\(\int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\) [373]

Optimal result

Integrand size = 27, antiderivative size = 45 \[ \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=-\frac {2 i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b} d} \]

[Out]

-2*I*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d/(a-I*b)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3618, 65, 214} \[ \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=-\frac {2 i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}} \]

[In]

Int[(1 + I*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]],x]

[Out]

((-2*I)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d)

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, 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 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {i \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{d} \\ & = -\frac {2 \text {Subst}\left (\int \frac {1}{-1-\frac {i a}{b}+\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b d} \\ & = -\frac {2 i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=-\frac {2 i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b} d} \]

[In]

Integrate[(1 + I*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]],x]

[Out]

((-2*I)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 728 vs. \(2 (36 ) = 72\).

Time = 0.10 (sec) , antiderivative size = 729, normalized size of antiderivative = 16.20

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (-i \sqrt {a^{2}+b^{2}}-i a +b \right ) \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (-i \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a +\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b -\frac {\left (-i \sqrt {a^{2}+b^{2}}-i a +b \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}+\frac {-\frac {\left (-2 i \sqrt {a^{2}+b^{2}}\, a^{2}-i \sqrt {a^{2}+b^{2}}\, b^{2}-2 i a^{3}-2 i a \,b^{2}+\sqrt {a^{2}+b^{2}}\, a b +a^{2} b +b^{3}\right ) \ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+i \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+i \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a b -\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b -\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{3}+\frac {\left (-2 i \sqrt {a^{2}+b^{2}}\, a^{2}-i \sqrt {a^{2}+b^{2}}\, b^{2}-2 i a^{3}-2 i a \,b^{2}+\sqrt {a^{2}+b^{2}}\, a b +a^{2} b +b^{3}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, \left (\sqrt {a^{2}+b^{2}}\, a +a^{2}+b^{2}\right )}}{d}\)	\(729\)
default	\(\frac {\frac {\frac {\left (-i \sqrt {a^{2}+b^{2}}-i a +b \right ) \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (-i \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a +\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b -\frac {\left (-i \sqrt {a^{2}+b^{2}}-i a +b \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}+\frac {-\frac {\left (-2 i \sqrt {a^{2}+b^{2}}\, a^{2}-i \sqrt {a^{2}+b^{2}}\, b^{2}-2 i a^{3}-2 i a \,b^{2}+\sqrt {a^{2}+b^{2}}\, a b +a^{2} b +b^{3}\right ) \ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+i \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+i \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a b -\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b -\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{3}+\frac {\left (-2 i \sqrt {a^{2}+b^{2}}\, a^{2}-i \sqrt {a^{2}+b^{2}}\, b^{2}-2 i a^{3}-2 i a \,b^{2}+\sqrt {a^{2}+b^{2}}\, a b +a^{2} b +b^{3}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, \left (\sqrt {a^{2}+b^{2}}\, a +a^{2}+b^{2}\right )}}{d}\)	\(729\)
parts	\(\text {Expression too large to display}\)	\(1890\)

[In]

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

[Out]

1/d*(1/(2*(a^2+b^2)^(1/2)+2*a)^(1/2)/(a^2+b^2)^(1/2)*(1/2*(-I*(a^2+b^2)^(1/2)-I*a+b)*ln(b*tan(d*x+c)+a+(a+b*ta
n(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))+2*(-I*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+(2*(a^2+b
^2)^(1/2)+2*a)^(1/2)*b-1/2*(-I*(a^2+b^2)^(1/2)-I*a+b)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(
1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)))+1/(2*(a^2
+b^2)^(1/2)+2*a)^(1/2)/(a^2+b^2)^(1/2)/((a^2+b^2)^(1/2)*a+a^2+b^2)*(-1/2*(-2*I*(a^2+b^2)^(1/2)*a^2-I*(a^2+b^2)
^(1/2)*b^2-2*I*a^3-2*I*a*b^2+(a^2+b^2)^(1/2)*a*b+a^2*b+b^3)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^
(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))+2*(I*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^2+I*(2*(a^2+b^2)^(1
/2)+2*a)^(1/2)*a^3+I*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a*b^2-(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a*b-(2*
(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2*b-(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*b^3+1/2*(-2*I*(a^2+b^2)^(1/2)*a^2-I*(a^2+b^2)^(
1/2)*b^2-2*I*a^3-2*I*a*b^2+(a^2+b^2)^(1/2)*a*b+a^2*b+b^3)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*
a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (33) = 66\).

