 \(\int e^{-3 i \text {ArcTan}(a+b x)} x^3 \, dx\) [208]

Optimal. Leaf size=249 \[ \frac {2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 \left (17+44 i a-36 a^2-8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}-\frac {9 x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {i (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (29 i-54 a-22 i a^2+2 (11+10 i a) b x\right )}{8 b^4}+\frac {3 \left (17 i-44 a-36 i a^2+8 a^3\right ) \sinh ^{-1}(a+b x)}{8 b^4} \]

[Out]

3/8*(17*I-44*a-36*I*a^2+8*a^3)*arcsinh(b*x+a)/b^4+2*I*x^3*(1-I*a-I*b*x)^(3/2)/b/(1+I*a+I*b*x)^(1/2)-9/4*x^2*(1
-I*a-I*b*x)^(3/2)*(1+I*a+I*b*x)^(1/2)/b^2-1/8*I*(1-I*a-I*b*x)^(3/2)*(29*I-54*a-22*I*a^2+2*(11+10*I*a)*b*x)*(1+
I*a+I*b*x)^(1/2)/b^4+3/8*(17+44*I*a-36*a^2-8*I*a^3)*(1-I*a-I*b*x)^(1/2)*(1+I*a+I*b*x)^(1/2)/b^4

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Rubi [A]
time = 0.17, antiderivative size = 249, 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, 99, 158, 152, 52, 55, 633, 221} \begin {gather*} -\frac {i (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1} \left (-22 i a^2+2 (11+10 i a) b x-54 a+29 i\right )}{8 b^4}+\frac {3 \left (-8 i a^3-36 a^2+44 i a+17\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{8 b^4}+\frac {3 \left (8 a^3-36 i a^2-44 a+17 i\right ) \sinh ^{-1}(a+b x)}{8 b^4}-\frac {9 x^2 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}{4 b^2}+\frac {2 i x^3 (-i a-i b x+1)^{3/2}}{b \sqrt {i a+i b x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/E^((3*I)*ArcTan[a + b*x]),x]

[Out]

((2*I)*x^3*(1 - I*a - I*b*x)^(3/2))/(b*Sqrt[1 + I*a + I*b*x]) + (3*(17 + (44*I)*a - 36*a^2 - (8*I)*a^3)*Sqrt[1
- I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(8*b^4) - (9*x^2*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x])/(4*b^2)
- ((I/8)*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x]*(29*I - 54*a - (22*I)*a^2 + 2*(11 + (10*I)*a)*b*x))/b^
4 + (3*(17*I - 44*a - (36*I)*a^2 + 8*a^3)*ArcSinh[a + b*x])/(8*b^4)

