3.553 \(\int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx\)
Optimal. Leaf size=245 \[ \frac {5 b \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{7/2} d}-\frac {b \left (3 a^2+5 b^2\right )}{3 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {b \left (a^4+10 a^2 b^2+5 b^4\right )}{a^3 d \left (a^2+b^2\right )^2 \sqrt {a+b \tan (c+d x)}}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{5/2}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{5/2}}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}} \]
[Out]
5*b*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/a^(7/2)/d+I*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^
(5/2)/d-I*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/(a+I*b)^(5/2)/d-b*(a^4+10*a^2*b^2+5*b^4)/a^3/(a^2+b^2)
^2/d/(a+b*tan(d*x+c))^(1/2)-1/3*b*(3*a^2+5*b^2)/a^2/(a^2+b^2)/d/(a+b*tan(d*x+c))^(3/2)-cot(d*x+c)/a/d/(a+b*tan
(d*x+c))^(3/2)
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Rubi [A] time = 0.94, antiderivative size = 245, normalized size of antiderivative = 1.00,
number of steps used = 14, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used
= {3569, 3649, 3653, 3539, 3537, 63, 208, 3634} \[ -\frac {b \left (3 a^2+5 b^2\right )}{3 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {b \left (10 a^2 b^2+a^4+5 b^4\right )}{a^3 d \left (a^2+b^2\right )^2 \sqrt {a+b \tan (c+d x)}}+\frac {5 b \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{7/2} d}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{5/2}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{5/2}}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[Cot[c + d*x]^2/(a + b*Tan[c + d*x])^(5/2),x]
[Out]
(5*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(a^(7/2)*d) + (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b
]])/((a - I*b)^(5/2)*d) - (I*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/((a + I*b)^(5/2)*d) - (b*(3*a^2
+ 5*b^2))/(3*a^2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - Cot[c + d*x]/(a*d*(a + b*Tan[c + d*x])^(3/2)) - (
b*(a^4 + 10*a^2*b^2 + 5*b^4))/(a^3*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])
Rule 63
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 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 3537
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^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 3539
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 3569
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d)), x] + D
ist[1/((m + 1)*(a^2 + b^2)*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c -
a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && I
ntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || IntegerQ[m]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] &&
NeQ[a, 0])))
Rule 3634
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]
Rule 3649
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*(b*B - a*C))*(a + b*T
an[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
- b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Rule 3653
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[((c + d*Tan[e + f*x])^n*(1 + Tan[e + f*x]^2))/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e,
f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !GtQ[n, 0] && !LeQ[n, -
1]
Rubi steps
\begin {align*} \int \frac {\cot ^2(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx &=-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {\int \frac {\cot (c+d x) \left (\frac {5 b}{2}+a \tan (c+d x)+\frac {5}{2} b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^{5/2}} \, dx}{a}\\ &=-\frac {b \left (3 a^2+5 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {2 \int \frac {\cot (c+d x) \left (\frac {15}{4} b \left (a^2+b^2\right )+\frac {3}{2} a^3 \tan (c+d x)+\frac {3}{4} b \left (3 a^2+5 b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^{3/2}} \, dx}{3 a^2 \left (a^2+b^2\right )}\\ &=-\frac {b \left (3 a^2+5 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {b \left (a^4+10 a^2 b^2+5 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {4 \int \frac {\cot (c+d x) \left (\frac {15}{8} b \left (a^2+b^2\right )^2+\frac {3}{4} a^3 \left (a^2-b^2\right ) \tan (c+d x)+\frac {3}{8} b \left (a^4+10 a^2 b^2+5 b^4\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{3 a^3 \left (a^2+b^2\right )^2}\\ &=-\frac {b \left (3 a^2+5 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {b \left (a^4+10 a^2 b^2+5 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {(5 b) \int \frac {\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 a^3}-\frac {4 \int \frac {\frac {3}{4} a^3 \left (a^2-b^2\right )-\frac {3}{2} a^4 b \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{3 a^3 \left (a^2+b^2\right )^2}\\ &=-\frac {b \left (3 a^2+5 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {b \left (a^4+10 a^2 b^2+5 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {\int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (a-i b)^2}-\frac {\int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (a+i b)^2}-\frac {(5 b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 a^3 d}\\ &=-\frac {b \left (3 a^2+5 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {b \left (a^4+10 a^2 b^2+5 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{a^3 d}-\frac {i \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 (a-i b)^2 d}+\frac {i \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 (a+i b)^2 d}\\ &=\frac {5 b \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{7/2} d}-\frac {b \left (3 a^2+5 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {b \left (a^4+10 a^2 b^2+5 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {\operatorname {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 )}{(a-i b)^2 b d}+\frac {\operatorname {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 )}{(a+i b)^2 b d}\\ &=\frac {5 b \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{7/2} d}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}-\frac {b \left (3 a^2+5 b^2\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac {b \left (a^4+10 a^2 b^2+5 b^4\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 4.77, size = 232, normalized size = 0.95 \[ -\frac {-\frac {15 b \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {b \left (3 a^2+5 b^2\right )}{\left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-3 i a^2 \left (\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2}}\right )+\frac {3 b \left (a^4+10 a^2 b^2+5 b^4\right )}{a \left (a^2+b^2\right )^2 \sqrt {a+b \tan (c+d x)}}+\frac {3 a \cot (c+d x)}{(a+b \tan (c+d x))^{3/2}}}{3 a^2 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cot[c + d*x]^2/(a + b*Tan[c + d*x])^(5/2),x]
[Out]
-1/3*((-15*b*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/a^(3/2) - (3*I)*a^2*(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/
Sqrt[a - I*b]]/(a - I*b)^(5/2) - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]]/(a + I*b)^(5/2)) + (b*(3*a^2
+ 5*b^2))/((a^2 + b^2)*(a + b*Tan[c + d*x])^(3/2)) + (3*a*Cot[c + d*x])/(a + b*Tan[c + d*x])^(3/2) + (3*b*(a^4
+ 10*a^2*b^2 + 5*b^4))/(a*(a^2 + b^2)^2*Sqrt[a + b*Tan[c + d*x]]))/(a^2*d)
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^2/(a+b*tan(d*x+c))^(5/2),x, algorithm="fricas")
[Out]
Timed out
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^2/(a+b*tan(d*x+c))^(5/2),x, algorithm="giac")
[Out]
Timed out
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maple [C] time = 10.13, size = 175534, normalized size = 716.47 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(d*x+c)^2/(a+b*tan(d*x+c))^(5/2),x)
[Out]
result too large to display
________________________________________________________________________________________
maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)^2/(a+b*tan(d*x+c))^(5/2),x, algorithm="maxima")
[Out]
Timed out
________________________________________________________________________________________
mupad [B] time = 4.95, size = 14956, normalized size = 61.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cot(c + d*x)^2/(a + b*tan(c + d*x))^(5/2),x)
[Out]
((2*b^3)/(3*a*(a^2 + b^2)) + (2*b^3*(11*a^2 + 5*b^2)*(a + b*tan(c + d*x)))/(3*(a*b^2 + a^3)^2) - (b*(a + b*tan
(c + d*x))^2*(a^4 + 5*b^4 + 10*a^2*b^2))/(a^3*(a^2 + b^2)^2))/(d*(a + b*tan(c + d*x))^(5/2) - a*d*(a + b*tan(c
+ d*x))^(3/2)) + (log(400*a^22*b^39*d^4 - ((((((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4
- 25*a^8*b^2*d^4)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a
^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(90304*a^29*b^37*d^6 - 800*a^21*b^45*d^6 - 10400*a^23*b^43
*d^6 - 54400*a^25*b^41*d^6 - 121600*a^27*b^39*d^6 - (((a + b*tan(c + d*x))^(1/2)*(1600*a^22*b^46*d^7 + 28800*a
^24*b^44*d^7 + 244800*a^26*b^42*d^7 + 1304256*a^28*b^40*d^7 + 4880128*a^30*b^38*d^7 + 13627392*a^32*b^36*d^7 +
29476608*a^34*b^34*d^7 + 50615552*a^36*b^32*d^7 + 70152576*a^38*b^30*d^7 + 79329536*a^40*b^28*d^7 + 73600384*
a^42*b^26*d^7 + 56025216*a^44*b^24*d^7 + 34754304*a^46*b^22*d^7 + 17296384*a^48*b^20*d^7 + 6713088*a^50*b^18*d
^7 + 1934592*a^52*b^16*d^7 + 377408*a^54*b^14*d^7 + 39552*a^56*b^12*d^7 + 64*a^58*b^10*d^7 - 320*a^60*b^8*d^7)
- ((((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) - a^5*d^2 - 5*a*b
^4*d^2 + 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^
4))^(1/2)*(((((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) - a^5*d^2
- 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^