Time = 0.24 (sec) , antiderivative size = 249, normalized size of antiderivative = 5.53 \[ \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {1}{4} \, \sqrt {-\frac {4 i}{{\left (i \, a + b\right )} d^{2}}} \log \left ({\left ({\left ({\left (i \, a + b\right )} d e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (i \, a + b\right )} d\right )} \sqrt {\frac {{\left (a - i \, b\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + a + i \, b}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {4 i}{{\left (i \, a + b\right )} d^{2}}} + 2 \, {\left (a - i \, b\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 \, a\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) - \frac {1}{4} \, \sqrt {-\frac {4 i}{{\left (i \, a + b\right )} d^{2}}} \log \left ({\left ({\left ({\left (-i \, a - b\right )} d e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-i \, a - b\right )} d\right )} \sqrt {\frac {{\left (a - i \, b\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + a + i \, b}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {-\frac {4 i}{{\left (i \, a + b\right )} d^{2}}} + 2 \, {\left (a - i \, b\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 \, a\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) \]

[In]

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

[Out]

1/4*sqrt(-4*I/((I*a + b)*d^2))*log((((I*a + b)*d*e^(2*I*d*x + 2*I*c) + (I*a + b)*d)*sqrt(((a - I*b)*e^(2*I*d*x
+ 2*I*c) + a + I*b)/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-4*I/((I*a + b)*d^2)) + 2*(a - I*b)*e^(2*I*d*x + 2*I*c) +
2*a)*e^(-2*I*d*x - 2*I*c)) - 1/4*sqrt(-4*I/((I*a + b)*d^2))*log((((-I*a - b)*d*e^(2*I*d*x + 2*I*c) + (-I*a -
b)*d)*sqrt(((a - I*b)*e^(2*I*d*x + 2*I*c) + a + I*b)/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(-4*I/((I*a + b)*d^2)) + 2
*(a - I*b)*e^(2*I*d*x + 2*I*c) + 2*a)*e^(-2*I*d*x - 2*I*c))

Sympy [F]

\[ \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=i \left (\int \left (- \frac {i}{\sqrt {a + b \tan {\left (c + d x \right )}}}\right )\, dx + \int \frac {\tan {\left (c + d x \right )}}{\sqrt {a + b \tan {\left (c + d x \right )}}}\, dx\right ) \]

[In]

integrate((1+I*tan(d*x+c))/(a+b*tan(d*x+c))**(1/2),x)

[Out]

I*(Integral(-I/sqrt(a + b*tan(c + d*x)), x) + Integral(tan(c + d*x)/sqrt(a + b*tan(c + d*x)), x))

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 6480 vs. \(2 (33) = 66\).