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 99

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

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^{-3 i \tan ^{-1}(a+b x)} x^3 \, dx &=\int \frac {x^3 (1-i a-i b x)^{3/2}}{(1+i a+i b x)^{3/2}} \, dx\\ &=\frac {2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}-\frac {(2 i) \int \frac {x^2 \sqrt {1-i a-i b x} \left (3 (1-i a)-\frac {9 i b x}{2}\right )}{\sqrt {1+i a+i b x}} \, dx}{b}\\ &=\frac {2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}-\frac {9 x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {i \int \frac {x \sqrt {1-i a-i b x} \left (9 i \left (1+a^2\right ) b+\frac {3}{2} (11+10 i a) b^2 x\right )}{\sqrt {1+i a+i b x}} \, dx}{2 b^3}\\ &=\frac {2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}-\frac {9 x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {i (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (29 i-54 a-22 i a^2+2 (11+10 i a) b x\right )}{8 b^4}+\frac {\left (3 \left (17 i-44 a-36 i a^2+8 a^3\right )\right ) \int \frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}} \, dx}{8 b^3}\\ &=\frac {2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 \left (17+44 i a-36 a^2-8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}-\frac {9 x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {i (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (29 i-54 a-22 i a^2+2 (11+10 i a) b x\right )}{8 b^4}+\frac {\left (3 \left (17 i-44 a-36 i a^2+8 a^3\right )\right ) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{8 b^3}\\ &=\frac {2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 \left (17+44 i a-36 a^2-8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}-\frac {9 x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {i (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (29 i-54 a-22 i a^2+2 (11+10 i a) b x\right )}{8 b^4}+\frac {\left (3 \left (17 i-44 a-36 i a^2+8 a^3\right )\right ) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{8 b^3}\\ &=\frac {2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 \left (17+44 i a-36 a^2-8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}-\frac {9 x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {i (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (29 i-54 a-22 i a^2+2 (11+10 i a) b x\right )}{8 b^4}+\frac {\left (3 \left (17 i-44 a-36 i a^2+8 a^3\right )\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^5}\\ &=\frac {2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 \left (17+44 i a-36 a^2-8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}-\frac {9 x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {i (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (29 i-54 a-22 i a^2+2 (11+10 i a) b x\right )}{8 b^4}+\frac {3 \left (17 i-44 a-36 i a^2+8 a^3\right ) \sinh ^{-1}(a+b x)}{8 b^4}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 244, normalized size = 0.98 \begin {gather*} \frac {80-2 i a^5-51 i b x+40 b^2 x^2-17 i b^3 x^3-8 b^4 x^4+2 i b^5 x^5+a^4 (-76-2 i b x)-5 a^3 (-31 i+20 b x)+a^2 \left (4+265 i b x-12 b^2 x^2\right )+a \left (157 i+212 b x+53 i b^2 x^2+4 b^3 x^3+2 i b^4 x^4\right )}{8 b^4 \sqrt {1+a^2+2 a b x+b^2 x^2}}+\frac {3 \sqrt [4]{-1} \left (17 i-44 a-36 i a^2+8 a^3\right ) \sqrt {-i b} \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (i+a+b x)}}{\sqrt {-i b}}\right )}{4 b^{9/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(80 - (2*I)*a^5 - (51*I)*b*x + 40*b^2*x^2 - (17*I)*b^3*x^3 - 8*b^4*x^4 + (2*I)*b^5*x^5 + a^4*(-76 - (2*I)*b*x)
- 5*a^3*(-31*I + 20*b*x) + a^2*(4 + (265*I)*b*x - 12*b^2*x^2) + a*(157*I + 212*b*x + (53*I)*b^2*x^2 + 4*b^3*x
^3 + (2*I)*b^4*x^4))/(8*b^4*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]) + (3*(-1)^(1/4)*(17*I - 44*a - (36*I)*a^2 + 8*a
^3)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/(4*b^(9/2))

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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. 982 vs. \(2 (200 ) = 400\).
time = 0.19, size = 983, normalized size = 3.95

method	result	size
risch	\(-\frac {i \left (-2 b^{3} x^{3}+2 a \,b^{2} x^{2}-8 i b^{2} x^{2}-2 a^{2} b x +20 i a b x +2 a^{3}-44 i a^{2}+19 b x -93 a +48 i\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{8 b^{4}}-\frac {27 i \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right ) a^{2}}{2 b^{3} \sqrt {b^{2}}}+\frac {3 \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right ) a^{3}}{b^{3} \sqrt {b^{2}}}+\frac {51 i \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{3} \sqrt {b^{2}}}-\frac {33 \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right ) a}{2 b^{3} \sqrt {b^{2}}}-\frac {4 \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, a^{3}}{b^{5} \left (x -\frac {i}{b}+\frac {a}{b}\right )}+\frac {12 \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, a}{b^{5} \left (x -\frac {i}{b}+\frac {a}{b}\right )}+\frac {12 i \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\, a^{2}}{b^{5} \left (x -\frac {i}{b}+\frac {a}{b}\right )}-\frac {4 i \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{b^{5} \left (x -\frac {i}{b}+\frac {a}{b}\right )}\)	\(525\)
default	\(\frac {i \left (\frac {\left (2 b^{2} x +2 a b \right ) \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{8 b^{2}}+\frac {3 \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\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 )}{16 b^{2}}\right )}{b^{3}}+\frac {3 \left (-i a -1\right ) \left (\frac {\left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {3}{2}}}{3}+i b \left (\frac {\left (2 \left (x -\frac {i-a}{b}\right ) b^{2}+2 i b \right ) \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{4 b^{2}}+\frac {\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 )}{2 \sqrt {b^{2}}}\right )\right )}{b^{4}}+\frac {\left (-i a^{3}-3 a^{2}+3 i a +1\right ) \left (\frac {i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b \left (x -\frac {i-a}{b}\right )^{3}}-2 i b \left (-\frac {i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b \left (x -\frac {i-a}{b}\right )^{2}}+3 i b \left (\frac {\left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {3}{2}}}{3}+i b \left (\frac {\left (2 \left (x -\frac {i-a}{b}\right ) b^{2}+2 i b \right ) \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{4 b^{2}}+\frac {\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 )}{2 \sqrt {b^{2}}}\right )\right )\right )\right )}{b^{6}}+\frac {3 \left (i a^{2}+2 a -i\right ) \left (-\frac {i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b \left (x -\frac {i-a}{b}\right )^{2}}+3 i b \left (\frac {\left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {3}{2}}}{3}+i b \left (\frac {\left (2 \left (x -\frac {i-a}{b}\right ) b^{2}+2 i b \right ) \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{4 b^{2}}+\frac {\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 )}{2 \sqrt {b^{2}}}\right )\right )\right )}{b^{5}}\)	\(983\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(1+I*(b*x+a))^3*(1+(b*x+a)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