8*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(512*a^27*b^46*d^9 + 9984*a^29*b^44*d^9 + 92160*a^31*b^42*d^9 + 5
35296*a^33*b^40*d^9 + 2193408*a^35*b^38*d^9 + 6736896*a^37*b^36*d^9 + 16084992*a^39*b^34*d^9 + 30551040*a^41*b
^32*d^9 + 46844928*a^43*b^30*d^9 + 58499584*a^45*b^28*d^9 + 59744256*a^47*b^26*d^9 + 49900032*a^49*b^24*d^9 +
33945600*a^51*b^22*d^9 + 18643968*a^53*b^20*d^9 + 8146944*a^55*b^18*d^9 + 2767872*a^57*b^16*d^9 + 705024*a^59*
b^14*d^9 + 126720*a^61*b^12*d^9 + 14336*a^63*b^10*d^9 + 768*a^65*b^8*d^9))/2 + 1280*a^24*b^47*d^8 + 24320*a^26
*b^45*d^8 + 219008*a^28*b^43*d^8 + 1241984*a^30*b^41*d^8 + 4970496*a^32*b^39*d^8 + 14909440*a^34*b^37*d^8 + 34
746880*a^36*b^35*d^8 + 64356864*a^38*b^33*d^8 + 96092672*a^40*b^31*d^8 + 116633088*a^42*b^29*d^8 + 115498240*a
^44*b^27*d^8 + 93267200*a^46*b^25*d^8 + 61128704*a^48*b^23*d^8 + 32212992*a^50*b^21*d^8 + 13439488*a^52*b^19*d
^8 + 4334080*a^54*b^17*d^8 + 1040640*a^56*b^15*d^8 + 174848*a^58*b^13*d^8 + 18304*a^60*b^11*d^8 + 896*a^62*b^9
*d^8))/2)*(((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) - a^5*d^2 -
5*a*b^4*d^2 + 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*
b^2*d^4))^(1/2))/2 + 1465856*a^31*b^35*d^6 + 5014464*a^33*b^33*d^6 + 10323456*a^35*b^31*d^6 + 14661504*a^37*b^
29*d^6 + 14908608*a^39*b^27*d^6 + 10808512*a^41*b^25*d^6 + 5328128*a^43*b^23*d^6 + 1531712*a^45*b^21*d^6 + 878
08*a^47*b^19*d^6 - 85696*a^49*b^17*d^6 - 6144*a^51*b^15*d^6 + 15264*a^53*b^13*d^6 + 5856*a^55*b^11*d^6 + 704*a
^57*b^9*d^6))/2 + (a + b*tan(c + d*x))^(1/2)*(67232*a^27*b^36*d^5 - 3200*a^23*b^40*d^5 - 3200*a^25*b^38*d^5 -
400*a^21*b^42*d^5 + 437248*a^29*b^34*d^5 + 1458912*a^31*b^32*d^5 + 3214848*a^33*b^30*d^5 + 5065632*a^35*b^28*d
^5 + 5898464*a^37*b^26*d^5 + 5129696*a^39*b^24*d^5 + 3313024*a^41*b^22*d^5 + 1552096*a^43*b^20*d^5 + 500864*a^
45*b^18*d^5 + 99232*a^47*b^16*d^5 + 8448*a^49*b^14*d^5 - 288*a^51*b^12*d^5 + 48*a^53*b^10*d^5 + 32*a^55*b^8*d^
5))*(((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) - a^5*d^2 - 5*a*b
^4*d^2 + 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^
4))^(1/2))/2 + 5520*a^24*b^37*d^4 + 35280*a^26*b^35*d^4 + 138320*a^28*b^33*d^4 + 371280*a^30*b^31*d^4 + 720720
*a^32*b^29*d^4 + 1041040*a^34*b^27*d^4 + 1132560*a^36*b^25*d^4 + 926640*a^38*b^23*d^4 + 560560*a^40*b^21*d^4 +
240240*a^42*b^19*d^4 + 65520*a^44*b^17*d^4 + 7280*a^46*b^15*d^4 - 1680*a^48*b^13*d^4 - 720*a^50*b^11*d^4 - 80
*a^52*b^9*d^4)*(((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) - a^5*
d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5
*a^8*b^2*d^4))^(1/2))/2 + (log(400*a^22*b^39*d^4 - ((((-((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^
6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d
^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2)*(90304*a^29*b^37*d^6 - 800*a^21*b^45*d^6 - 10400*
a^23*b^43*d^6 - 54400*a^25*b^41*d^6 - 121600*a^27*b^39*d^6 - (((a + b*tan(c + d*x))^(1/2)*(1600*a^22*b^46*d^7
+ 28800*a^24*b^44*d^7 + 244800*a^26*b^42*d^7 + 1304256*a^28*b^40*d^7 + 4880128*a^30*b^38*d^7 + 13627392*a^32*b
^36*d^7 + 29476608*a^34*b^34*d^7 + 50615552*a^36*b^32*d^7 + 70152576*a^38*b^30*d^7 + 79329536*a^40*b^28*d^7 +
73600384*a^42*b^26*d^7 + 56025216*a^44*b^24*d^7 + 34754304*a^46*b^22*d^7 + 17296384*a^48*b^20*d^7 + 6713088*a^
50*b^18*d^7 + 1934592*a^52*b^16*d^7 + 377408*a^54*b^14*d^7 + 39552*a^56*b^12*d^7 + 64*a^58*b^10*d^7 - 320*a^60
*b^8*d^7) - ((-((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) + a^5*d
^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 + 5*
a^8*b^2*d^4))^(1/2)*(((-((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2
) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4
*d^4 + 5*a^8*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(512*a^27*b^46*d^9 + 9984*a^29*b^44*d^9 + 92160*a^31*b
^42*d^9 + 535296*a^33*b^40*d^9 + 2193408*a^35*b^38*d^9 + 6736896*a^37*b^36*d^9 + 16084992*a^39*b^34*d^9 + 3055
1040*a^41*b^32*d^9 + 46844928*a^43*b^30*d^9 + 58499584*a^45*b^28*d^9 + 59744256*a^47*b^26*d^9 + 49900032*a^49*
b^24*d^9 + 33945600*a^51*b^22*d^9 + 18643968*a^53*b^20*d^9 + 8146944*a^55*b^18*d^9 + 2767872*a^57*b^16*d^9 + 7
05024*a^59*b^14*d^9 + 126720*a^61*b^12*d^9 + 14336*a^63*b^10*d^9 + 768*a^65*b^8*d^9))/2 + 1280*a^24*b^47*d^8 +
24320*a^26*b^45*d^8 + 219008*a^28*b^43*d^8 + 1241984*a^30*b^41*d^8 + 4970496*a^32*b^39*d^8 + 14909440*a^34*b^
37*d^8 + 34746880*a^36*b^35*d^8 + 64356864*a^38*b^33*d^8 + 96092672*a^40*b^31*d^8 + 116633088*a^42*b^29*d^8 +
115498240*a^44*b^27*d^8 + 93267200*a^46*b^25*d^8 + 61128704*a^48*b^23*d^8 + 32212992*a^50*b^21*d^8 + 13439488*
a^52*b^19*d^8 + 4334080*a^54*b^17*d^8 + 1040640*a^56*b^15*d^8 + 174848*a^58*b^13*d^8 + 18304*a^60*b^11*d^8 + 8
96*a^62*b^9*d^8))/2)*(-((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2)
+ a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*
