\(\int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\) [374]

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 = 0.04 (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. 740 vs. \(2 (36 ) = 72\).

Time = 0.09 (sec) , antiderivative size = 741, normalized size of antiderivative = 16.47

method	result	size
derivativedivides	\(\frac {-\frac {\frac {\left (-i \sqrt {a^{2}+b^{2}}-i a -b \right ) \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (-i \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a -\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b -\frac {\left (-i \sqrt {a^{2}+b^{2}}-i a -b \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}-\frac {-\frac {\left (-2 i \sqrt {a^{2}+b^{2}}\, a^{2}-i \sqrt {a^{2}+b^{2}}\, b^{2}-2 i a^{3}-2 i a \,b^{2}-\sqrt {a^{2}+b^{2}}\, a b -a^{2} b -b^{3}\right ) \ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+i \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+i \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a b +\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b +\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{3}+\frac {\left (-2 i \sqrt {a^{2}+b^{2}}\, a^{2}-i \sqrt {a^{2}+b^{2}}\, b^{2}-2 i a^{3}-2 i a \,b^{2}-\sqrt {a^{2}+b^{2}}\, a b -a^{2} b -b^{3}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, \left (\sqrt {a^{2}+b^{2}}\, a +a^{2}+b^{2}\right )}}{d}\)	\(741\)
default	\(\frac {-\frac {\frac {\left (-i \sqrt {a^{2}+b^{2}}-i a -b \right ) \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (-i \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a -\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b -\frac {\left (-i \sqrt {a^{2}+b^{2}}-i a -b \right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}-\frac {-\frac {\left (-2 i \sqrt {a^{2}+b^{2}}\, a^{2}-i \sqrt {a^{2}+b^{2}}\, b^{2}-2 i a^{3}-2 i a \,b^{2}-\sqrt {a^{2}+b^{2}}\, a b -a^{2} b -b^{3}\right ) \ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (i \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+i \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}+i \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a b +\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b +\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{3}+\frac {\left (-2 i \sqrt {a^{2}+b^{2}}\, a^{2}-i \sqrt {a^{2}+b^{2}}\, b^{2}-2 i a^{3}-2 i a \,b^{2}-\sqrt {a^{2}+b^{2}}\, a b -a^{2} b -b^{3}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, \left (\sqrt {a^{2}+b^{2}}\, a +a^{2}+b^{2}\right )}}{d}\)	\(741\)
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*t
an(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. 267 vs. \(2 (33) = 66\).

Time = 0.24 (sec) , antiderivative size = 267, normalized size of antiderivative = 5.93 \[ \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=-\frac {1}{4} \, \sqrt {\frac {4 i}{{\left (-i \, a + b\right )} d^{2}}} \log \left (\frac {{\left ({\left ({\left (i \, a - b\right )} d e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (i \, a - b\right )} d\right )} \sqrt {\frac {{\left (a - i \, b\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + a + i \, b}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {4 i}{{\left (-i \, a + b\right )} d^{2}}} + 2 \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 2 \, a + 2 i \, b\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, a + b\right )} d}\right ) + \frac {1}{4} \, \sqrt {\frac {4 i}{{\left (-i \, a + b\right )} d^{2}}} \log \left (\frac {{\left ({\left ({\left (-i \, a + b\right )} d e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-i \, a + b\right )} d\right )} \sqrt {\frac {{\left (a - i \, b\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + a + i \, b}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {4 i}{{\left (-i \, a + b\right )} d^{2}}} + 2 \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 2 \, a + 2 i \, b\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, a + b\right )} d}\right ) \]

[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*e^(2*I*d*x + 2*I*c) + 2*a +
2*I*b)*e^(-2*I*d*x - 2*I*c)/((-I*a + b)*d)) + 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*e^(2*I*d*x + 2*I*c) + 2*a + 2*I*b)*e^(-2*I*d*x - 2*I*c)/((-I*a + b)*d))

Sympy [F]

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

[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 [F(-2)]

Exception generated. \[ \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is

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 = 0.40 (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 = 9.22 (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]

(log(d*(-1/(d^2*(a - b*1i)))^(1/2)*(a + b*tan(c + d*x))^(1/2) + 1i)*(-1/(a*d^2 - b*d^2*1i))^(1/2))/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))/((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*256
i)/(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/(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)