3.2.87 \(\int \frac {e^{3 i \text {ArcTan}(a+b x)}}{x^3} \, dx\) [187]

Optimal. Leaf size=264 \[ \frac {3 (3 i-2 a) b^2 \sqrt {1+i a+i b x}}{(1+i a) (i+a)^3 \sqrt {1-i a-i b x}}+\frac {(3 i-2 a) b (1+i a+i b x)^{3/2}}{2 (1+i a) (i+a)^2 x \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1-i a-i b x}}+\frac {3 (3+2 i a) b^2 \tanh ^{-1}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{\sqrt {i-a} (i+a)^{7/2}} \]

[Out]

3*(3+2*I*a)*b^2*arctanh((I+a)^(1/2)*(1+I*a+I*b*x)^(1/2)/(I-a)^(1/2)/(1-I*a-I*b*x)^(1/2))/(I+a)^(7/2)/(I-a)^(1/
2)+1/2*(3*I-2*a)*b*(1+I*a+I*b*x)^(3/2)/(1+I*a)/(I+a)^2/x/(1-I*a-I*b*x)^(1/2)-1/2*(1+I*a+I*b*x)^(5/2)/(a^2+1)/x
^2/(1-I*a-I*b*x)^(1/2)+3*(3*I-2*a)*b^2*(1+I*a+I*b*x)^(1/2)/(1+I*a)/(I+a)^3/(1-I*a-I*b*x)^(1/2)

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Rubi [A]
time = 0.13, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5203, 98, 96, 95, 214} \begin {gather*} -\frac {(i a+i b x+1)^{5/2}}{2 \left (a^2+1\right ) x^2 \sqrt {-i a-i b x+1}}+\frac {3 (-2 a+3 i) b^2 \sqrt {i a+i b x+1}}{(1+i a) (a+i)^3 \sqrt {-i a-i b x+1}}+\frac {3 (3+2 i a) b^2 \tanh ^{-1}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{\sqrt {-a+i} (a+i)^{7/2}}+\frac {(-2 a+3 i) b (i a+i b x+1)^{3/2}}{2 (1+i a) (a+i)^2 x \sqrt {-i a-i b x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((3*I)*ArcTan[a + b*x])/x^3,x]

[Out]

(3*(3*I - 2*a)*b^2*Sqrt[1 + I*a + I*b*x])/((1 + I*a)*(I + a)^3*Sqrt[1 - I*a - I*b*x]) + ((3*I - 2*a)*b*(1 + I*
a + I*b*x)^(3/2))/(2*(1 + I*a)*(I + a)^2*x*Sqrt[1 - I*a - I*b*x]) - (1 + I*a + I*b*x)^(5/2)/(2*(1 + a^2)*x^2*S
qrt[1 - I*a - I*b*x]) + (3*(3 + (2*I)*a)*b^2*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 -
I*a - I*b*x])])/(Sqrt[I - a]*(I + a)^(7/2))

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 96

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 + 1)/((m + 1)*(b*e - a*f))), x] - Dist[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 98

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)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