I/b^3*(1/8*(2*b^2*x+2*a*b)/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)+3/16*(4*b^2*(a^2+1)-4*a^2*b^2)/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)))+3*(-I*a-1)/b^4*(1/3*((x-(I-a)/b)^2*b^2+2*I*b*(x-(I-a)/b))^(3/2)+I*b*(1/4*
(2*(x-(I-a)/b)*b^2+2*I*b)/b^2*((x-(I-a)/b)^2*b^2+2*I*b*(x-(I-a)/b))^(1/2)+1/2*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)))+(-I*a^3-3*a^2+3*I*a+1)/b^6*(I/b/(x-(I-a)/b)^3*
((x-(I-a)/b)^2*b^2+2*I*b*(x-(I-a)/b))^(5/2)-2*I*b*(-I/b/(x-(I-a)/b)^2*((x-(I-a)/b)^2*b^2+2*I*b*(x-(I-a)/b))^(5
/2)+3*I*b*(1/3*((x-(I-a)/b)^2*b^2+2*I*b*(x-(I-a)/b))^(3/2)+I*b*(1/4*(2*(x-(I-a)/b)*b^2+2*I*b)/b^2*((x-(I-a)/b)
^2*b^2+2*I*b*(x-(I-a)/b))^(1/2)+1/2*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)))))+3/b^5*(I*a^2-I+2*a)*(-I/b/(x-(I-a)/b)^2*((x-(I-a)/b)^2*b^2+2*I*b*(x-(I-a)/b))^(5/2)+3*
I*b*(1/3*((x-(I-a)/b)^2*b^2+2*I*b*(x-(I-a)/b))^(3/2)+I*b*(1/4*(2*(x-(I-a)/b)*b^2+2*I*b)/b^2*((x-(I-a)/b)^2*b^2
+2*I*b*(x-(I-a)/b))^(1/2)+1/2*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. 979 vs. \(2 (175) = 350\).
time = 0.48, size = 979, normalized size = 3.93 \begin {gather*} -\frac {i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a^{3}}{b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4} - 2 i \, b^{5} x - 2 i \, a b^{4} - b^{4}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4} - 2 i \, b^{5} x - 2 i \, a b^{4} - b^{4}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{2 i \, b^{5} x + 2 i \, a b^{4} + 2 \, b^{4}} - \frac {6 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3}}{i \, b^{5} x + i \, a b^{4} + b^{4}} + \frac {3 i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a}{b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4} - 2 i \, b^{5} x - 2 i \, a b^{4} - b^{4}} + \frac {6 i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a}{2 i \, b^{5} x + 2 i \, a b^{4} + 2 \, b^{4}} - \frac {18 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2}}{i \, b^{5} x + i \, a b^{4} + b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4} - 2 i \, b^{5} x - 2 i \, a b^{4} - b^{4}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{2 i \, b^{5} x + 2 i \, a b^{4} + 2 \, b^{4}} + \frac {18 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{i \, b^{5} x + i \, a b^{4} + b^{4}} + \frac {3 \, a^{3} \operatorname {arsinh}\left (b x + a\right )}{b^{4}} + \frac {6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{i \, b^{5} x + i \, a b^{4} + b^{4}} + \frac {i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} x}{4 \, b^{3}} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 4 i \, b x + 4 i \, a + 3} a x}{2 \, b^{3}} - \frac {27 i \, a^{2} \operatorname {arsinh}\left (b x + a\right )}{2 \, b^{4}} - \frac {3 i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a}{4 \, b^{4}} - \frac {9 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2}}{2 \, b^{4}} + \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 4 i \, b x + 4 i \, a + 3} a^{2}}{2 \, b^{4}} + \frac {3 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x}{8 \, b^{3}} - \frac {3 i \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 4 i \, b x + 4 i \, a + 3} x}{2 \, b^{3}} - \frac {3 \, a \arcsin \left (i \, b x + i \, a + 2\right )}{2 \, b^{4}} - \frac {18 \, a \operatorname {arsinh}\left (b x + a\right )}{b^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{b^{4}} + \frac {75 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{8 \, b^{4}} - \frac {9 i \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 4 i \, b x + 4 i \, a + 3} a}{2 \, b^{4}} + \frac {3 i \, \arcsin \left (i \, b x + i \, a + 2\right )}{2 \, b^{4}} + \frac {63 i \, \operatorname {arsinh}\left (b x + a\right )}{8 \, b^{4}} + \frac {9 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{2 \, b^{4}} - \frac {3 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 4 i \, b x + 4 i \, a + 3}}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(1+I*(b*x+a))^3*(1+(b*x+a)^2)^(3/2),x, algorithm="maxima")

