3.2.85 \(\int \frac {e^{3 i \text {ArcTan}(a+b x)}}{x} \, dx\) [185]

Optimal. Leaf size=134 \[ \frac {4 \sqrt {1+i a+i b x}}{(1-i a) \sqrt {1-i a-i b x}}-i \sinh ^{-1}(a+b x)-\frac {2 (i-a)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{(i+a)^{3/2}} \]

[Out]

-I*arcsinh(b*x+a)-2*(I-a)^(3/2)*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)
^(3/2)+4*(1+I*a+I*b*x)^(1/2)/(1-I*a)/(1-I*a-I*b*x)^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 134, 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, 100, 163, 55, 633, 221, 95, 214} \begin {gather*} \frac {4 \sqrt {i a+i b x+1}}{(1-i a) \sqrt {-i a-i b x+1}}-i \sinh ^{-1}(a+b x)-\frac {2 (-a+i)^{3/2} \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 )}{(a+i)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((3*I)*ArcTan[a + b*x])/x,x]

[Out]

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

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 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 100

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

Rule 163

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

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 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 \frac {e^{3 i \tan ^{-1}(a+b x)}}{x} \, dx &=\int \frac {(1+i a+i b x)^{3/2}}{x (1-i a-i b x)^{3/2}} \, dx\\ &=\frac {4 \sqrt {1+i a+i b x}}{(1-i a) \sqrt {1-i a-i b x}}-\frac {2 \int \frac {\frac {1}{2} i (i-a)^2 b-\frac {1}{2} (1-i a) b^2 x}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{(i+a) b}\\ &=\frac {4 \sqrt {1+i a+i b x}}{(1-i a) \sqrt {1-i a-i b x}}-\frac {(i-a)^2 \int \frac {1}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{1-i a}-(i b) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx\\ &=\frac {4 \sqrt {1+i a+i b x}}{(1-i a) \sqrt {1-i a-i b x}}-\frac {\left (2 (i-a)^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 )}{1-i a}-(i b) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx\\ &=\frac {4 \sqrt {1+i a+i b x}}{(1-i a) \sqrt {1-i a-i b x}}-\frac {2 (i-a)^{3/2} \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 )}{(i+a)^{3/2}}-\frac {i \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4 b^2}}} \, dx,x,2 a b+2 b^2 x\right )}{2 b}\\ &=\frac {4 \sqrt {1+i a+i b x}}{(1-i a) \sqrt {1-i a-i b x}}-i \sinh ^{-1}(a+b x)-\frac {2 (i-a)^{3/2} \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 )}{(i+a)^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.53, size = 196, normalized size = 1.46 \begin {gather*} \frac {2 \left (\frac {2 i \sqrt {1+i a+i b x}}{\sqrt {-i (i+a+b x)}}+\frac {\sqrt [4]{-1} (i+a) (-i b)^{3/2} \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (i+a+b x)}}{\sqrt {-i b}}\right )}{b^{3/2}}+\frac {\sqrt {-1-i a} (-i+a) \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}}\right )}{i+a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(((2*I)*Sqrt[1 + I*a + I*b*x])/Sqrt[(-I)*(I + a + b*x)] + ((-1)^(1/4)*(I + a)*((-I)*b)^(3/2)*ArcSinh[((1/2
+ I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/b^(3/2) + (Sqrt[-1 - I*a]*(-I + a)*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)

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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. 484 vs. \(2 (104 ) = 208\).
time = 0.12, size = 485, normalized size = 3.62

method	result	size
default	\(-i b^{3} \left (-\frac {x}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {a \left (-\frac {1}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a \left (2 b^{2} x +2 a b \right )}{b \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right )}{b}+\frac {\ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b^{2} \sqrt {b^{2}}}\right )-3 \left (i a +1\right ) b^{2} \left (-\frac {1}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a \left (2 b^{2} x +2 a b \right )}{b \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right )+\frac {6 i \left (i a +1\right )^{2} b \left (2 b^{2} x +2 a b \right )}{\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\left (-i a^{3}-3 a^{2}+3 i a +1\right ) \left (\frac {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a b \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\right ) \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )\)	\(485\)