d^4 + 5*a^8*b^2*d^4))^(1/2))/2 + 1465856*a^31*b^35*d^6 + 5014464*a^33*b^33*d^6 + 10323456*a^35*b^31*d^6 + 1466
1504*a^37*b^29*d^6 + 14908608*a^39*b^27*d^6 + 10808512*a^41*b^25*d^6 + 5328128*a^43*b^23*d^6 + 1531712*a^45*b^
21*d^6 + 87808*a^47*b^19*d^6 - 85696*a^49*b^17*d^6 - 6144*a^51*b^15*d^6 + 15264*a^53*b^13*d^6 + 5856*a^55*b^11
*d^6 + 704*a^57*b^9*d^6))/2 + (a + b*tan(c + d*x))^(1/2)*(67232*a^27*b^36*d^5 - 3200*a^23*b^40*d^5 - 3200*a^25
*b^38*d^5 - 400*a^21*b^42*d^5 + 437248*a^29*b^34*d^5 + 1458912*a^31*b^32*d^5 + 3214848*a^33*b^30*d^5 + 5065632
*a^35*b^28*d^5 + 5898464*a^37*b^26*d^5 + 5129696*a^39*b^24*d^5 + 3313024*a^41*b^22*d^5 + 1552096*a^43*b^20*d^5
+ 500864*a^45*b^18*d^5 + 99232*a^47*b^16*d^5 + 8448*a^49*b^14*d^5 - 288*a^51*b^12*d^5 + 48*a^53*b^10*d^5 + 32
*a^55*b^8*d^5))*(-((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) + a^
5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a^6*b^4*d^4 +
5*a^8*b^2*d^4))^(1/2))/2 + 5520*a^24*b^37*d^4 + 35280*a^26*b^35*d^4 + 138320*a^28*b^33*d^4 + 371280*a^30*b^31
*d^4 + 720720*a^32*b^29*d^4 + 1041040*a^34*b^27*d^4 + 1132560*a^36*b^25*d^4 + 926640*a^38*b^23*d^4 + 560560*a^
40*b^21*d^4 + 240240*a^42*b^19*d^4 + 65520*a^44*b^17*d^4 + 7280*a^46*b^15*d^4 - 1680*a^48*b^13*d^4 - 720*a^50*
b^11*d^4 - 80*a^52*b^9*d^4)*(-((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4
)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/(a^10*d^4 + b^10*d^4 + 5*a^2*b^8*d^4 + 10*a^4*b^6*d^4 + 10*a
^6*b^4*d^4 + 5*a^8*b^2*d^4))^(1/2))/2 - log(400*a^22*b^39*d^4 - ((((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^
4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/(4*a^10*d^4 + 4*b^10*d^4
+ 20*a^2*b^8*d^4 + 40*a^4*b^6*d^4 + 40*a^6*b^4*d^4 + 20*a^8*b^2*d^4))^(1/2)*(((a + b*tan(c + d*x))^(1/2)*(160
0*a^22*b^46*d^7 + 28800*a^24*b^44*d^7 + 244800*a^26*b^42*d^7 + 1304256*a^28*b^40*d^7 + 4880128*a^30*b^38*d^7 +
13627392*a^32*b^36*d^7 + 29476608*a^34*b^34*d^7 + 50615552*a^36*b^32*d^7 + 70152576*a^38*b^30*d^7 + 79329536*
a^40*b^28*d^7 + 73600384*a^42*b^26*d^7 + 56025216*a^44*b^24*d^7 + 34754304*a^46*b^22*d^7 + 17296384*a^48*b^20*
d^7 + 6713088*a^50*b^18*d^7 + 1934592*a^52*b^16*d^7 + 377408*a^54*b^14*d^7 + 39552*a^56*b^12*d^7 + 64*a^58*b^1
0*d^7 - 320*a^60*b^8*d^7) + (((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)
^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/(4*a^10*d^4 + 4*b^10*d^4 + 20*a^2*b^8*d^4 + 40*a^4*b^6*d^4 +
40*a^6*b^4*d^4 + 20*a^8*b^2*d^4))^(1/2)*(1280*a^24*b^47*d^8 - (((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 +
100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/(4*a^10*d^4 + 4*b^10*d^4 +
20*a^2*b^8*d^4 + 40*a^4*b^6*d^4 + 40*a^6*b^4*d^4 + 20*a^8*b^2*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(512*a^27
*b^46*d^9 + 9984*a^29*b^44*d^9 + 92160*a^31*b^42*d^9 + 535296*a^33*b^40*d^9 + 2193408*a^35*b^38*d^9 + 6736896*
a^37*b^36*d^9 + 16084992*a^39*b^34*d^9 + 30551040*a^41*b^32*d^9 + 46844928*a^43*b^30*d^9 + 58499584*a^45*b^28*
d^9 + 59744256*a^47*b^26*d^9 + 49900032*a^49*b^24*d^9 + 33945600*a^51*b^22*d^9 + 18643968*a^53*b^20*d^9 + 8146
944*a^55*b^18*d^9 + 2767872*a^57*b^16*d^9 + 705024*a^59*b^14*d^9 + 126720*a^61*b^12*d^9 + 14336*a^63*b^10*d^9
+ 768*a^65*b^8*d^9) + 24320*a^26*b^45*d^8 + 219008*a^28*b^43*d^8 + 1241984*a^30*b^41*d^8 + 4970496*a^32*b^39*d
^8 + 14909440*a^34*b^37*d^8 + 34746880*a^36*b^35*d^8 + 64356864*a^38*b^33*d^8 + 96092672*a^40*b^31*d^8 + 11663
3088*a^42*b^29*d^8 + 115498240*a^44*b^27*d^8 + 93267200*a^46*b^25*d^8 + 61128704*a^48*b^23*d^8 + 32212992*a^50
*b^21*d^8 + 13439488*a^52*b^19*d^8 + 4334080*a^54*b^17*d^8 + 1040640*a^56*b^15*d^8 + 174848*a^58*b^13*d^8 + 18
304*a^60*b^11*d^8 + 896*a^62*b^9*d^8))*(((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a
^8*b^2*d^4)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/(4*a^10*d^4 + 4*b^10*d^4 + 20*a^2*b^8*d^4 + 40*a^4
*b^6*d^4 + 40*a^6*b^4*d^4 + 20*a^8*b^2*d^4))^(1/2) - 800*a^21*b^45*d^6 - 10400*a^23*b^43*d^6 - 54400*a^25*b^41
*d^6 - 121600*a^27*b^39*d^6 + 90304*a^29*b^37*d^6 + 1465856*a^31*b^35*d^6 + 5014464*a^33*b^33*d^6 + 10323456*a
^35*b^31*d^6 + 14661504*a^37*b^29*d^6 + 14908608*a^39*b^27*d^6 + 10808512*a^41*b^25*d^6 + 5328128*a^43*b^23*d^
6 + 1531712*a^45*b^21*d^6 + 87808*a^47*b^19*d^6 - 85696*a^49*b^17*d^6 - 6144*a^51*b^15*d^6 + 15264*a^53*b^13*d
^6 + 5856*a^55*b^11*d^6 + 704*a^57*b^9*d^6) - (a + b*tan(c + d*x))^(1/2)*(67232*a^27*b^36*d^5 - 3200*a^23*b^40
*d^5 - 3200*a^25*b^38*d^5 - 400*a^21*b^42*d^5 + 437248*a^29*b^34*d^5 + 1458912*a^31*b^32*d^5 + 3214848*a^33*b^
30*d^5 + 5065632*a^35*b^28*d^5 + 5898464*a^37*b^26*d^5 + 5129696*a^39*b^24*d^5 + 3313024*a^41*b^22*d^5 + 15520
96*a^43*b^20*d^5 + 500864*a^45*b^18*d^5 + 99232*a^47*b^16*d^5 + 8448*a^49*b^14*d^5 - 288*a^51*b^12*d^5 + 48*a^