[Out]

-I*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*a^3/(b^6*x^2 + 2*a*b^5*x + a^2*b^4 - 2*I*b^5*x - 2*I*a*b^4 - b^4) - 3*(
b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*a^2/(b^6*x^2 + 2*a*b^5*x + a^2*b^4 - 2*I*b^5*x - 2*I*a*b^4 - b^4) - 3*(b^2*
x^2 + 2*a*b*x + a^2 + 1)^(3/2)*a^2/(2*I*b^5*x + 2*I*a*b^4 + 2*b^4) - 6*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^3
/(I*b^5*x + I*a*b^4 + b^4) + 3*I*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*a/(b^6*x^2 + 2*a*b^5*x + a^2*b^4 - 2*I*b^
5*x - 2*I*a*b^4 - b^4) + 6*I*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*a/(2*I*b^5*x + 2*I*a*b^4 + 2*b^4) - 18*sqrt(b
^2*x^2 + 2*a*b*x + a^2 + 1)*a^2/(I*b^5*x + I*a*b^4 + b^4) + (b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)/(b^6*x^2 + 2*a
*b^5*x + a^2*b^4 - 2*I*b^5*x - 2*I*a*b^4 - b^4) + 3*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)/(2*I*b^5*x + 2*I*a*b^4
+ 2*b^4) + 18*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a/(I*b^5*x + I*a*b^4 + b^4) + 3*a^3*arcsinh(b*x + a)/b^4 +
6*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(I*b^5*x + I*a*b^4 + b^4) + 1/4*I*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*x/b^
3 + 3/2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 4*I*b*x + 4*I*a + 3)*a*x/b^3 - 27/2*I*a^2*arcsinh(b*x + a)/b^4 - 3/4*I
*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*a/b^4 - 9/2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^2/b^4 + 3/2*sqrt(-b^2*x^2
- 2*a*b*x - a^2 + 4*I*b*x + 4*I*a + 3)*a^2/b^4 + 3/8*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*x/b^3 - 3/2*I*sqrt(-
b^2*x^2 - 2*a*b*x - a^2 + 4*I*b*x + 4*I*a + 3)*x/b^3 - 3/2*a*arcsin(I*b*x + I*a + 2)/b^4 - 18*a*arcsinh(b*x +
a)/b^4 - (b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)/b^4 + 75/8*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a/b^4 - 9/2*I*sqrt
(-b^2*x^2 - 2*a*b*x - a^2 + 4*I*b*x + 4*I*a + 3)*a/b^4 + 3/2*I*arcsin(I*b*x + I*a + 2)/b^4 + 63/8*I*arcsinh(b*
x + a)/b^4 + 9/2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/b^4 - 3*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 4*I*b*x + 4*I*a + 3
)/b^4