Time = 0.64 (sec) , antiderivative size = 6480, normalized size of antiderivative = 144.00 \[ \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/4*(sqrt(2*a^2 + 2*b^2)*sqrt(a + sqrt(a^2 + b^2))*(2*arctan2((b^2*cos(2*d*x + 2*c) - a*b*sin(2*d*x + 2*c) +
((a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 + (a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a^2*b^2 + b
^4 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4*(a^3*b + a*b^3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*b^2 - b^4)
*cos(2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + a^2*b^2
)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^3*b + a*b^
3)*cos(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))^(1/4)*b*sin(1/2*arctan2(-2*(a*b*
cos(2*d*x + 2*c)^2 - a*b*sin(2*d*x + 2*c)^2 + a*b*cos(2*d*x + 2*c) - (a^2 + (a^2 - b^2)*cos(2*d*x + 2*c))*sin(
2*d*x + 2*c))/b^2, (2*a^2*cos(2*d*x + 2*c) + (a^2 - b^2)*cos(2*d*x + 2*c)^2 - (a^2 - b^2)*sin(2*d*x + 2*c)^2 +
a^2 + b^2 + 2*(2*a*b*cos(2*d*x + 2*c) + a*b)*sin(2*d*x + 2*c))/b^2)))/b^2, -(a*cos(2*d*x + 2*c) + b*sin(2*d*x
+ 2*c) - ((a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 + (a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a
^2*b^2 + b^4 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4*(a^3*b + a*b^3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*
b^2 - b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4
+ a^2*b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^
3*b + a*b^3)*cos(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))^(1/4)*cos(1/2*arctan2(
-2*(a*b*cos(2*d*x + 2*c)^2 - a*b*sin(2*d*x + 2*c)^2 + a*b*cos(2*d*x + 2*c) - (a^2 + (a^2 - b^2)*cos(2*d*x + 2*
c))*sin(2*d*x + 2*c))/b^2, (2*a^2*cos(2*d*x + 2*c) + (a^2 - b^2)*cos(2*d*x + 2*c)^2 - (a^2 - b^2)*sin(2*d*x +
2*c)^2 + a^2 + b^2 + 2*(2*a*b*cos(2*d*x + 2*c) + a*b)*sin(2*d*x + 2*c))/b^2)) + a)/b) - I*log((2*a^2*cos(2*d*x
+ 2*c) + (a^2 + b^2)*cos(2*d*x + 2*c)^2 + 2*a*b*sin(2*d*x + 2*c) + (a^2 + b^2)*sin(2*d*x + 2*c)^2 + sqrt((a^4
+ 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 + (a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a^2*b^2 + b^4 +
4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4*(a^3*b + a*b^3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*b^2 - b^4)*cos(
2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + a^2*b^2)*cos
(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^3*b + a*b^3)*co
s(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))*cos(1/2*arctan2(-2*(a*b*cos(2*d*x + 2
*c)^2 - a*b*sin(2*d*x + 2*c)^2 + a*b*cos(2*d*x + 2*c) - (a^2 + (a^2 - b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))
/b^2, (2*a^2*cos(2*d*x + 2*c) + (a^2 - b^2)*cos(2*d*x + 2*c)^2 - (a^2 - b^2)*sin(2*d*x + 2*c)^2 + a^2 + b^2 +
2*(2*a*b*cos(2*d*x + 2*c) + a*b)*sin(2*d*x + 2*c))/b^2))^2 + sqrt((a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 +
(a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4
*(a^3*b + a*b^3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*b^2 - b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^
4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + a^2*b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a
^4 + a^2*b^2)*cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^3*b + a*b^3)*cos(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos
(2*d*x + 2*c))*sin(2*d*x + 2*c))*sin(1/2*arctan2(-2*(a*b*cos(2*d*x + 2*c)^2 - a*b*sin(2*d*x + 2*c)^2 + a*b*cos
(2*d*x + 2*c) - (a^2 + (a^2 - b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))/b^2, (2*a^2*cos(2*d*x + 2*c) + (a^2 - b
^2)*cos(2*d*x + 2*c)^2 - (a^2 - b^2)*sin(2*d*x + 2*c)^2 + a^2 + b^2 + 2*(2*a*b*cos(2*d*x + 2*c) + a*b)*sin(2*d
