\(\int (a+b \tan (c+d x))^{5/2} \, dx\) [522]

Optimal result

Integrand size = 14, antiderivative size = 134 \[ \int (a+b \tan (c+d x))^{5/2} \, dx=-\frac {i (a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {i (a+i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {4 a b \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b (a+b \tan (c+d x))^{3/2}}{3 d} \]

[Out]

-I*(a-I*b)^(5/2)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d+I*(a+I*b)^(5/2)*arctanh((a+b*tan(d*x+c))^(1/2
)/(a+I*b)^(1/2))/d+4*a*b*(a+b*tan(d*x+c))^(1/2)/d+2/3*b*(a+b*tan(d*x+c))^(3/2)/d

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3563, 3609, 3620, 3618, 65, 214} \[ \int (a+b \tan (c+d x))^{5/2} \, dx=-\frac {i (a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {i (a+i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 b (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {4 a b \sqrt {a+b \tan (c+d x)}}{d} \]

[In]

Int[(a + b*Tan[c + d*x])^(5/2),x]

[Out]

((-I)*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + (I*(a + I*b)^(5/2)*ArcTanh[Sqrt[a +
b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (4*a*b*Sqrt[a + b*Tan[c + d*x]])/d + (2*b*(a + b*Tan[c + d*x])^(3/2))/(3*
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 3563

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1))
), x] + Int[(a^2 - b^2 + 2*a*b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n - 2), x] /; FreeQ[{a, b, c, d}, x] && NeQ
[a^2 + b^2, 0] && GtQ[n, 1]

Rule 3609

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*
((a + b*Tan[e + f*x])^m/(f*m)), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]

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]

Rule 3620

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {2 b (a+b \tan (c+d x))^{3/2}}{3 d}+\int \sqrt {a+b \tan (c+d x)} \left (a^2-b^2+2 a b \tan (c+d x)\right ) \, dx \\ & = \frac {4 a b \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b (a+b \tan (c+d x))^{3/2}}{3 d}+\int \frac {a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = \frac {4 a b \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {1}{2} (a-i b)^3 \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} (a+i b)^3 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = \frac {4 a b \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {(i a-b)^3 \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 d}-\frac {(i a+b)^3 \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 d} \\ & = \frac {4 a b \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b (a+b \tan (c+d x))^{3/2}}{3 d}-\frac {(a-i b)^3 \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 {(a+i b)^3 \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 {i (a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {i (a+i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {4 a b \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 b (a+b \tan (c+d x))^{3/2}}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.90 \[ \int (a+b \tan (c+d x))^{5/2} \, dx=\frac {-3 i (a-i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+3 i (a+i b)^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+2 b \sqrt {a+b \tan (c+d x)} (7 a+b \tan (c+d x))}{3 d} \]

[In]

Integrate[(a + b*Tan[c + d*x])^(5/2),x]

[Out]

((-3*I)*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + (3*I)*(a + I*b)^(5/2)*ArcTanh[Sqrt[a
+ b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*b*Sqrt[a + b*Tan[c + d*x]]*(7*a + b*Tan[c + d*x]))/(3*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1173\) vs. \(2(110)=220\).

Time = 0.08 (sec) , antiderivative size = 1174, normalized size of antiderivative = 8.76

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1174\)
default	\(\text {Expression too large to display}\)	\(1174\)

[In]

int((a+b*tan(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/3*b*(a+b*tan(d*x+c))^(3/2)/d+4*a*b*(a+b*tan(d*x+c))^(1/2)/d+1/4/d/b*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*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^2-1/4/d*b*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*(a^2+b^2)^(1/2)+2*a)^(1
/2)*(a^2+b^2)^(1/2)-1/4/d/b*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*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3+3/4/d*b*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*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+2/d*b/(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))*(a^2+b^2)^(1/2)*a-3/d*b/(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))*a^2+1/d*b^3/(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))-1/4/d/b*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1
/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^2+1/4/d*b*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+
c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)+1/4/d/b
*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*
a)^(1/2)*a^3-3/4/d*b*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(
2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a-2/d*b/(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))*(a^2+b^2)^(1/2)*a+3/d*b/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arct
an((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))*a^2-1/d*b^3/(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))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1125 vs. \(2 (104) = 208\).