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Fricas [A]
time = 3.92, size = 216, normalized size = 0.87 \begin {gather*} \frac {-15 i \, a^{5} - 495 \, a^{4} + 1664 i \, a^{3} + {\left (-15 i \, a^{4} - 480 \, a^{3} + 1184 i \, a^{2} + 968 \, a - 256 i\right )} b x + 2152 \, a^{2} - 24 \, {\left (8 \, a^{4} - 44 i \, a^{3} + {\left (8 \, a^{3} - 36 i \, a^{2} - 44 \, a + 17 i\right )} b x - 80 \, a^{2} + 61 i \, a + 17\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 8 \, {\left (-2 i \, b^{4} x^{4} + 6 \, b^{3} x^{3} - {\left (10 \, a - 11 i\right )} b^{2} x^{2} + 2 i \, a^{4} + 78 \, a^{3} + {\left (22 \, a^{2} - 54 i \, a - 29\right )} b x - 233 i \, a^{2} - 237 \, a + 80 i\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 1224 i \, a - 256}{64 \, {\left (b^{5} x + {\left (a - i\right )} b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(1+I*(b*x+a))^3*(1+(b*x+a)^2)^(3/2),x, algorithm="fricas")

[Out]

1/64*(-15*I*a^5 - 495*a^4 + 1664*I*a^3 + (-15*I*a^4 - 480*a^3 + 1184*I*a^2 + 968*a - 256*I)*b*x + 2152*a^2 - 2
4*(8*a^4 - 44*I*a^3 + (8*a^3 - 36*I*a^2 - 44*a + 17*I)*b*x - 80*a^2 + 61*I*a + 17)*log(-b*x - a + sqrt(b^2*x^2
+ 2*a*b*x + a^2 + 1)) - 8*(-2*I*b^4*x^4 + 6*b^3*x^3 - (10*a - 11*I)*b^2*x^2 + 2*I*a^4 + 78*a^3 + (22*a^2 - 54
*I*a - 29)*b*x - 233*I*a^2 - 237*a + 80*I)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) - 1224*I*a - 256)/(b^5*x + (a - I
)*b^4)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(1+I*(b*x+a))**3*(1+(b*x+a)**2)**(3/2),x)

[Out]

Timed out

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Giac [A]
time = 0.45, size = 285, normalized size = 1.14 \begin {gather*} -\frac {1}{8} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left ({\left (2 \, x {\left (-\frac {i \, x}{b} - \frac {-i \, a b^{11} - 4 \, b^{11}}{b^{13}}\right )} - \frac {2 i \, a^{2} b^{10} + 20 \, a b^{10} - 19 i \, b^{10}}{b^{13}}\right )} x - \frac {-2 i \, a^{3} b^{9} - 44 \, a^{2} b^{9} + 93 i \, a b^{9} + 48 \, b^{9}}{b^{13}}\right )} - \frac {{\left (8 \, a^{3} - 36 i \, a^{2} - 44 \, a + 17 i\right )} \log \left (3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} {\left | b \right |} - 2 i \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b - 2 i \, a^{2} b + 4 \, {\left (-i \, x {\left | b \right |} + i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a {\left | b \right |} - a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{8 \, b^{3} {\left | b \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(1+I*(b*x+a))^3*(1+(b*x+a)^2)^(3/2),x, algorithm="giac")

[Out]

-1/8*sqrt((b*x + a)^2 + 1)*((2*x*(-I*x/b - (-I*a*b^11 - 4*b^11)/b^13) - (2*I*a^2*b^10 + 20*a*b^10 - 19*I*b^10)
/b^13)*x - (-2*I*a^3*b^9 - 44*a^2*b^9 + 93*I*a*b^9 + 48*b^9)/b^13) - 1/8*(8*a^3 - 36*I*a^2 - 44*a + 17*I)*log(
3*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*a*b + a^3*b + (x*abs(b) - sqrt((b*x + a)^2 + 1))^3*abs(b) + 3*(x*abs(b)
- sqrt((b*x + a)^2 + 1))*a^2*abs(b) - 2*I*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*b - 2*I*a^2*b + 4*(-I*x*abs(b)
+ I*sqrt((b*x + a)^2 + 1))*a*abs(b) - a*b - (x*abs(b) - sqrt((b*x + a)^2 + 1))*abs(b))/(b^3*abs(b))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}}{{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*((a + b*x)^2 + 1)^(3/2))/(a*1i + b*x*1i + 1)^3,x)

[Out]

int((x^3*((a + b*x)^2 + 1)^(3/2))/(a*1i + b*x*1i + 1)^3, x)

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