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 \frac {e^{3 i \tan ^{-1}(a+b x)}}{x^3} \, dx &=\int \frac {(1+i a+i b x)^{3/2}}{x^3 (1-i a-i b x)^{3/2}} \, dx\\ &=-\frac {(1+i a+i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1-i a-i b x}}+\frac {((3 i-2 a) b) \int \frac {(1+i a+i b x)^{3/2}}{x^2 (1-i a-i b x)^{3/2}} \, dx}{2 \left (1+a^2\right )}\\ &=-\frac {(3 i-2 a) b (1+i a+i b x)^{3/2}}{2 (1-i a) \left (1+a^2\right ) x \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1-i a-i b x}}-\frac {\left (3 (3 i-2 a) b^2\right ) \int \frac {\sqrt {1+i a+i b x}}{x (1-i a-i b x)^{3/2}} \, dx}{2 (i+a) \left (1+a^2\right )}\\ &=-\frac {3 (3 i-2 a) b^2 \sqrt {1+i a+i b x}}{(i-a) (1-i a)^3 \sqrt {1-i a-i b x}}-\frac {(3 i-2 a) b (1+i a+i b x)^{3/2}}{2 (1-i a) \left (1+a^2\right ) x \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1-i a-i b x}}+\frac {\left (3 (3 i-2 a) b^2\right ) \int \frac {1}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{2 (i+a)^3}\\ &=-\frac {3 (3 i-2 a) b^2 \sqrt {1+i a+i b x}}{(i-a) (1-i a)^3 \sqrt {1-i a-i b x}}-\frac {(3 i-2 a) b (1+i a+i b x)^{3/2}}{2 (1-i a) \left (1+a^2\right ) x \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1-i a-i b x}}+\frac {\left (3 (3 i-2 a) b^2\right ) \text {Subst}\left (\int \frac {1}{-1-i a-(-1+i a) x^2} \, dx,x,\frac {\sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}}\right )}{(i+a)^3}\\ &=-\frac {3 (3 i-2 a) b^2 \sqrt {1+i a+i b x}}{(i-a) (1-i a)^3 \sqrt {1-i a-i b x}}-\frac {(3 i-2 a) b (1+i a+i b x)^{3/2}}{2 (1-i a) \left (1+a^2\right ) x \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1-i a-i b x}}+\frac {3 (3+2 i a) b^2 \tanh ^{-1}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{\sqrt {i-a} (i+a)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 194, normalized size = 0.73 \begin {gather*} \frac {\frac {\sqrt {1+i a+i b x} \left (i+a+i a^2+a^3-5 b x+5 i a b x+14 i b^2 x^2-a b^2 x^2\right )}{x^2 \sqrt {-i (i+a+b x)}}-\frac {6 i \sqrt {-1-i a} (-3 i+2 a) b^2 \tanh ^{-1}\left (\frac {\sqrt {-1-i a} \sqrt {-i (i+a+b x)}}{\sqrt {-1+i a} \sqrt {1+i a+i b x}}\right )}{\sqrt {-1+i a} (-i+a)}}{2 (i+a)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[1 + I*a + I*b*x]*(I + a + I*a^2 + a^3 - 5*b*x + (5*I)*a*b*x + (14*I)*b^2*x^2 - a*b^2*x^2))/(x^2*Sqrt[(-
I)*(I + a + b*x)]) - ((6*I)*Sqrt[-1 - I*a]*(-3*I + 2*a)*b^2*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)])/
(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])])/(Sqrt[-1 + I*a]*(-I + a)))/(2*(I + a)^3)

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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. 945 vs. \(2 (209 ) = 418\).
time = 0.21, size = 946, normalized size = 3.58