*x + 2*c))/b^2))^2 + a^2 - 2*((a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 + (a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x +
2*c)^4 + a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4*(a^3*b + a*b^3)*sin(2*d*x + 2*c)^3
+ 2*(3*a^4 + 2*a^2*b^2 - b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*
x + 2*c)^2 + 2*(a^4 + a^2*b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c) + 4*(
a^3*b + a*b^3 + (a^3*b + a*b^3)*cos(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))^(1/
4)*(a*b*cos(2*d*x + 2*c) + b^2*sin(2*d*x + 2*c) + a*b)*cos(1/2*arctan2(-2*(a*b*cos(2*d*x + 2*c)^2 - a*b*sin(2*
d*x + 2*c)^2 + a*b*cos(2*d*x + 2*c) - (a^2 + (a^2 - b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))/b^2, (2*a^2*cos(2
*d*x + 2*c) + (a^2 - b^2)*cos(2*d*x + 2*c)^2 - (a^2 - b^2)*sin(2*d*x + 2*c)^2 + a^2 + b^2 + 2*(2*a*b*cos(2*d*x
+ 2*c) + a*b)*sin(2*d*x + 2*c))/b^2))/b + 2*((a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 + (a^4 + 2*a^2*b^2 +
b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4*(a^3*b + a*b^3)*sin
(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*b^2 - b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a^2*b^2
+ b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + a^2*b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a^4 + a^2*b^2)*cos(2*
d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^3*b + a*b^3)*cos(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos(2*d*x + 2*c))*sin(2
*d*x + 2*c))^(1/4)*(b^2*cos(2*d*x + 2*c) - a*b*sin(2*d*x + 2*c))*sin(1/2*arctan2(-2*(a*b*cos(2*d*x + 2*c)^2 -
a*b*sin(2*d*x + 2*c)^2 + a*b*cos(2*d*x + 2*c) - (a^2 + (a^2 - b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))/b^2, (2
*a^2*cos(2*d*x + 2*c) + (a^2 - b^2)*cos(2*d*x + 2*c)^2 - (a^2 - b^2)*sin(2*d*x + 2*c)^2 + a^2 + b^2 + 2*(2*a*b
*cos(2*d*x + 2*c) + a*b)*sin(2*d*x + 2*c))/b^2))/b)/b^2)) - sqrt(2*a^2 + 2*b^2)*sqrt(-a + sqrt(a^2 + b^2))*(-2
*I*arctan2((b^2*cos(2*d*x + 2*c) - a*b*sin(2*d*x + 2*c) + ((a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 + (a^4 +
2*a^2*b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4*(a^3*b
+ a*b^3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*b^2 - b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + (a^
4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + a^2*b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a^4 + a^
2*b^2)*cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^3*b + a*b^3)*cos(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos(2*d*x
+ 2*c))*sin(2*d*x + 2*c))^(1/4)*b*sin(1/2*arctan2(-2*(a*b*cos(2*d*x + 2*c)^2 - a*b*sin(2*d*x + 2*c)^2 + a*b*co
s(2*d*x + 2*c) - (a^2 + (a^2 - b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))/b^2, (2*a^2*cos(2*d*x + 2*c) + (a^2 -
b^2)*cos(2*d*x + 2*c)^2 - (a^2 - b^2)*sin(2*d*x + 2*c)^2 + a^2 + b^2 + 2*(2*a*b*cos(2*d*x + 2*c) + a*b)*sin(2*
d*x + 2*c))/b^2)))/b^2, -(a*cos(2*d*x + 2*c) + b*sin(2*d*x + 2*c) - ((a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^
4 + (a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3
+ 4*(a^3*b + a*b^3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*b^2 - b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 +
b^4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + a^2*b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4
*(a^4 + a^2*b^2)*cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^3*b + a*b^3)*cos(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*
cos(2*d*x + 2*c))*sin(2*d*x + 2*c))^(1/4)*cos(1/2*arctan2(-2*(a*b*cos(2*d*x + 2*c)^2 - a*b*sin(2*d*x + 2*c)^2
+ a*b*cos(2*d*x + 2*c) - (a^2 + (a^2 - b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))/b^2, (2*a^2*cos(2*d*x + 2*c) +
(a^2 - b^2)*cos(2*d*x + 2*c)^2 - (a^2 - b^2)*sin(2*d*x + 2*c)^2 + a^2 + b^2 + 2*(2*a*b*cos(2*d*x + 2*c) + a*b
