3.6.34 \(\int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\) [534]

Optimal. Leaf size=116 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b} d} \]

[Out]

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

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Rubi [A]
time = 0.18, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3655, 3620, 3618, 65, 214, 3715} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]/Sqrt[a + b*Tan[c + d*x]],x]

[Out]

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

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]

Rule 3655

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/
(c^2 + d^2), Int[(a + b*Tan[e + f*x])^m*(c - d*Tan[e + f*x]), x], x] + Dist[d^2/(c^2 + d^2), Int[(a + b*Tan[e
+ f*x])^m*((1 + Tan[e + f*x]^2)/(c + d*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]

Rule 3715

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

Rubi steps

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

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Mathematica [A]
time = 0.16, size = 111, normalized size = 0.96 \begin {gather*} \frac {-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b}}}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]/Sqrt[a + b*Tan[c + d*x]],x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 1.03, size = 20194, normalized size = 174.09

method	result	size
default	\(\text {Expression too large to display}\)	\(20194\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cot(d*x + c)/sqrt(b*tan(d*x + c) + a), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2186 vs. \(2 (90) = 180\).
time = 1.43, size = 4447, normalized size = 38.34 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*sqrt(2)*(a^3 + a*b^2)*d^5*sqrt(-((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*sqrt(b^
2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*(1/((a^2 + b^2)*d^4))^(3/4)*arctan(-((a^4 + 2*a^2*b^2 + b^4)*d^4*sqrt(b^2/((a
^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt(1/((a^2 + b^2)*d^4)) + (a^3 + a*b^2)*d^2*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^
4)) - sqrt(2)*((a^5 + 2*a^3*b^2 + a*b^4)*d^7*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt(1/((a^2 + b^2)*d^4))
+ (a^4 + 2*a^2*b^2 + b^4)*d^5*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(((a^2 + b^2)*d^2*sqrt(1/((a^2 + b
^2)*d^4))*cos(d*x + c) + sqrt(2)*((a^2 + b^2)*d^3*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) + a*d*cos(d*x + c))*s
qrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(-((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 -
b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(1/4) + a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(-((a^3 + a*b^2)*d
^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(3/4) + sqrt(2)*((a^5 + 2*a^3*b^2 + a*b^4
)*d^7*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt(1/((a^2 + b^2)*d^4)) + (a^4 + 2*a^2*b^2 + b^4)*d^5*sqrt(b^2
/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(-((a^3 + a*b^2)*d^2
*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(3/4))/b^2) + 4*sqrt(2)*(a^3 + a*b^2)*d^5*s
qrt(-((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*(1
/((a^2 + b^2)*d^4))^(3/4)*arctan(((a^4 + 2*a^2*b^2 + b^4)*d^4*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt(1/(
(a^2 + b^2)*d^4)) + (a^3 + a*b^2)*d^2*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + sqrt(2)*((a^5 + 2*a^3*b^2 + a*
b^4)*d^7*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt(1/((a^2 + b^2)*d^4)) + (a^4 + 2*a^2*b^2 + b^4)*d^5*sqrt(
b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(((a^2 + b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) - sqrt(2)*((a
^2 + b^2)*d^3*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) + a*d*cos(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c)
)/cos(d*x + c))*sqrt(-((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(1/
4) + a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(-((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 -
b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(3/4) - sqrt(2)*((a^5 + 2*a^3*b^2 + a*b^4)*d^7*sqrt(b^2/((a^4 + 2*a^2*b^2 + b
^4)*d^4))*sqrt(1/((a^2 + b^2)*d^4)) + (a^4 + 2*a^2*b^2 + b^4)*d^5*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqr
t((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(-((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b
^2)/b^2)*(1/((a^2 + b^2)*d^4))^(3/4))/b^2) + sqrt(2)*(a^2*d^3*sqrt(1/((a^2 + b^2)*d^4)) + a*d)*sqrt(-((a^3 + a
*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(1/4)*log(((a^2 + b^2)*d^2*sqrt(1/
((a^2 + b^2)*d^4))*cos(d*x + c) + sqrt(2)*((a^2 + b^2)*d^3*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) + a*d*cos(d*
x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(-((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)
) - a^2 - b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(1/4) + a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c)) - sqrt(2)*(a^
2*d^3*sqrt(1/((a^2 + b^2)*d^4)) + a*d)*sqrt(-((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*(1
/((a^2 + b^2)*d^4))^(1/4)*log(((a^2 + b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) - sqrt(2)*((a^2 + b^2)*d
^3*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) + a*d*cos(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x +
c))*sqrt(-((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(1/4) + a*cos(
d*x + c) + b*sin(d*x + c))/cos(d*x + c)) + 2*sqrt(a)*log(-(8*a*b*cos(d*x + c)*sin(d*x + c) + (8*a^2 - b^2)*cos
