3.1.88 \(\int \cot ^4(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\) [88]

Optimal. Leaf size=217 \[ \frac {a^{5/2} (45 i A+46 B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{8 d}-\frac {4 \sqrt {2} a^{5/2} (i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a^2 (19 A-18 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a^2 (3 i A+2 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d} \]

[Out]

1/8*a^(5/2)*(45*I*A+46*B)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/d-4*a^(5/2)*(I*A+B)*arctanh(1/2*(a+I*a*tan
(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)/d+1/8*a^2*(19*A-18*I*B)*cot(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/d-1/4*a^2*
(3*I*A+2*B)*cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(1/2)/d-1/3*a*A*cot(d*x+c)^3*(a+I*a*tan(d*x+c))^(3/2)/d

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Rubi [A]
time = 0.53, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3674, 3679, 3681, 3561, 212, 3680, 65, 214} \begin {gather*} \frac {a^{5/2} (46 B+45 i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{8 d}-\frac {4 \sqrt {2} a^{5/2} (B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {a^2 (2 B+3 i A) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}+\frac {a^2 (19 A-18 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]

[Out]

(a^(5/2)*((45*I)*A + 46*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(8*d) - (4*Sqrt[2]*a^(5/2)*(I*A + B)*A
rcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (a^2*(19*A - (18*I)*B)*Cot[c + d*x]*Sqrt[a + I*a*Tan
[c + d*x]])/(8*d) - (a^2*((3*I)*A + 2*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(4*d) - (a*A*Cot[c + d*x]^
3*(a + I*a*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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 3561

Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(2*a - x^2), x], x, Sq
rt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]

Rule 3674

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

Rule 3679

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

Rule 3680

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[b*(B/f), Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x
]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && EqQ[A*b + a*B,
0]

Rule 3681

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

Rubi steps

\begin {align*} \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx &=-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac {1}{3} \int \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2} \left (\frac {3}{2} a (3 i A+2 B)-\frac {3}{2} a (A-2 i B) \tan (c+d x)\right ) \, dx\\ &=-\frac {a^2 (3 i A+2 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac {1}{6} \int \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{4} a^2 (19 A-18 i B)-\frac {3}{4} a^2 (13 i A+14 B) \tan (c+d x)\right ) \, dx\\ &=\frac {a^2 (19 A-18 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a^2 (3 i A+2 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{8} a^3 (45 i A+46 B)+\frac {3}{8} a^3 (19 A-18 i B) \tan (c+d x)\right ) \, dx}{6 a}\\ &=\frac {a^2 (19 A-18 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a^2 (3 i A+2 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}+\left (4 a^2 (A-i B)\right ) \int \sqrt {a+i a \tan (c+d x)} \, dx-\frac {1}{16} (a (45 i A+46 B)) \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \, dx\\ &=\frac {a^2 (19 A-18 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a^2 (3 i A+2 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {\left (8 a^3 (i A+B)\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{d}-\frac {\left (a^3 (45 i A+46 B)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{16 d}\\ &=-\frac {4 \sqrt {2} a^{5/2} (i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a^2 (19 A-18 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a^2 (3 i A+2 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}-\frac {\left (a^2 (45 A-46 i B)\right ) \text {Subst}\left (\int \frac {1}{i-\frac {i x^2}{a}} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{8 d}\\ &=\frac {a^{5/2} (45 i A+46 B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{8 d}-\frac {4 \sqrt {2} a^{5/2} (i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a^2 (19 A-18 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {a^2 (3 i A+2 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{4 d}-\frac {a A \cot ^3(c+d x) (a+i a \tan (c+d x))^{3/2}}{3 d}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(634\) vs. \(2(217)=434\).
time = 8.67, size = 634, normalized size = 2.92 \begin {gather*} -\frac {i e^{-2 i c} \sqrt {e^{i d x}} \left (256 (A-i B) \sinh ^{-1}\left (e^{i (c+d x)}\right )+\sqrt {2} (45 A-46 i B) \left (\log \left (\left (-1+e^{i (c+d x)}\right )^2\right )-\log \left (\left (1+e^{i (c+d x)}\right )^2\right )+\log \left (3+3 e^{2 i (c+d x)}+2 \sqrt {2} \sqrt {1+e^{2 i (c+d x)}}-2 e^{i (c+d x)} \left (1+\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}\right )\right )-\log \left (3+3 e^{2 i (c+d x)}+2 \sqrt {2} \sqrt {1+e^{2 i (c+d x)}}+2 e^{i (c+d x)} \left (1+\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}\right )\right )\right )\right ) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{32 \sqrt {2} d \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \sec ^{\frac {7}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{5/2} (A \cos (c+d x)+B \sin (c+d x))}+\frac {\cos ^3(c+d x) \left (\csc (c) (65 A \cos (c)-54 i B \cos (c)+26 i A \sin (c)+12 B \sin (c)) \left (\frac {1}{24} \cos (2 c)-\frac {1}{24} i \sin (2 c)\right )+\csc (c) \csc ^2(c+d x) (-4 A \cos (c)-13 i A \sin (c)-6 B \sin (c)) \left (\frac {1}{12} \cos (2 c)-\frac {1}{12} i \sin (2 c)\right )+A \csc (c) \csc ^3(c+d x) \left (\frac {1}{3} \cos (2 c)-\frac {1}{3} i \sin (2 c)\right ) \sin (d x)+\csc (c) \csc (c+d x) \left (\frac {1}{24} \cos (2 c)-\frac {1}{24} i \sin (2 c)\right ) (-65 A \sin (d x)+54 i B \sin (d x))\right ) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]