)*sin(2*d*x + 2*c))/b^2)) + a)/b) - log((2*a^2*cos(2*d*x + 2*c) + (a^2 + b^2)*cos(2*d*x + 2*c)^2 + 2*a*b*sin(2
*d*x + 2*c) + (a^2 + b^2)*sin(2*d*x + 2*c)^2 + sqrt((a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 + (a^4 + 2*a^2*
b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4*(a^3*b + a*b^
3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*b^2 - b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a
^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + a^2*b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a^4 + a^2*b^2)*
cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^3*b + a*b^3)*cos(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos(2*d*x + 2*c))
*sin(2*d*x + 2*c))*cos(1/2*arctan2(-2*(a*b*cos(2*d*x + 2*c)^2 - a*b*sin(2*d*x + 2*c)^2 + a*b*cos(2*d*x + 2*c)
- (a^2 + (a^2 - b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))/b^2, (2*a^2*cos(2*d*x + 2*c) + (a^2 - b^2)*cos(2*d*x
+ 2*c)^2 - (a^2 - b^2)*sin(2*d*x + 2*c)^2 + a^2 + b^2 + 2*(2*a*b*cos(2*d*x + 2*c) + a*b)*sin(2*d*x + 2*c))/b^2
))^2 + sqrt((a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 + (a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*
a^2*b^2 + b^4 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4*(a^3*b + a*b^3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2
*b^2 - b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^
4 + a^2*b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a
^3*b + a*b^3)*cos(2*d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))*sin(1/2*arctan2(-2*(a
*b*cos(2*d*x + 2*c)^2 - a*b*sin(2*d*x + 2*c)^2 + a*b*cos(2*d*x + 2*c) - (a^2 + (a^2 - b^2)*cos(2*d*x + 2*c))*s
in(2*d*x + 2*c))/b^2, (2*a^2*cos(2*d*x + 2*c) + (a^2 - b^2)*cos(2*d*x + 2*c)^2 - (a^2 - b^2)*sin(2*d*x + 2*c)^
2 + a^2 + b^2 + 2*(2*a*b*cos(2*d*x + 2*c) + a*b)*sin(2*d*x + 2*c))/b^2))^2 + a^2 - 2*((a^4 + 2*a^2*b^2 + b^4)*
cos(2*d*x + 2*c)^4 + (a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + a^2*b^2)*co
s(2*d*x + 2*c)^3 + 4*(a^3*b + a*b^3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*b^2 - b^4)*cos(2*d*x + 2*c)^2 + 2*(
a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + a^2*b^2)*cos(2*d*x + 2*c))*sin(2
*d*x + 2*c)^2 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^3*b + a*b^3)*cos(2*d*x + 2*c)^2 + 2
*(a^3*b + a*b^3)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))^(1/4)*(a*b*cos(2*d*x + 2*c) + b^2*sin(2*d*x + 2*c) + a*b)
*cos(1/2*arctan2(-2*(a*b*cos(2*d*x + 2*c)^2 - a*b*sin(2*d*x + 2*c)^2 + a*b*cos(2*d*x + 2*c) - (a^2 + (a^2 - b^
2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))/b^2, (2*a^2*cos(2*d*x + 2*c) + (a^2 - b^2)*cos(2*d*x + 2*c)^2 - (a^2 -
b^2)*sin(2*d*x + 2*c)^2 + a^2 + b^2 + 2*(2*a*b*cos(2*d*x + 2*c) + a*b)*sin(2*d*x + 2*c))/b^2))/b + 2*((a^4 + 2
*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^4 + (a^4 + 2*a^2*b^2 + b^4)*sin(2*d*x + 2*c)^4 + a^4 + 2*a^2*b^2 + b^4 + 4*(a
^4 + a^2*b^2)*cos(2*d*x + 2*c)^3 + 4*(a^3*b + a*b^3)*sin(2*d*x + 2*c)^3 + 2*(3*a^4 + 2*a^2*b^2 - b^4)*cos(2*d*
x + 2*c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + (a^4 + 2*a^2*b^2 + b^4)*cos(2*d*x + 2*c)^2 + 2*(a^4 + a^2*b^2)*cos(2*d
*x + 2*c))*sin(2*d*x + 2*c)^2 + 4*(a^4 + a^2*b^2)*cos(2*d*x + 2*c) + 4*(a^3*b + a*b^3 + (a^3*b + a*b^3)*cos(2*
d*x + 2*c)^2 + 2*(a^3*b + a*b^3)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))^(1/4)*(b^2*cos(2*d*x + 2*c) - a*b*sin(2*d
*x + 2*c))*sin(1/2*arctan2(-2*(a*b*cos(2*d*x + 2*c)^2 - a*b*sin(2*d*x + 2*c)^2 + a*b*cos(2*d*x + 2*c) - (a^2 +
(a^2 - b^2)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))/b^2, (2*a^2*cos(2*d*x + 2*c) + (a^2 - b^2)*cos(2*d*x + 2*c)^2
- (a^2 - b^2)*sin(2*d*x + 2*c)^2 + a^2 + b^2 + 2*(2*a*b*cos(2*d*x + 2*c) + a*b)*sin(2*d*x + 2*c))/b^2))/b)/b^
2)))/((a^2 + b^2)*d)