method	result	size
risch	\(\frac {i \left (-a \,b^{3} x^{3}+6 i b^{3} x^{3}-a^{2} b^{2} x^{2}+12 i a \,b^{2} x^{2}+a^{3} b x +6 i x \,a^{2} b +a^{4}+b^{2} x^{2}+a b x +6 i b x +2 a^{2}+1\right )}{2 x^{2} \left (i+a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {3 i b^{2} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right ) a}{2 \left (a^{3}+3 i a^{2}-3 a -i\right ) \left (i+a \right ) \sqrt {a^{2}+1}}+\frac {3 b^{2} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right ) a^{2}}{\left (a^{3}+3 i a^{2}-3 a -i\right ) \left (i+a \right ) \sqrt {a^{2}+1}}+\frac {9 b^{2} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{2 \left (a^{3}+3 i a^{2}-3 a -i\right ) \left (i+a \right ) \sqrt {a^{2}+1}}-\frac {4 b \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}\, a}{\left (a^{3}+3 i a^{2}-3 a -i\right ) \left (i+a \right ) \left (x +\frac {i}{b}+\frac {a}{b}\right )}-\frac {4 i b \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}}{\left (a^{3}+3 i a^{2}-3 a -i\right ) \left (i+a \right ) \left (x +\frac {i}{b}+\frac {a}{b}\right )}\)	\(507\)
default	\(-\frac {2 i b^{3} \left (2 b^{2} x +2 a b \right )}{\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\left (-i a^{3}-3 a^{2}+3 i a +1\right ) \left (-\frac {1}{2 \left (a^{2}+1\right ) x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {5 a b \left (-\frac {1}{\left (a^{2}+1\right ) x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {3 a b \left (\frac {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a b \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{a^{2}+1}-\frac {4 b^{2} \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right )}{2 \left (a^{2}+1\right )}-\frac {3 b^{2} \left (\frac {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a b \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+1\right )}\right )-3 \left (i a +1\right ) b^{2} \left (\frac {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a b \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )-3 b \left (i a^{2}+2 a -i\right ) \left (-\frac {1}{\left (a^{2}+1\right ) x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {3 a b \left (\frac {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a b \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{a^{2}+1}-\frac {4 b^{2} \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right )\)	\(946\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*I*b^3*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+(-I*a^3-3*a^2+3*I*a+1)*(-1/2/
(a^2+1)/x^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-5/2*a*b/(a^2+1)*(-1/(a^2+1)/x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-3*a*b/(a
^2+1)*(1/(a^2+1)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a*b/(a^2+1)*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^
2+2*a*b*x+a^2+1)^(1/2)-1/(a^2+1)^(3/2)*ln((2*a^2+2+2*a*b*x+2*(a^2+1)^(1/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x))-
4*b^2/(a^2+1)*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))-3/2*b^2/(a^2+1)*(1/(a^2
+1)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a*b/(a^2+1)*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2
+1)^(1/2)-1/(a^2+1)^(3/2)*ln((2*a^2+2+2*a*b*x+2*(a^2+1)^(1/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x)))-3*(1+I*a)*b^
2*(1/(a^2+1)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a*b/(a^2+1)*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*
a*b*x+a^2+1)^(1/2)-1/(a^2+1)^(3/2)*ln((2*a^2+2+2*a*b*x+2*(a^2+1)^(1/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x))-3*b*
(I*a^2-I+2*a)*(-1/(a^2+1)/x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-3*a*b/(a^2+1)*(1/(a^2+1)/(b^2*x^2+2*a*b*x+a^2+1)^(1/
2)-2*a*b/(a^2+1)*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-1/(a^2+1)^(3/2)*ln((2
*a^2+2+2*a*b*x+2*(a^2+1)^(1/2)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/x))-4*b^2/(a^2+1)*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)
-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(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. 1536 vs. \(2 (182) = 364\).
time = 0.27, size = 1536, normalized size = 5.82 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a^3*b^5*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2
+ 1)^3) + 15/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a^4*b^4/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 +
1)*(a^2 + 1)^3) + 9*(I*a^2*b + 2*a*b - I*b)*a^2*b^4*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2
+ 1)*(a^2 + 1)^2) + I*b^5*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 13/2*(-I*a^3 - 3*a
^2 + 3*I*a + 1)*a*b^5*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2) + 9*(I*a^2*b
+ 2*a*b - I*b)*a^3*b^3/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2) + I*a*b^4/((
a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 13/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a^2*b^4/((a^2*
b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2) - 3*(I*a*b^2 + b^2)*a*b^3*x/((a^2*b^2 - (a
^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)) - 6*(I*a^2*b + 2*a*b - I*b)*b^4*x/((a^2*b^2 - (a^2 +
1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)) - 3*(I*a*b^2 + b^2)*a^2*b^2/((a^2*b^2 - (a^2 + 1)*b^2)*s
qrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)) - 6*(I*a^2*b + 2*a*b - I*b)*a*b^3/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(
b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)) - 15/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a^2*b^2*arcsinh(2*a*b*x/(sqrt(-4*a
^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2/(sqrt(-4*a^2*b^2 + 4
*(a^2 + 1)*b^2)*abs(x)))/(a^2 + 1)^(7/2) + 15/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a^2*b^2/(sqrt(b^2*x^2 + 2*a*b*x +
a^2 + 1)*(a^2 + 1)^3) - 9*(I*a^2*b + 2*a*b - I*b)*a*b*arcsinh(2*a*b*x/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs
(x)) + 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)))/(a^2
+ 1)^(5/2) + 3/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*b^2*arcsinh(2*a*b*x/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x))
+ 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)))/(a^2 + 1