[Out]

((-1/32*I)*Sqrt[E^(I*d*x)]*(256*(A - I*B)*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*(45*A - (46*I)*B)*(Log[(-1 + E^(I
*(c + d*x)))^2] - Log[(1 + E^(I*(c + d*x)))^2] + Log[3 + 3*E^((2*I)*(c + d*x)) + 2*Sqrt[2]*Sqrt[1 + E^((2*I)*(
c + d*x))] - 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - Log[3 + 3*E^((2*I)*(c + d*x)) +
2*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))] + 2*E^(I*(c + d*x))*(1 + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])]))*(a
+ I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]))/(Sqrt[2]*d*E^((2*I)*c)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c +
d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])^(5/2)*(A*Cos[c + d*x] + B*Si
n[c + d*x])) + (Cos[c + d*x]^3*(Csc[c]*(65*A*Cos[c] - (54*I)*B*Cos[c] + (26*I)*A*Sin[c] + 12*B*Sin[c])*(Cos[2*
c]/24 - (I/24)*Sin[2*c]) + Csc[c]*Csc[c + d*x]^2*(-4*A*Cos[c] - (13*I)*A*Sin[c] - 6*B*Sin[c])*(Cos[2*c]/12 - (
I/12)*Sin[2*c]) + A*Csc[c]*Csc[c + d*x]^3*(Cos[2*c]/3 - (I/3)*Sin[2*c])*Sin[d*x] + Csc[c]*Csc[c + d*x]*(Cos[2*
c]/24 - (I/24)*Sin[2*c])*(-65*A*Sin[d*x] + (54*I)*B*Sin[d*x]))*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x
]))/(d*(Cos[d*x] + I*Sin[d*x])^2*(A*Cos[c + d*x] + B*Sin[c + d*x]))

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2505 vs. \(2 (179 ) = 358\).
time = 0.63, size = 2506, normalized size = 11.55