53*b^10*d^5 + 32*a^55*b^8*d^5))*(((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*
d^4)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/(4*a^10*d^4 + 4*b^10*d^4 + 20*a^2*b^8*d^4 + 40*a^4*b^6*d^
4 + 40*a^6*b^4*d^4 + 20*a^8*b^2*d^4))^(1/2) + 5520*a^24*b^37*d^4 + 35280*a^26*b^35*d^4 + 138320*a^28*b^33*d^4
+ 371280*a^30*b^31*d^4 + 720720*a^32*b^29*d^4 + 1041040*a^34*b^27*d^4 + 1132560*a^36*b^25*d^4 + 926640*a^38*b^
23*d^4 + 560560*a^40*b^21*d^4 + 240240*a^42*b^19*d^4 + 65520*a^44*b^17*d^4 + 7280*a^46*b^15*d^4 - 1680*a^48*b^
13*d^4 - 720*a^50*b^11*d^4 - 80*a^52*b^9*d^4)*(((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4
- 25*a^8*b^2*d^4)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/(4*a^10*d^4 + 4*b^10*d^4 + 20*a^2*b^8*d^4 +
40*a^4*b^6*d^4 + 40*a^6*b^4*d^4 + 20*a^8*b^2*d^4))^(1/2) - log(400*a^22*b^39*d^4 - ((-((20*a^2*b^8*d^4 - b^10
*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/(4*
a^10*d^4 + 4*b^10*d^4 + 20*a^2*b^8*d^4 + 40*a^4*b^6*d^4 + 40*a^6*b^4*d^4 + 20*a^8*b^2*d^4))^(1/2)*(((a + b*tan
(c + d*x))^(1/2)*(1600*a^22*b^46*d^7 + 28800*a^24*b^44*d^7 + 244800*a^26*b^42*d^7 + 1304256*a^28*b^40*d^7 + 48
80128*a^30*b^38*d^7 + 13627392*a^32*b^36*d^7 + 29476608*a^34*b^34*d^7 + 50615552*a^36*b^32*d^7 + 70152576*a^38
*b^30*d^7 + 79329536*a^40*b^28*d^7 + 73600384*a^42*b^26*d^7 + 56025216*a^44*b^24*d^7 + 34754304*a^46*b^22*d^7
+ 17296384*a^48*b^20*d^7 + 6713088*a^50*b^18*d^7 + 1934592*a^52*b^16*d^7 + 377408*a^54*b^14*d^7 + 39552*a^56*b
^12*d^7 + 64*a^58*b^10*d^7 - 320*a^60*b^8*d^7) + (-((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4
*d^4 - 25*a^8*b^2*d^4)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/(4*a^10*d^4 + 4*b^10*d^4 + 20*a^2*b^8*d
^4 + 40*a^4*b^6*d^4 + 40*a^6*b^4*d^4 + 20*a^8*b^2*d^4))^(1/2)*(1280*a^24*b^47*d^8 - (-((20*a^2*b^8*d^4 - b^10*
d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/(4*a
^10*d^4 + 4*b^10*d^4 + 20*a^2*b^8*d^4 + 40*a^4*b^6*d^4 + 40*a^6*b^4*d^4 + 20*a^8*b^2*d^4))^(1/2)*(a + b*tan(c
+ d*x))^(1/2)*(512*a^27*b^46*d^9 + 9984*a^29*b^44*d^9 + 92160*a^31*b^42*d^9 + 535296*a^33*b^40*d^9 + 2193408*a
^35*b^38*d^9 + 6736896*a^37*b^36*d^9 + 16084992*a^39*b^34*d^9 + 30551040*a^41*b^32*d^9 + 46844928*a^43*b^30*d^
9 + 58499584*a^45*b^28*d^9 + 59744256*a^47*b^26*d^9 + 49900032*a^49*b^24*d^9 + 33945600*a^51*b^22*d^9 + 186439
68*a^53*b^20*d^9 + 8146944*a^55*b^18*d^9 + 2767872*a^57*b^16*d^9 + 705024*a^59*b^14*d^9 + 126720*a^61*b^12*d^9
+ 14336*a^63*b^10*d^9 + 768*a^65*b^8*d^9) + 24320*a^26*b^45*d^8 + 219008*a^28*b^43*d^8 + 1241984*a^30*b^41*d^
8 + 4970496*a^32*b^39*d^8 + 14909440*a^34*b^37*d^8 + 34746880*a^36*b^35*d^8 + 64356864*a^38*b^33*d^8 + 9609267
2*a^40*b^31*d^8 + 116633088*a^42*b^29*d^8 + 115498240*a^44*b^27*d^8 + 93267200*a^46*b^25*d^8 + 61128704*a^48*b
^23*d^8 + 32212992*a^50*b^21*d^8 + 13439488*a^52*b^19*d^8 + 4334080*a^54*b^17*d^8 + 1040640*a^56*b^15*d^8 + 17
4848*a^58*b^13*d^8 + 18304*a^60*b^11*d^8 + 896*a^62*b^9*d^8))*(-((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4
+ 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/(4*a^10*d^4 + 4*b^10*d^4 +
20*a^2*b^8*d^4 + 40*a^4*b^6*d^4 + 40*a^6*b^4*d^4 + 20*a^8*b^2*d^4))^(1/2) - 800*a^21*b^45*d^6 - 10400*a^23*b^
43*d^6 - 54400*a^25*b^41*d^6 - 121600*a^27*b^39*d^6 + 90304*a^29*b^37*d^6 + 1465856*a^31*b^35*d^6 + 5014464*a^
33*b^33*d^6 + 10323456*a^35*b^31*d^6 + 14661504*a^37*b^29*d^6 + 14908608*a^39*b^27*d^6 + 10808512*a^41*b^25*d^
6 + 5328128*a^43*b^23*d^6 + 1531712*a^45*b^21*d^6 + 87808*a^47*b^19*d^6 - 85696*a^49*b^17*d^6 - 6144*a^51*b^15
*d^6 + 15264*a^53*b^13*d^6 + 5856*a^55*b^11*d^6 + 704*a^57*b^9*d^6) - (a + b*tan(c + d*x))^(1/2)*(67232*a^27*b
^36*d^5 - 3200*a^23*b^40*d^5 - 3200*a^25*b^38*d^5 - 400*a^21*b^42*d^5 + 437248*a^29*b^34*d^5 + 1458912*a^31*b^
32*d^5 + 3214848*a^33*b^30*d^5 + 5065632*a^35*b^28*d^5 + 5898464*a^37*b^26*d^5 + 5129696*a^39*b^24*d^5 + 33130
24*a^41*b^22*d^5 + 1552096*a^43*b^20*d^5 + 500864*a^45*b^18*d^5 + 99232*a^47*b^16*d^5 + 8448*a^49*b^14*d^5 - 2
88*a^51*b^12*d^5 + 48*a^53*b^10*d^5 + 32*a^55*b^8*d^5))*(-((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*
a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/(4*a^10*d^4 + 4*b^10*d^4 + 20*a^
2*b^8*d^4 + 40*a^4*b^6*d^4 + 40*a^6*b^4*d^4 + 20*a^8*b^2*d^4))^(1/2) + 5520*a^24*b^37*d^4 + 35280*a^26*b^35*d^
4 + 138320*a^28*b^33*d^4 + 371280*a^30*b^31*d^4 + 720720*a^32*b^29*d^4 + 1041040*a^34*b^27*d^4 + 1132560*a^36*
b^25*d^4 + 926640*a^38*b^23*d^4 + 560560*a^40*b^21*d^4 + 240240*a^42*b^19*d^4 + 65520*a^44*b^17*d^4 + 7280*a^4
6*b^15*d^4 - 1680*a^48*b^13*d^4 - 720*a^50*b^11*d^4 - 80*a^52*b^9*d^4)*(-((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4
*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/(4*a^10*d^4 + 4*b