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,x,method=_RETURNVERBOSE)

[Out]

-I*b^3*(-x/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-a/b*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(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/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*(1+I*a)*b^2*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(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))+6*I*(1+I*a)^2*b*(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/(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))

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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. 733 vs. \(2 (90) = 180\).
time = 0.28, size = 733, normalized size = 5.47 \begin {gather*} \frac {2 i \, a^{2} b^{3} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} + \frac {{\left (-i \, a^{2} - i\right )} b^{3} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} + \frac {{\left (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\right )} a b^{3} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )}} + \frac {i \, {\left (a^{2} + 1\right )} a b^{2}}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} + \frac {{\left (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\right )} a^{2} b^{2}}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )}} - \frac {3 \, {\left (i \, a b^{2} + b^{2}\right )} a b x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} + \frac {3 \, {\left (i \, a^{2} b + 2 \, a b - i \, b\right )} b^{2} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} - \frac {3 \, {\left (i \, a b^{2} + b^{2}\right )} a^{2}}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} + \frac {3 \, {\left (i \, a^{2} b + 2 \, a b - i \, b\right )} a b}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} - \frac {{\left (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\right )} \operatorname {arsinh}\left (\frac {2 \, a b x}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2 \, a^{2}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}} + \frac {2}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}} {\left | x \right |}}\right )}{{\left (a^{2} + 1\right )}^{\frac {3}{2}}} + \frac {-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )}} + \frac {3 \, {\left (i \, a b^{2} + b^{2}\right )}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b^{2}} - i \, \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 ) \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,x, algorithm="maxima")

[Out]

2*I*a^2*b^3*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + (-I*a^2 - I)*b^3*x/((a^2*b^2 - (
a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + (-I*a^3 - 3*a^2 + 3*I*a + 1)*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)) + I*(a^2 + 1)*a*b^2/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2
+ 2*a*b*x + a^2 + 1)) + (-I*a^3 - 3*a^2 + 3*I*a + 1)*a^2*b^2/((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*b*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))
+ 3*(I*a^2*b + 2*a*b - I*b)*b^2*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 3*(I*a*b^2
+ b^2)*a^2/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + 3*(I*a^2*b + 2*a*b - I*b)*a*b/((a^2
*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - (-I*a^3 - 3*a^2 + 3*I*a + 1)*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) + (-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)) + 3*(I*a*b^2 + b^2)/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - I*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^
2*b^2 + 4*(a^2 + 1)*b^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. 356 vs. \(2 (90) = 180\).
time = 1.92, size = 356, normalized size = 2.66 \begin {gather*} -\frac {{\left ({\left (a + i\right )} b x + a^{2} + 2 i \, a - 1\right )} \sqrt {-\frac {a^{3} - 3 i \, a^{2} - 3 \, a + i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}} \log \left (-\frac {{\left (a - i\right )} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a - i\right )} - {\left (i \, a^{2} - 2 \, a - i\right )} \sqrt {-\frac {a^{3} - 3 i \, a^{2} - 3 \, a + i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}}}{a - i}\right ) - {\left ({\left (a + i\right )} b x + a^{2} + 2 i \, a - 1\right )} \sqrt {-\frac {a^{3} - 3 i \, a^{2} - 3 \, a + i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}} \log \left (-\frac {{\left (a - i\right )} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a - i\right )} - {\left (-i \, a^{2} + 2 \, a + i\right )} \sqrt {-\frac {a^{3} - 3 i \, a^{2} - 3 \, a + i}{a^{3} + 3 i \, a^{2} - 3 \, a - i}}}{a - i}\right ) + 4 \, b x + {\left ({\left (-i \, a + 1\right )} b x - i \, a^{2} + 2 \, a + i\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + 4 \, a + 4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 4 i}{{\left (a + i\right )} b x + a^{2} + 2 i \, a - 1} \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,x, algorithm="fricas")