method	result	size
default	\(\text {Expression too large to display}\)	\(2506\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-1/48/d*a^2*(52*A*cos(d*x+c)^2*sin(d*x+c)-114*A*cos(d*x+c)*sin(d*x+c)-192*I*A*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+
c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*sin(d*x+c)-108*B*
cos(d*x+c)^3-132*B*cos(d*x+c)^4+132*B*cos(d*x+c)^2+108*B*cos(d*x+c)-192*A*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1
))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)*sin(d*x+c)-192*B*2^(1/2)*(-2*cos(
d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*c
os(d*x+c)*sin(d*x+c)-132*I*B*cos(d*x+c)^3*sin(d*x+c)-24*I*B*cos(d*x+c)^2*sin(d*x+c)+108*I*B*cos(d*x+c)*sin(d*x
+c)+114*I*A*cos(d*x+c)+166*I*A*cos(d*x+c)^2-130*I*A*cos(d*x+c)^3-182*I*A*cos(d*x+c)^4+192*I*A*2^(1/2)*(-2*cos(
d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*c
os(d*x+c)^3*sin(d*x+c)-192*I*B*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/
(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^3*sin(d*x+c)+192*I*A*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/
2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*cos(d*x+c)^2*sin(d*x+c)-192*I*B*2^(1/2)*
(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^2*sin
(d*x+c)-192*I*A*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1)
)^(1/2)*sin(d*x+c)/cos(d*x+c))*cos(d*x+c)*sin(d*x+c)+192*I*B*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arct
an(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)*sin(d*x+c)+192*I*B*2^(1/2)*(-2*cos(d*x+c)/(cos
(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*sin(d*x+c)+192*A*2^(1/2)*(-2*cos(d*
x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^3*sin(d*x+c)+19
2*B*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(
d*x+c)/cos(d*x+c))*cos(d*x+c)^3*sin(d*x+c)+192*A*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/
2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^2*sin(d*x+c)+192*B*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^
(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*cos(d*x+c)^2*sin(d*x+c)+
135*I*A*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln((sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-cos(d*x+c)+1)
/sin(d*x+c))*cos(d*x+c)^3*sin(d*x+c)-138*I*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos
(d*x+c)+1))^(1/2))*cos(d*x+c)^3*sin(d*x+c)+135*I*A*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln((sin(d*x+c)*(-2*cos
(d*x+c)/(cos(d*x+c)+1))^(1/2)-cos(d*x+c)+1)/sin(d*x+c))*cos(d*x+c)^2*sin(d*x+c)-138*I*B*(-2*cos(d*x+c)/(cos(d*
x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^2*sin(d*x+c)-135*I*A*(-2*cos(d*x+c)/(
cos(d*x+c)+1))^(1/2)*ln((sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-cos(d*x+c)+1)/sin(d*x+c))*cos(d*x+c)*
sin(d*x+c)+138*I*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x
+c)*sin(d*x+c)-192*A*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c
)+1))^(1/2))*sin(d*x+c)-135*A*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/
2))*cos(d*x+c)*sin(d*x+c)-192*B*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c
)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*sin(d*x+c)-138*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln((sin(d
*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-cos(d*x+c)+1)/sin(d*x+c))*cos(d*x+c)*sin(d*x+c)-135*A*(-2*cos(d*x+c
)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*sin(d*x+c)-138*B*(-2*cos(d*x+c)/(cos(d*
x+c)+1))^(1/2)*ln((sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-cos(d*x+c)+1)/sin(d*x+c))*sin(d*x+c)+182*A*
cos(d*x+c)^3*sin(d*x+c)+135*A*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/
2))*cos(d*x+c)^3*sin(d*x+c)+138*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln((sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+
c)+1))^(1/2)-cos(d*x+c)+1)/sin(d*x+c))*cos(d*x+c)^3*sin(d*x+c)+135*A*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arct
an(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^2*sin(d*x+c)+138*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*
ln((sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-cos(d*x+c)+1)/sin(d*x+c))*cos(d*x+c)^2*sin(d*x+c)-135*I*A*
(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln((sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-cos(d*x+c)+1)/sin(d*x
+c))*sin(d*x+c)+138*I*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*si
n(d*x+c))*((I*sin(d*x+c)+cos(d*x+c))*a/cos(d*x+c))^(1/2)/(-1+cos(d*x+c))/(I*sin(d*x+c)+cos(d*x+c)-1)/(cos(d*x+
c)+1)^2