^10*d^4 + 20*a^2*b^8*d^4 + 40*a^4*b^6*d^4 + 40*a^6*b^4*d^4 + 20*a^8*b^2*d^4))^(1/2) - (b*atan((a^13*b^51*d^4*(
a + b*tan(c + d*x))^(1/2)*100000i)/((a^7)^(1/2)*(100000*a^10*b^51*d^4 + 1900000*a^12*b^49*d^4 + 17100000*a^14*
b^47*d^4 + 96900000*a^16*b^45*d^4 + 387640000*a^18*b^43*d^4 + 1163280000*a^20*b^41*d^4 + 2715728000*a^22*b^39*
d^4 + 5046192160*a^24*b^37*d^4 + 7569850240*a^26*b^35*d^4 + 9240726560*a^28*b^33*d^4 + 9205826240*a^30*b^31*d^
4 + 7471416160*a^32*b^29*d^4 + 4908704320*a^34*b^27*d^4 + 2580976480*a^36*b^25*d^4 + 1067253120*a^38*b^23*d^4
+ 338576480*a^40*b^21*d^4 + 79748320*a^42*b^19*d^4 + 13452160*a^44*b^17*d^4 + 1606240*a^46*b^15*d^4 + 146560*a
^48*b^13*d^4 + 10240*a^50*b^11*d^4 + 160*a^52*b^9*d^4)) + (a^15*b^49*d^4*(a + b*tan(c + d*x))^(1/2)*1900000i)/
((a^7)^(1/2)*(100000*a^10*b^51*d^4 + 1900000*a^12*b^49*d^4 + 17100000*a^14*b^47*d^4 + 96900000*a^16*b^45*d^4 +
387640000*a^18*b^43*d^4 + 1163280000*a^20*b^41*d^4 + 2715728000*a^22*b^39*d^4 + 5046192160*a^24*b^37*d^4 + 75
69850240*a^26*b^35*d^4 + 9240726560*a^28*b^33*d^4 + 9205826240*a^30*b^31*d^4 + 7471416160*a^32*b^29*d^4 + 4908
704320*a^34*b^27*d^4 + 2580976480*a^36*b^25*d^4 + 1067253120*a^38*b^23*d^4 + 338576480*a^40*b^21*d^4 + 7974832
0*a^42*b^19*d^4 + 13452160*a^44*b^17*d^4 + 1606240*a^46*b^15*d^4 + 146560*a^48*b^13*d^4 + 10240*a^50*b^11*d^4
+ 160*a^52*b^9*d^4)) + (a^17*b^47*d^4*(a + b*tan(c + d*x))^(1/2)*17100000i)/((a^7)^(1/2)*(100000*a^10*b^51*d^4
+ 1900000*a^12*b^49*d^4 + 17100000*a^14*b^47*d^4 + 96900000*a^16*b^45*d^4 + 387640000*a^18*b^43*d^4 + 1163280
000*a^20*b^41*d^4 + 2715728000*a^22*b^39*d^4 + 5046192160*a^24*b^37*d^4 + 7569850240*a^26*b^35*d^4 + 924072656
0*a^28*b^33*d^4 + 9205826240*a^30*b^31*d^4 + 7471416160*a^32*b^29*d^4 + 4908704320*a^34*b^27*d^4 + 2580976480*
a^36*b^25*d^4 + 1067253120*a^38*b^23*d^4 + 338576480*a^40*b^21*d^4 + 79748320*a^42*b^19*d^4 + 13452160*a^44*b^
17*d^4 + 1606240*a^46*b^15*d^4 + 146560*a^48*b^13*d^4 + 10240*a^50*b^11*d^4 + 160*a^52*b^9*d^4)) + (a^19*b^45*
d^4*(a + b*tan(c + d*x))^(1/2)*96900000i)/((a^7)^(1/2)*(100000*a^10*b^51*d^4 + 1900000*a^12*b^49*d^4 + 1710000
0*a^14*b^47*d^4 + 96900000*a^16*b^45*d^4 + 387640000*a^18*b^43*d^4 + 1163280000*a^20*b^41*d^4 + 2715728000*a^2
2*b^39*d^4 + 5046192160*a^24*b^37*d^4 + 7569850240*a^26*b^35*d^4 + 9240726560*a^28*b^33*d^4 + 9205826240*a^30*
b^31*d^4 + 7471416160*a^32*b^29*d^4 + 4908704320*a^34*b^27*d^4 + 2580976480*a^36*b^25*d^4 + 1067253120*a^38*b^
23*d^4 + 338576480*a^40*b^21*d^4 + 79748320*a^42*b^19*d^4 + 13452160*a^44*b^17*d^4 + 1606240*a^46*b^15*d^4 + 1
46560*a^48*b^13*d^4 + 10240*a^50*b^11*d^4 + 160*a^52*b^9*d^4)) + (a^21*b^43*d^4*(a + b*tan(c + d*x))^(1/2)*387
640000i)/((a^7)^(1/2)*(100000*a^10*b^51*d^4 + 1900000*a^12*b^49*d^4 + 17100000*a^14*b^47*d^4 + 96900000*a^16*b
^45*d^4 + 387640000*a^18*b^43*d^4 + 1163280000*a^20*b^41*d^4 + 2715728000*a^22*b^39*d^4 + 5046192160*a^24*b^37
*d^4 + 7569850240*a^26*b^35*d^4 + 9240726560*a^28*b^33*d^4 + 9205826240*a^30*b^31*d^4 + 7471416160*a^32*b^29*d
^4 + 4908704320*a^34*b^27*d^4 + 2580976480*a^36*b^25*d^4 + 1067253120*a^38*b^23*d^4 + 338576480*a^40*b^21*d^4
+ 79748320*a^42*b^19*d^4 + 13452160*a^44*b^17*d^4 + 1606240*a^46*b^15*d^4 + 146560*a^48*b^13*d^4 + 10240*a^50*
b^11*d^4 + 160*a^52*b^9*d^4)) + (a^23*b^41*d^4*(a + b*tan(c + d*x))^(1/2)*1163280000i)/((a^7)^(1/2)*(100000*a^
10*b^51*d^4 + 1900000*a^12*b^49*d^4 + 17100000*a^14*b^47*d^4 + 96900000*a^16*b^45*d^4 + 387640000*a^18*b^43*d^
4 + 1163280000*a^20*b^41*d^4 + 2715728000*a^22*b^39*d^4 + 5046192160*a^24*b^37*d^4 + 7569850240*a^26*b^35*d^4
+ 9240726560*a^28*b^33*d^4 + 9205826240*a^30*b^31*d^4 + 7471416160*a^32*b^29*d^4 + 4908704320*a^34*b^27*d^4 +
2580976480*a^36*b^25*d^4 + 1067253120*a^38*b^23*d^4 + 338576480*a^40*b^21*d^4 + 79748320*a^42*b^19*d^4 + 13452
160*a^44*b^17*d^4 + 1606240*a^46*b^15*d^4 + 146560*a^48*b^13*d^4 + 10240*a^50*b^11*d^4 + 160*a^52*b^9*d^4)) +
(a^25*b^39*d^4*(a + b*tan(c + d*x))^(1/2)*2715728000i)/((a^7)^(1/2)*(100000*a^10*b^51*d^4 + 1900000*a^12*b^49*
d^4 + 17100000*a^14*b^47*d^4 + 96900000*a^16*b^45*d^4 + 387640000*a^18*b^43*d^4 + 1163280000*a^20*b^41*d^4 + 2
715728000*a^22*b^39*d^4 + 5046192160*a^24*b^37*d^4 + 7569850240*a^26*b^35*d^4 + 9240726560*a^28*b^33*d^4 + 920
5826240*a^30*b^31*d^4 + 7471416160*a^32*b^29*d^4 + 4908704320*a^34*b^27*d^4 + 2580976480*a^36*b^25*d^4 + 10672
53120*a^38*b^23*d^4 + 338576480*a^40*b^21*d^4 + 79748320*a^42*b^19*d^4 + 13452160*a^44*b^17*d^4 + 1606240*a^46
*b^15*d^4 + 146560*a^48*b^13*d^4 + 10240*a^50*b^11*d^4 + 160*a^52*b^9*d^4)) + (a^27*b^37*d^4*(a + b*tan(c + d*
x))^(1/2)*5046192160i)/((a^7)^(1/2)*(100000*a^10*b^51*d^4 + 1900000*a^12*b^49*d^4 + 17100000*a^14*b^47*d^4 + 9
6900000*a^16*b^45*d^4 + 387640000*a^18*b^43*d^4 + 1163280000*a^20*b^41*d^4 + 2715728000*a^22*b^39*d^4 + 504619