)^(5/2) + 9*(I*a^2*b + 2*a*b - I*b)*a*b/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2) - 3/2*(-I*a^3 - 3*a^2
+ 3*I*a + 1)*b^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2) + 3*(I*a*b^2 + b^2)*arcsinh(2*a*b*x/(sqrt(-4*
a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2/(sqrt(-4*a^2*b^2 +
4*(a^2 + 1)*b^2)*abs(x)))/(a^2 + 1)^(3/2) - 3*(I*a*b^2 + b^2)/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)) +
5/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a*b/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2*x) + 3*(I*a^2*b + 2*a*b -
I*b)/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)*x) - 1/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)/(sqrt(b^2*x^2 + 2*a*b*
x + a^2 + 1)*(a^2 + 1)*x^2)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 574 vs. \(2 (182) = 364\).
time = 6.05, size = 574, normalized size = 2.17 \begin {gather*} \frac {{\left (-i \, a - 14\right )} b^{3} x^{3} + {\left (-i \, a^{2} - 13 \, a - 14 i\right )} b^{2} x^{2} - 3 \, {\left ({\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} b x^{3} + {\left (a^{4} + 4 i \, a^{3} - 6 \, a^{2} - 4 i \, a + 1\right )} x^{2}\right )} \sqrt {\frac {{\left (4 \, a^{2} - 12 i \, a - 9\right )} b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}} \log \left (-\frac {{\left (2 \, a - 3 i\right )} b^{3} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 \, a - 3 i\right )} b^{2} + {\left (a^{5} + 3 i \, a^{4} - 2 \, a^{3} + 2 i \, a^{2} - 3 \, a - i\right )} \sqrt {\frac {{\left (4 \, a^{2} - 12 i \, a - 9\right )} b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}}}{{\left (2 \, a - 3 i\right )} b^{2}}\right ) + 3 \, {\left ({\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} b x^{3} + {\left (a^{4} + 4 i \, a^{3} - 6 \, a^{2} - 4 i \, a + 1\right )} x^{2}\right )} \sqrt {\frac {{\left (4 \, a^{2} - 12 i \, a - 9\right )} b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}} \log \left (-\frac {{\left (2 \, a - 3 i\right )} b^{3} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 \, a - 3 i\right )} b^{2} - {\left (a^{5} + 3 i \, a^{4} - 2 \, a^{3} + 2 i \, a^{2} - 3 \, a - i\right )} \sqrt {\frac {{\left (4 \, a^{2} - 12 i \, a - 9\right )} b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}}}{{\left (2 \, a - 3 i\right )} b^{2}}\right ) + {\left ({\left (-i \, a - 14\right )} b^{2} x^{2} + i \, a^{3} - 5 \, {\left (a + i\right )} b x - a^{2} + i \, a - 1\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{2 \, {\left ({\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} b x^{3} + {\left (a^{4} + 4 i \, a^{3} - 6 \, a^{2} - 4 i \, a + 1\right )} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((-I*a - 14)*b^3*x^3 + (-I*a^2 - 13*a - 14*I)*b^2*x^2 - 3*((a^3 + 3*I*a^2 - 3*a - I)*b*x^3 + (a^4 + 4*I*a^
3 - 6*a^2 - 4*I*a + 1)*x^2)*sqrt((4*a^2 - 12*I*a - 9)*b^4/(a^8 + 6*I*a^7 - 14*a^6 - 14*I*a^5 - 14*I*a^3 + 14*a
^2 + 6*I*a - 1))*log(-((2*a - 3*I)*b^3*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(2*a - 3*I)*b^2 + (a^5 + 3*I*a^4
- 2*a^3 + 2*I*a^2 - 3*a - I)*sqrt((4*a^2 - 12*I*a - 9)*b^4/(a^8 + 6*I*a^7 - 14*a^6 - 14*I*a^5 - 14*I*a^3 + 14*
a^2 + 6*I*a - 1)))/((2*a - 3*I)*b^2)) + 3*((a^3 + 3*I*a^2 - 3*a - I)*b*x^3 + (a^4 + 4*I*a^3 - 6*a^2 - 4*I*a +
1)*x^2)*sqrt((4*a^2 - 12*I*a - 9)*b^4/(a^8 + 6*I*a^7 - 14*a^6 - 14*I*a^5 - 14*I*a^3 + 14*a^2 + 6*I*a - 1))*log
(-((2*a - 3*I)*b^3*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(2*a - 3*I)*b^2 - (a^5 + 3*I*a^4 - 2*a^3 + 2*I*a^2 -
3*a - I)*sqrt((4*a^2 - 12*I*a - 9)*b^4/(a^8 + 6*I*a^7 - 14*a^6 - 14*I*a^5 - 14*I*a^3 + 14*a^2 + 6*I*a - 1)))/(
(2*a - 3*I)*b^2)) + ((-I*a - 14)*b^2*x^2 + I*a^3 - 5*(a + I)*b*x - a^2 + I*a - 1)*sqrt(b^2*x^2 + 2*a*b*x + a^2
+ 1))/((a^3 + 3*I*a^2 - 3*a - I)*b*x^3 + (a^4 + 4*I*a^3 - 6*a^2 - 4*I*a + 1)*x^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - i \left (\int \frac {i}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3}}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2}}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \left (- \frac {3 b x}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {b^{3} x^{3}}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i b^{2} x^{2}}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {3 a b^{2} x^{2}}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} b x}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {6 i a b x}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*(Integral(I/(a**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 +
1) + b**2*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integra
l(-3*a/(a**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**
2*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(a**3/(
a**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**5*s
qrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*I*a**2/(a**
2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**5*sqrt
(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*b*x/(a**2*x**3
*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**5*sqrt(a**2
+ 2*a*b*x + b**2*x**2 + 1) + x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(b**3*x**3/(a**2*x**3*sq
rt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**5*sqrt(a**2 + 2
*a*b*x + b**2*x**2 + 1) + x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*I*b**2*x**2/(a**2*x**3*
sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**5*sqrt(a**2 +
2*a*b*x + b**2*x**2 + 1) + x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(3*a*b**2*x**2/(a**2*x**3
*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**5*sqrt(a**2
+ 2*a*b*x + b**2*x**2 + 1) + x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(3*a**2*b*x/(a**2*x**3*s
qrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**5*sqrt(a**2 +
2*a*b*x + b**2*x**2 + 1) + x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-6*I*a*b*x/(a**2*x**3*sqr
t(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**5*sqrt(a**2 + 2*
a*b*x + b**2*x**2 + 1) + x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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