[Out]

-(((a + I)*b*x + a^2 + 2*I*a - 1)*sqrt(-(a^3 - 3*I*a^2 - 3*a + I)/(a^3 + 3*I*a^2 - 3*a - I))*log(-((a - I)*b*x
- sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a - I) - (I*a^2 - 2*a - I)*sqrt(-(a^3 - 3*I*a^2 - 3*a + I)/(a^3 + 3*I*a^
2 - 3*a - I)))/(a - I)) - ((a + I)*b*x + a^2 + 2*I*a - 1)*sqrt(-(a^3 - 3*I*a^2 - 3*a + I)/(a^3 + 3*I*a^2 - 3*a
- I))*log(-((a - I)*b*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a - I) - (-I*a^2 + 2*a + I)*sqrt(-(a^3 - 3*I*a^2
- 3*a + I)/(a^3 + 3*I*a^2 - 3*a - I)))/(a - I)) + 4*b*x + ((-I*a + 1)*b*x - I*a^2 + 2*a + I)*log(-b*x - a + s
qrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + 4*a + 4*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 4*I)/((a + I)*b*x + a^2 + 2*I*
a - 1)

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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 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a}{a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3}}{a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2}}{a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \left (- \frac {3 b x}{a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {b^{3} x^{3}}{a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \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 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \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 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} b x}{a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {6 i a b x}{a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x \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,x)

[Out]

-I*(Integral(I/(a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)
+ b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*a
/(a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**3*sq
rt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(a**3/(a**2*x*sqrt(
a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**3*sqrt(a**2 + 2*a*
b*x + b**2*x**2 + 1) + x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*I*a**2/(a**2*x*sqrt(a**2 + 2*
a*b*x + b**2*x**2 + 1) + 2*a*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**3*sqrt(a**2 + 2*a*b*x + b**
2*x**2 + 1) + x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*b*x/(a**2*x*sqrt(a**2 + 2*a*b*x + b**2
*x**2 + 1) + 2*a*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)
+ x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(b**3*x**3/(a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1
) + 2*a*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x*sqrt(
a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*I*b**2*x**2/(a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) +
2*a*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x*sqrt(a**2
+ 2*a*b*x + b**2*x**2 + 1)), x) + Integral(3*a*b**2*x**2/(a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b
*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x*sqrt(a**2 + 2*
a*b*x + b**2*x**2 + 1)), x) + Integral(3*a**2*b*x/(a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**2*sq
rt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x*sqrt(a**2 + 2*a*b*x +
b**2*x**2 + 1)), x) + Integral(-6*I*a*b*x/(a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**2*sqrt(a**2
+ 2*a*b*x + b**2*x**2 + 1) + b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x*sqrt(a**2 + 2*a*b*x + b**2*x**
2 + 1)), x))

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (90) = 180\).
time = 0.56, size = 252, normalized size = 1.88 \begin {gather*} \frac {i \, b \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 )}{3 \, {\left | b \right |}} - \frac {{\left (i \, a^{2} + 2 \, a - i\right )} \log \left (\frac {{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {{\left (b x + a\right )}^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {{\left (b x + a\right )}^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{\sqrt {a^{2} + 1} {\left (a + i\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,x, algorithm="giac")

[Out]

1/3*I*b*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))/abs(b) - (
I*a^2 + 2*a - I)*log(abs(-2*x*abs(b) + 2*sqrt((b*x + a)^2 + 1) - 2*sqrt(a^2 + 1))/abs(-2*x*abs(b) + 2*sqrt((b*
x + a)^2 + 1) + 2*sqrt(a^2 + 1)))/(sqrt(a^2 + 1)*(a + I))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3}{x\,{\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*((a + b*x)^2 + 1)^(3/2)),x)

[Out]

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

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