2160*a^24*b^37*d^4 + 7569850240*a^26*b^35*d^4 + 9240726560*a^28*b^33*d^4 + 9205826240*a^30*b^31*d^4 + 74714161
60*a^32*b^29*d^4 + 4908704320*a^34*b^27*d^4 + 2580976480*a^36*b^25*d^4 + 1067253120*a^38*b^23*d^4 + 338576480*
a^40*b^21*d^4 + 79748320*a^42*b^19*d^4 + 13452160*a^44*b^17*d^4 + 1606240*a^46*b^15*d^4 + 146560*a^48*b^13*d^4
+ 10240*a^50*b^11*d^4 + 160*a^52*b^9*d^4)) + (a^29*b^35*d^4*(a + b*tan(c + d*x))^(1/2)*7569850240i)/((a^7)^(1
/2)*(100000*a^10*b^51*d^4 + 1900000*a^12*b^49*d^4 + 17100000*a^14*b^47*d^4 + 96900000*a^16*b^45*d^4 + 38764000
0*a^18*b^43*d^4 + 1163280000*a^20*b^41*d^4 + 2715728000*a^22*b^39*d^4 + 5046192160*a^24*b^37*d^4 + 7569850240*
a^26*b^35*d^4 + 9240726560*a^28*b^33*d^4 + 9205826240*a^30*b^31*d^4 + 7471416160*a^32*b^29*d^4 + 4908704320*a^
34*b^27*d^4 + 2580976480*a^36*b^25*d^4 + 1067253120*a^38*b^23*d^4 + 338576480*a^40*b^21*d^4 + 79748320*a^42*b^
19*d^4 + 13452160*a^44*b^17*d^4 + 1606240*a^46*b^15*d^4 + 146560*a^48*b^13*d^4 + 10240*a^50*b^11*d^4 + 160*a^5
2*b^9*d^4)) + (a^31*b^33*d^4*(a + b*tan(c + d*x))^(1/2)*9240726560i)/((a^7)^(1/2)*(100000*a^10*b^51*d^4 + 1900
000*a^12*b^49*d^4 + 17100000*a^14*b^47*d^4 + 96900000*a^16*b^45*d^4 + 387640000*a^18*b^43*d^4 + 1163280000*a^2
0*b^41*d^4 + 2715728000*a^22*b^39*d^4 + 5046192160*a^24*b^37*d^4 + 7569850240*a^26*b^35*d^4 + 9240726560*a^28*
b^33*d^4 + 9205826240*a^30*b^31*d^4 + 7471416160*a^32*b^29*d^4 + 4908704320*a^34*b^27*d^4 + 2580976480*a^36*b^
25*d^4 + 1067253120*a^38*b^23*d^4 + 338576480*a^40*b^21*d^4 + 79748320*a^42*b^19*d^4 + 13452160*a^44*b^17*d^4
+ 1606240*a^46*b^15*d^4 + 146560*a^48*b^13*d^4 + 10240*a^50*b^11*d^4 + 160*a^52*b^9*d^4)) + (a^33*b^31*d^4*(a
+ b*tan(c + d*x))^(1/2)*9205826240i)/((a^7)^(1/2)*(100000*a^10*b^51*d^4 + 1900000*a^12*b^49*d^4 + 17100000*a^1
4*b^47*d^4 + 96900000*a^16*b^45*d^4 + 387640000*a^18*b^43*d^4 + 1163280000*a^20*b^41*d^4 + 2715728000*a^22*b^3
9*d^4 + 5046192160*a^24*b^37*d^4 + 7569850240*a^26*b^35*d^4 + 9240726560*a^28*b^33*d^4 + 9205826240*a^30*b^31*
d^4 + 7471416160*a^32*b^29*d^4 + 4908704320*a^34*b^27*d^4 + 2580976480*a^36*b^25*d^4 + 1067253120*a^38*b^23*d^
4 + 338576480*a^40*b^21*d^4 + 79748320*a^42*b^19*d^4 + 13452160*a^44*b^17*d^4 + 1606240*a^46*b^15*d^4 + 146560
*a^48*b^13*d^4 + 10240*a^50*b^11*d^4 + 160*a^52*b^9*d^4)) + (a^35*b^29*d^4*(a + b*tan(c + d*x))^(1/2)*74714161
60i)/((a^7)^(1/2)*(100000*a^10*b^51*d^4 + 1900000*a^12*b^49*d^4 + 17100000*a^14*b^47*d^4 + 96900000*a^16*b^45*
d^4 + 387640000*a^18*b^43*d^4 + 1163280000*a^20*b^41*d^4 + 2715728000*a^22*b^39*d^4 + 5046192160*a^24*b^37*d^4
+ 7569850240*a^26*b^35*d^4 + 9240726560*a^28*b^33*d^4 + 9205826240*a^30*b^31*d^4 + 7471416160*a^32*b^29*d^4 +
4908704320*a^34*b^27*d^4 + 2580976480*a^36*b^25*d^4 + 1067253120*a^38*b^23*d^4 + 338576480*a^40*b^21*d^4 + 79
748320*a^42*b^19*d^4 + 13452160*a^44*b^17*d^4 + 1606240*a^46*b^15*d^4 + 146560*a^48*b^13*d^4 + 10240*a^50*b^11
*d^4 + 160*a^52*b^9*d^4)) + (a^37*b^27*d^4*(a + b*tan(c + d*x))^(1/2)*4908704320i)/((a^7)^(1/2)*(100000*a^10*b
^51*d^4 + 1900000*a^12*b^49*d^4 + 17100000*a^14*b^47*d^4 + 96900000*a^16*b^45*d^4 + 387640000*a^18*b^43*d^4 +
1163280000*a^20*b^41*d^4 + 2715728000*a^22*b^39*d^4 + 5046192160*a^24*b^37*d^4 + 7569850240*a^26*b^35*d^4 + 92
40726560*a^28*b^33*d^4 + 9205826240*a^30*b^31*d^4 + 7471416160*a^32*b^29*d^4 + 4908704320*a^34*b^27*d^4 + 2580
976480*a^36*b^25*d^4 + 1067253120*a^38*b^23*d^4 + 338576480*a^40*b^21*d^4 + 79748320*a^42*b^19*d^4 + 13452160*
a^44*b^17*d^4 + 1606240*a^46*b^15*d^4 + 146560*a^48*b^13*d^4 + 10240*a^50*b^11*d^4 + 160*a^52*b^9*d^4)) + (a^3
9*b^25*d^4*(a + b*tan(c + d*x))^(1/2)*2580976480i)/((a^7)^(1/2)*(100000*a^10*b^51*d^4 + 1900000*a^12*b^49*d^4
+ 17100000*a^14*b^47*d^4 + 96900000*a^16*b^45*d^4 + 387640000*a^18*b^43*d^4 + 1163280000*a^20*b^41*d^4 + 27157
28000*a^22*b^39*d^4 + 5046192160*a^24*b^37*d^4 + 7569850240*a^26*b^35*d^4 + 9240726560*a^28*b^33*d^4 + 9205826
240*a^30*b^31*d^4 + 7471416160*a^32*b^29*d^4 + 4908704320*a^34*b^27*d^4 + 2580976480*a^36*b^25*d^4 + 106725312
0*a^38*b^23*d^4 + 338576480*a^40*b^21*d^4 + 79748320*a^42*b^19*d^4 + 13452160*a^44*b^17*d^4 + 1606240*a^46*b^1
5*d^4 + 146560*a^48*b^13*d^4 + 10240*a^50*b^11*d^4 + 160*a^52*b^9*d^4)) + (a^41*b^23*d^4*(a + b*tan(c + d*x))^
(1/2)*1067253120i)/((a^7)^(1/2)*(100000*a^10*b^51*d^4 + 1900000*a^12*b^49*d^4 + 17100000*a^14*b^47*d^4 + 96900
000*a^16*b^45*d^4 + 387640000*a^18*b^43*d^4 + 1163280000*a^20*b^41*d^4 + 2715728000*a^22*b^39*d^4 + 5046192160
*a^24*b^37*d^4 + 7569850240*a^26*b^35*d^4 + 9240726560*a^28*b^33*d^4 + 9205826240*a^30*b^31*d^4 + 7471416160*a
^32*b^29*d^4 + 4908704320*a^34*b^27*d^4 + 2580976480*a^36*b^25*d^4 + 1067253120*a^38*b^23*d^4 + 338576480*a^40
*b^21*d^4 + 79748320*a^42*b^19*d^4 + 13452160*a^44*b^17*d^4 + 1606240*a^46*b^15*d^4 + 146560*a^48*b^13*d^4 + 1