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (33) = 66\).

Time = 0.41 (sec) , antiderivative size = 155, normalized size of antiderivative = 3.44 \[ \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {4 i \, \arctan \left (\frac {2 \, {\left (\sqrt {b \tan \left (d x + c\right ) + a} a - \sqrt {a^{2} + b^{2}} \sqrt {b \tan \left (d x + c\right ) + a}\right )}}{a \sqrt {-2 \, a + 2 \, \sqrt {a^{2} + b^{2}}} - i \, \sqrt {-2 \, a + 2 \, \sqrt {a^{2} + b^{2}}} b - \sqrt {a^{2} + b^{2}} \sqrt {-2 \, a + 2 \, \sqrt {a^{2} + b^{2}}}}\right )}{\sqrt {-2 \, a + 2 \, \sqrt {a^{2} + b^{2}}} d {\left (-\frac {i \, b}{a - \sqrt {a^{2} + b^{2}}} + 1\right )}} \]

[In]

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

[Out]

4*I*arctan(2*(sqrt(b*tan(d*x + c) + a)*a - sqrt(a^2 + b^2)*sqrt(b*tan(d*x + c) + a))/(a*sqrt(-2*a + 2*sqrt(a^2
+ b^2)) - I*sqrt(-2*a + 2*sqrt(a^2 + b^2))*b - sqrt(a^2 + b^2)*sqrt(-2*a + 2*sqrt(a^2 + b^2))))/(sqrt(-2*a +
2*sqrt(a^2 + b^2))*d*(-I*b/(a - sqrt(a^2 + b^2)) + 1))

Mupad [B] (verification not implemented)

Time = 10.39 (sec) , antiderivative size = 1410, normalized size of antiderivative = 31.33 \[ \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Too large to display} \]

[In]

int((tan(c + d*x)*1i + 1)/(a + b*tan(c + d*x))^(1/2),x)

[Out]