Time = 0.26 (sec) , antiderivative size = 1125, normalized size of antiderivative = 8.40 \[ \int (a+b \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*tan(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

-1/6*(3*d*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 +
b^10)/d^4))/d^2)*log((5*a^8*b - 14*a^4*b^5 - 8*a^2*b^7 + b^9)*sqrt(b*tan(d*x + c) + a) + ((a^2 - b^2)*d^3*sqr
t(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) + 2*(5*a^5*b^2 - 10*a^3*b^4 + a*b^6)*d)*s
qrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4)
)/d^2)) - 3*d*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b
^8 + b^10)/d^4))/d^2)*log((5*a^8*b - 14*a^4*b^5 - 8*a^2*b^7 + b^9)*sqrt(b*tan(d*x + c) + a) - ((a^2 - b^2)*d^3
*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) + 2*(5*a^5*b^2 - 10*a^3*b^4 + a*b^6)*
d)*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 + d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/
d^4))/d^2)) - 3*d*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a
^2*b^8 + b^10)/d^4))/d^2)*log((5*a^8*b - 14*a^4*b^5 - 8*a^2*b^7 + b^9)*sqrt(b*tan(d*x + c) + a) + ((a^2 - b^2)
*d^3*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) - 2*(5*a^5*b^2 - 10*a^3*b^4 + a*b
^6)*d)*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^
10)/d^4))/d^2)) + 3*d*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 -
20*a^2*b^8 + b^10)/d^4))/d^2)*log((5*a^8*b - 14*a^4*b^5 - 8*a^2*b^7 + b^9)*sqrt(b*tan(d*x + c) + a) - ((a^2 -
b^2)*d^3*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) - 2*(5*a^5*b^2 - 10*a^3*b^4 +
a*b^6)*d)*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 - d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8
+ b^10)/d^4))/d^2)) - 4*(b^2*tan(d*x + c) + 7*a*b)*sqrt(b*tan(d*x + c) + a))/d

Sympy [F]

\[ \int (a+b \tan (c+d x))^{5/2} \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]

[In]

integrate((a+b*tan(d*x+c))**(5/2),x)

[Out]

Integral((a + b*tan(c + d*x))**(5/2), x)

Maxima [F(-2)]

Exception generated. \[ \int (a+b \tan (c+d x))^{5/2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*tan(d*x+c))^(5/2),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 [F(-1)]

Timed out. \[ \int (a+b \tan (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate((a+b*tan(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 7.76 (sec) , antiderivative size = 2100, normalized size of antiderivative = 15.67 \[ \int (a+b \tan (c+d x))^{5/2} \, dx=\text {Too large to display} \]

[In]

int((a + b*tan(c + d*x))^(5/2),x)

[Out]

(2*b*(a + b*tan(c + d*x))^(3/2))/(3*d) - atan(((((8*(8*a*b^5*d^2 + 8*a^3*b^3*d^2))/d^3 - 64*a*b^2*(a + b*tan(c
+ d*x))^(1/2)*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2))*(-(5*a*b^4 + a
^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(b^8 - 15*a
^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))
^(1/2)*1i - (((8*(8*a*b^5*d^2 + 8*a^3*b^3*d^2))/d^3 + 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*(-(5*a*b^4 + a^4*b*5
i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2))*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10
i - 10*a^3*b^2)/(4*d^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(b^8 - 15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2
)*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2)*1i)/((((8*(8*a*b^5*d^2 + 8*a
^3*b^3*d^2))/d^3 - 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10
*a^3*b^2)/(4*d^2))^(1/2))*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) + (1
6*(a + b*tan(c + d*x))^(1/2)*(b^8 - 15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5
*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) - (16*(6*a^4*b^7 - b^11 + 8*a^6*b^5 + 3*a^8*b^3))/d^3 + (((8*(8
*a*b^5*d^2 + 8*a^3*b^3*d^2))/d^3 + 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i -
a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2))*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*
d^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(b^8 - 15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*(-(5*a*b^4 + a^4*
b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2)))*(-(5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^
3*10i - 10*a^3*b^2)/(4*d^2))^(1/2)*2i - atan(((((8*(8*a*b^5*d^2 + 8*a^3*b^3*d^2))/d^3 - 64*a*b^2*(a + b*tan(c
+ d*x))^(1/2)*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2))*(-(5*a*b^4 - a^
4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(b^8 - 15*a^
2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^
(1/2)*1i - (((8*(8*a*b^5*d^2 + 8*a^3*b^3*d^2))/d^3 + 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*(-(5*a*b^4 - a^4*b*5i
+ a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2))*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i
- 10*a^3*b^2)/(4*d^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(b^8 - 15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)
*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2)*1i)/((((8*(8*a*b^5*d^2 + 8*a^
3*b^3*d^2))/d^3 - 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*
a^3*b^2)/(4*d^2))^(1/2))*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) + (16
*(a + b*tan(c + d*x))^(1/2)*(b^8 - 15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*
1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) - (16*(6*a^4*b^7 - b^11 + 8*a^6*b^5 + 3*a^8*b^3))/d^3 + (((8*(8*
a*b^5*d^2 + 8*a^3*b^3*d^2))/d^3 + 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a
^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2))*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d
^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(b^8 - 15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*(-(5*a*b^4 - a^4*b
*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2)))*(-(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3
*10i - 10*a^3*b^2)/(4*d^2))^(1/2)*2i + (4*a*b*(a + b*tan(c + d*x))^(1/2))/d