(d*x + c)^2 + b^2 - 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c)*sin(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b*sin(
d*x + c))/cos(d*x + c)))/(cos(d*x + c)^2 - 1)))/(a*d), 1/4*(4*sqrt(2)*(a^3 + a*b^2)*d^5*sqrt(-((a^3 + a*b^2)*d
^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*(1/((a^2 + b^2)*d^4))^(
3/4)*arctan(-((a^4 + 2*a^2*b^2 + b^4)*d^4*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt(1/((a^2 + b^2)*d^4)) +
(a^3 + a*b^2)*d^2*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - sqrt(2)*((a^5 + 2*a^3*b^2 + a*b^4)*d^7*sqrt(b^2/((
a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt(1/((a^2 + b^2)*d^4)) + (a^4 + 2*a^2*b^2 + b^4)*d^5*sqrt(b^2/((a^4 + 2*a^2*b^
2 + b^4)*d^4)))*sqrt(((a^2 + b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4))*cos(d*x + c) + sqrt(2)*((a^2 + b^2)*d^3*sqrt(1
/((a^2 + b^2)*d^4))*cos(d*x + c) + a*d*cos(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt
(-((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*(1/((a^2 + b^2)*d^4))^(1/4) + a*cos(d*x + c)
+ b*sin(d*x + c))/cos(d*x + c))*sqrt(-((a^3 + a*b^2)*d^2*sqrt(1/((a^2 + b^2)*d^4)) - a^2 - b^2)/b^2)*(1/((a^2
+ b^2)*d^4))^(3/4) + sqrt(2)*((a^5 + 2*a^3*b^2 + a*b^4)*d^7*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt(1/((a
^2 + b^2)*d^4)) + (a^4 + 2*a^2*b^2 + b^4)*d^5*sqrt(b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt((a*cos(d*x + c) +
b*sin(d*x + c))/cos(d*x + c))*sqrt(-((a^3 + a*b...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot {\left (c + d x \right )}}{\sqrt {a + b \tan {\left (c + d x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [B]
time = 4.64, size = 2028, normalized size = 17.48 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- atan((((((((((1/(a*d^2 - b*d^2*1i))^(1/2)*((32*(16*b^10*d^2 + 12*a^2*b^8*d^2))/d^3 - (16*(1/(a*d^2 - b*d^2*1
i))^(1/2)*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^4))/2 + (576*a*b^8*(a + b*tan(c + d*x))
^(1/2))/d^2)*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 - (96*a*b^8)/d^3)*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 - (96*b^8*(a +
b*tan(c + d*x))^(1/2))/d^4)*(1/(a*d^2 - b*d^2*1i))^(1/2)*1i)/2 - ((((((((1/(a*d^2 - b*d^2*1i))^(1/2)*((32*(16*
b^10*d^2 + 12*a^2*b^8*d^2))/d^3 + (16*(1/(a*d^2 - b*d^2*1i))^(1/2)*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c
+ d*x))^(1/2))/d^4))/2 - (576*a*b^8*(a + b*tan(c + d*x))^(1/2))/d^2)*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 - (96*a*
b^8)/d^3)*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 + (96*b^8*(a + b*tan(c + d*x))^(1/2))/d^4)*(1/(a*d^2 - b*d^2*1i))^(1
/2)*1i)/2)/(((((((((1/(a*d^2 - b*d^2*1i))^(1/2)*((32*(16*b^10*d^2 + 12*a^2*b^8*d^2))/d^3 - (16*(1/(a*d^2 - b*d
^2*1i))^(1/2)*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^4))/2 + (576*a*b^8*(a + b*tan(c + d
*x))^(1/2))/d^2)*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 - (96*a*b^8)/d^3)*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 - (96*b^8*(
a + b*tan(c + d*x))^(1/2))/d^4)*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 + ((((((((1/(a*d^2 - b*d^2*1i))^(1/2)*((32*(16
*b^10*d^2 + 12*a^2*b^8*d^2))/d^3 + (16*(1/(a*d^2 - b*d^2*1i))^(1/2)*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(
c + d*x))^(1/2))/d^4))/2 - (576*a*b^8*(a + b*tan(c + d*x))^(1/2))/d^2)*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 - (96*a
*b^8)/d^3)*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 + (96*b^8*(a + b*tan(c + d*x))^(1/2))/d^4)*(1/(a*d^2 - b*d^2*1i))^(
1/2))/2))*(1/(a*d^2 - b*d^2*1i))^(1/2)*1i - atan(((((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(((a - b*1i)/(4*
a^2*d^2 + 4*b^2*d^2))^(1/2)*(((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((32*(16*b^10*d^2 + 12*a^2*b^8*d^2))/d
^3 - (32*((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))
/d^4) + (576*a*b^8*(a + b*tan(c + d*x))^(1/2))/d^2) - (96*a*b^8)/d^3) - (96*b^8*(a + b*tan(c + d*x))^(1/2))/d^
4)*((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*1i - (((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(((a - b*1i)/(4
*a^2*d^2 + 4*b^2*d^2))^(1/2)*(((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((32*(16*b^10*d^2 + 12*a^2*b^8*d^2))/
d^3 + (32*((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)
)/d^4) - (576*a*b^8*(a + b*tan(c + d*x))^(1/2))/d^2) - (96*a*b^8)/d^3) + (96*b^8*(a + b*tan(c + d*x))^(1/2))/d
^4)*((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*1i)/((((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(((a - b*1i)/(
4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((32*(16*b^10*d^2 + 12*a^2*b^8*d^2))
/d^3 - (32*((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2
))/d^4) + (576*a*b^8*(a + b*tan(c + d*x))^(1/2))/d^2) - (96*a*b^8)/d^3) - (96*b^8*(a + b*tan(c + d*x))^(1/2))/
d^4)*((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2) + (((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(((a - b*1i)/(4*
a^2*d^2 + 4*b^2*d^2))^(1/2)*(((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((32*(16*b^10*d^2 + 12*a^2*b^8*d^2))/d
^3 + (32*((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))
/d^4) - (576*a*b^8*(a + b*tan(c + d*x))^(1/2))/d^2) - (96*a*b^8)/d^3) + (96*b^8*(a + b*tan(c + d*x))^(1/2))/d^
4)*((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)))*((a - b*1i)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*2i - (2*atanh((576*
b^8*(a + b*tan(c + d*x))^(1/2))/(a^(1/2)*(576*b^8 + (1024*b^10)/a^2)) + (1024*b^10*(a + b*tan(c + d*x))^(1/2))
/(a^(5/2)*(576*b^8 + (1024*b^10)/a^2))))/(a^(1/2)*d)

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