0240*a^50*b^11*d^4 + 160*a^52*b^9*d^4)) + (a^43*b^21*d^4*(a + b*tan(c + d*x))^(1/2)*338576480i)/((a^7)^(1/2)*(
100000*a^10*b^51*d^4 + 1900000*a^12*b^49*d^4 + 17100000*a^14*b^47*d^4 + 96900000*a^16*b^45*d^4 + 387640000*a^1
8*b^43*d^4 + 1163280000*a^20*b^41*d^4 + 2715728000*a^22*b^39*d^4 + 5046192160*a^24*b^37*d^4 + 7569850240*a^26*
b^35*d^4 + 9240726560*a^28*b^33*d^4 + 9205826240*a^30*b^31*d^4 + 7471416160*a^32*b^29*d^4 + 4908704320*a^34*b^
27*d^4 + 2580976480*a^36*b^25*d^4 + 1067253120*a^38*b^23*d^4 + 338576480*a^40*b^21*d^4 + 79748320*a^42*b^19*d^
4 + 13452160*a^44*b^17*d^4 + 1606240*a^46*b^15*d^4 + 146560*a^48*b^13*d^4 + 10240*a^50*b^11*d^4 + 160*a^52*b^9
*d^4)) + (a^45*b^19*d^4*(a + b*tan(c + d*x))^(1/2)*79748320i)/((a^7)^(1/2)*(100000*a^10*b^51*d^4 + 1900000*a^1
2*b^49*d^4 + 17100000*a^14*b^47*d^4 + 96900000*a^16*b^45*d^4 + 387640000*a^18*b^43*d^4 + 1163280000*a^20*b^41*
d^4 + 2715728000*a^22*b^39*d^4 + 5046192160*a^24*b^37*d^4 + 7569850240*a^26*b^35*d^4 + 9240726560*a^28*b^33*d^
4 + 9205826240*a^30*b^31*d^4 + 7471416160*a^32*b^29*d^4 + 4908704320*a^34*b^27*d^4 + 2580976480*a^36*b^25*d^4
+ 1067253120*a^38*b^23*d^4 + 338576480*a^40*b^21*d^4 + 79748320*a^42*b^19*d^4 + 13452160*a^44*b^17*d^4 + 16062
40*a^46*b^15*d^4 + 146560*a^48*b^13*d^4 + 10240*a^50*b^11*d^4 + 160*a^52*b^9*d^4)) + (a^47*b^17*d^4*(a + b*tan
(c + d*x))^(1/2)*13452160i)/((a^7)^(1/2)*(100000*a^10*b^51*d^4 + 1900000*a^12*b^49*d^4 + 17100000*a^14*b^47*d^
4 + 96900000*a^16*b^45*d^4 + 387640000*a^18*b^43*d^4 + 1163280000*a^20*b^41*d^4 + 2715728000*a^22*b^39*d^4 + 5
046192160*a^24*b^37*d^4 + 7569850240*a^26*b^35*d^4 + 9240726560*a^28*b^33*d^4 + 9205826240*a^30*b^31*d^4 + 747
1416160*a^32*b^29*d^4 + 4908704320*a^34*b^27*d^4 + 2580976480*a^36*b^25*d^4 + 1067253120*a^38*b^23*d^4 + 33857
6480*a^40*b^21*d^4 + 79748320*a^42*b^19*d^4 + 13452160*a^44*b^17*d^4 + 1606240*a^46*b^15*d^4 + 146560*a^48*b^1
3*d^4 + 10240*a^50*b^11*d^4 + 160*a^52*b^9*d^4)) + (a^49*b^15*d^4*(a + b*tan(c + d*x))^(1/2)*1606240i)/((a^7)^
(1/2)*(100000*a^10*b^51*d^4 + 1900000*a^12*b^49*d^4 + 17100000*a^14*b^47*d^4 + 96900000*a^16*b^45*d^4 + 387640
000*a^18*b^43*d^4 + 1163280000*a^20*b^41*d^4 + 2715728000*a^22*b^39*d^4 + 5046192160*a^24*b^37*d^4 + 756985024
0*a^26*b^35*d^4 + 9240726560*a^28*b^33*d^4 + 9205826240*a^30*b^31*d^4 + 7471416160*a^32*b^29*d^4 + 4908704320*
a^34*b^27*d^4 + 2580976480*a^36*b^25*d^4 + 1067253120*a^38*b^23*d^4 + 338576480*a^40*b^21*d^4 + 79748320*a^42*
b^19*d^4 + 13452160*a^44*b^17*d^4 + 1606240*a^46*b^15*d^4 + 146560*a^48*b^13*d^4 + 10240*a^50*b^11*d^4 + 160*a
^52*b^9*d^4)) + (a^51*b^13*d^4*(a + b*tan(c + d*x))^(1/2)*146560i)/((a^7)^(1/2)*(100000*a^10*b^51*d^4 + 190000
0*a^12*b^49*d^4 + 17100000*a^14*b^47*d^4 + 96900000*a^16*b^45*d^4 + 387640000*a^18*b^43*d^4 + 1163280000*a^20*
b^41*d^4 + 2715728000*a^22*b^39*d^4 + 5046192160*a^24*b^37*d^4 + 7569850240*a^26*b^35*d^4 + 9240726560*a^28*b^
33*d^4 + 9205826240*a^30*b^31*d^4 + 7471416160*a^32*b^29*d^4 + 4908704320*a^34*b^27*d^4 + 2580976480*a^36*b^25
*d^4 + 1067253120*a^38*b^23*d^4 + 338576480*a^40*b^21*d^4 + 79748320*a^42*b^19*d^4 + 13452160*a^44*b^17*d^4 +
1606240*a^46*b^15*d^4 + 146560*a^48*b^13*d^4 + 10240*a^50*b^11*d^4 + 160*a^52*b^9*d^4)) + (a^53*b^11*d^4*(a +
b*tan(c + d*x))^(1/2)*10240i)/((a^7)^(1/2)*(100000*a^10*b^51*d^4 + 1900000*a^12*b^49*d^4 + 17100000*a^14*b^47*
d^4 + 96900000*a^16*b^45*d^4 + 387640000*a^18*b^43*d^4 + 1163280000*a^20*b^41*d^4 + 2715728000*a^22*b^39*d^4 +
5046192160*a^24*b^37*d^4 + 7569850240*a^26*b^35*d^4 + 9240726560*a^28*b^33*d^4 + 9205826240*a^30*b^31*d^4 + 7
471416160*a^32*b^29*d^4 + 4908704320*a^34*b^27*d^4 + 2580976480*a^36*b^25*d^4 + 1067253120*a^38*b^23*d^4 + 338
576480*a^40*b^21*d^4 + 79748320*a^42*b^19*d^4 + 13452160*a^44*b^17*d^4 + 1606240*a^46*b^15*d^4 + 146560*a^48*b
^13*d^4 + 10240*a^50*b^11*d^4 + 160*a^52*b^9*d^4)) + (a^55*b^9*d^4*(a + b*tan(c + d*x))^(1/2)*160i)/((a^7)^(1/
2)*(100000*a^10*b^51*d^4 + 1900000*a^12*b^49*d^4 + 17100000*a^14*b^47*d^4 + 96900000*a^16*b^45*d^4 + 387640000
*a^18*b^43*d^4 + 1163280000*a^20*b^41*d^4 + 2715728000*a^22*b^39*d^4 + 5046192160*a^24*b^37*d^4 + 7569850240*a
^26*b^35*d^4 + 9240726560*a^28*b^33*d^4 + 9205826240*a^30*b^31*d^4 + 7471416160*a^32*b^29*d^4 + 4908704320*a^3
4*b^27*d^4 + 2580976480*a^36*b^25*d^4 + 1067253120*a^38*b^23*d^4 + 338576480*a^40*b^21*d^4 + 79748320*a^42*b^1
9*d^4 + 13452160*a^44*b^17*d^4 + 1606240*a^46*b^15*d^4 + 146560*a^48*b^13*d^4 + 10240*a^50*b^11*d^4 + 160*a^52
*b^9*d^4)))*5i)/(d*(a^7)^(1/2))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{2}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cot(d*x+c)**2/(a+b*tan(d*x+c))**(5/2),x)
[Out]
Integral(cot(c + d*x)**2/(a + b*tan(c + d*x))**(5/2), x)
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