2*atanh((32*b^2*((b*1i)/(4*a^2*d^2 + 4*b^2*d^2) - a/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))
/((a^2*b^2*d^2*64i)/(4*a^2*d^3 + 4*b^2*d^3) - (b^2*16i)/d + (64*a*b^3*d^2)/(4*a^2*d^3 + 4*b^2*d^3)) - (128*a^2
*b^2*((b*1i)/(4*a^2*d^2 + 4*b^2*d^2) - a/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((a^2*b^4*
d^2*256i)/(4*a^2*d^3 + 4*b^2*d^3) - (a^2*b^2*64i)/d - (b^4*64i)/d + (256*a^3*b^3*d^2)/(4*a^2*d^3 + 4*b^2*d^3)
+ (a^4*b^2*d^2*256i)/(4*a^2*d^3 + 4*b^2*d^3) + (256*a*b^5*d^2)/(4*a^2*d^3 + 4*b^2*d^3)) + (a*b^3*((b*1i)/(4*a^
2*d^2 + 4*b^2*d^2) - a/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*128i)/((a^2*b^4*d^2*256i)/(4*
a^2*d^3 + 4*b^2*d^3) - (a^2*b^2*64i)/d - (b^4*64i)/d + (256*a^3*b^3*d^2)/(4*a^2*d^3 + 4*b^2*d^3) + (a^4*b^2*d^
2*256i)/(4*a^2*d^3 + 4*b^2*d^3) + (256*a*b^5*d^2)/(4*a^2*d^3 + 4*b^2*d^3)))*(-(a - b*1i)/(4*a^2*d^2 + 4*b^2*d^
2))^(1/2) + (log(d*(-1/(d^2*(a - b*1i)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*1i + 1)*(-1/(a*d^2 - b*d^2*1i))^(1/2
))/2 - log(d*(-1/(d^2*(a - b*1i)))^(1/2)*(a + b*tan(c + d*x))^(1/2) + 1i)*(-1/(4*(a*d^2 - b*d^2*1i)))^(1/2) +
(log(16*b^2*(a + b*tan(c + d*x))^(1/2) + 16*b^3*d*(-1/(d^2*(a - b*1i)))^(1/2) - (16*a*b^2*(a + b*tan(c + d*x))
^(1/2))/(a - b*1i))*(-1/(a*d^2 - b*d^2*1i))^(1/2))/2 - log(16*b^3*d*(-1/(d^2*(a - b*1i)))^(1/2) - 16*b^2*(a +
b*tan(c + d*x))^(1/2) + (16*a*b^2*(a + b*tan(c + d*x))^(1/2))/(a - b*1i))*(-1/(4*(a*d^2 - b*d^2*1i)))^(1/2) +
2*atanh((32*b^2*((b*1i)/(4*a^2*d^2 + 4*b^2*d^2) - a/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))
/((b^4*d^2*64i)/(4*a^2*d^3 + 4*b^2*d^3) - (64*a*b^3*d^2)/(4*a^2*d^3 + 4*b^2*d^3)) + (a*b^3*((b*1i)/(4*a^2*d^2
+ 4*b^2*d^2) - a/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*128i)/((b^6*d^2*256i)/(4*a^2*d^3 +
4*b^2*d^3) + (a^2*b^4*d^2*256i)/(4*a^2*d^3 + 4*b^2*d^3) - (256*a^3*b^3*d^2)/(4*a^2*d^3 + 4*b^2*d^3) - (256*a*b
^5*d^2)/(4*a^2*d^3 + 4*b^2*d^3)) - (128*a^2*b^2*((b*1i)/(4*a^2*d^2 + 4*b^2*d^2) - a/(4*a^2*d^2 + 4*b^2*d^2))^(
1/2)*(a + b*tan(c + d*x))^(1/2))/((b^6*d^2*256i)/(4*a^2*d^3 + 4*b^2*d^3) + (a^2*b^4*d^2*256i)/(4*a^2*d^3 + 4*b
^2*d^3) - (256*a^3*b^3*d^2)/(4*a^2*d^3 + 4*b^2*d^3) - (256*a*b^5*d^2)/(4*a^2*d^3 + 4*b^2*d^3)))*(-(a - b*1i)/(
4*a^2*d^2 + 4*b^2*d^2))^(1/2)