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Maxima [A]
time = 0.50, size = 249, normalized size = 1.15 \begin {gather*} \frac {i \, {\left (\frac {96 \, \sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{\sqrt {a}} - \frac {3 \, {\left (45 \, A - 46 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{\sqrt {a}} + \frac {2 \, {\left (3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} {\left (19 \, A - 18 i \, B\right )} - 8 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} {\left (11 \, A - 12 i \, B\right )} a + 3 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} {\left (13 \, A - 14 i \, B\right )} a^{2}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} - 3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a + 3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{2} - a^{3}}\right )} a^{3}}{48 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/48*I*(96*sqrt(2)*(A - I*B)*log(-(sqrt(2)*sqrt(a) - sqrt(I*a*tan(d*x + c) + a))/(sqrt(2)*sqrt(a) + sqrt(I*a*t
an(d*x + c) + a)))/sqrt(a) - 3*(45*A - 46*I*B)*log((sqrt(I*a*tan(d*x + c) + a) - sqrt(a))/(sqrt(I*a*tan(d*x +
c) + a) + sqrt(a)))/sqrt(a) + 2*(3*(I*a*tan(d*x + c) + a)^(5/2)*(19*A - 18*I*B) - 8*(I*a*tan(d*x + c) + a)^(3/
2)*(11*A - 12*I*B)*a + 3*sqrt(I*a*tan(d*x + c) + a)*(13*A - 14*I*B)*a^2)/((I*a*tan(d*x + c) + a)^3 - 3*(I*a*ta
n(d*x + c) + a)^2*a + 3*(I*a*tan(d*x + c) + a)*a^2 - a^3))*a^3/d

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 868 vs. \(2 (170) = 340\).
time = 2.00, size = 868, normalized size = 4.00 \begin {gather*} \frac {192 \, \sqrt {2} \sqrt {-\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac {4 \, {\left ({\left (-i \, A - B\right )} a^{3} e^{\left (i \, d x + i \, c\right )} + \sqrt {-\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{{\left (-i \, A - B\right )} a^{2}}\right ) - 192 \, \sqrt {2} \sqrt {-\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac {4 \, {\left ({\left (-i \, A - B\right )} a^{3} e^{\left (i \, d x + i \, c\right )} - \sqrt {-\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{{\left (-i \, A - B\right )} a^{2}}\right ) - 3 \, \sqrt {-\frac {{\left (2025 \, A^{2} - 4140 i \, A B - 2116 \, B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (-\frac {16 \, {\left (3 \, {\left (-45 i \, A - 46 \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-45 i \, A - 46 \, B\right )} a^{3} + 2 \, \sqrt {2} \sqrt {-\frac {{\left (2025 \, A^{2} - 4140 i \, A B - 2116 \, B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (3 i \, d x + 3 i \, c\right )} + d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (45 i \, A + 46 \, B\right )} a}\right ) + 3 \, \sqrt {-\frac {{\left (2025 \, A^{2} - 4140 i \, A B - 2116 \, B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (-\frac {16 \, {\left (3 \, {\left (-45 i \, A - 46 \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-45 i \, A - 46 \, B\right )} a^{3} - 2 \, \sqrt {2} \sqrt {-\frac {{\left (2025 \, A^{2} - 4140 i \, A B - 2116 \, B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (3 i \, d x + 3 i \, c\right )} + d e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (45 i \, A + 46 \, B\right )} a}\right ) + 4 \, \sqrt {2} {\left ({\left (91 i \, A + 66 \, B\right )} a^{2} e^{\left (7 i \, d x + 7 i \, c\right )} - 7 \, {\left (i \, A + 6 \, B\right )} a^{2} e^{\left (5 i \, d x + 5 i \, c\right )} + {\left (-59 i \, A - 66 \, B\right )} a^{2} e^{\left (3 i \, d x + 3 i \, c\right )} - 3 \, {\left (-13 i \, A - 14 \, B\right )} a^{2} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{96 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} - 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/96*(192*sqrt(2)*sqrt(-(A^2 - 2*I*A*B - B^2)*a^5/d^2)*(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x + 4*I*c) + 3*d*
e^(2*I*d*x + 2*I*c) - d)*log(4*((-I*A - B)*a^3*e^(I*d*x + I*c) + sqrt(-(A^2 - 2*I*A*B - B^2)*a^5/d^2)*(d*e^(2*
I*d*x + 2*I*c) + d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-I*d*x - I*c)/((-I*A - B)*a^2)) - 192*sqrt(2)*sqrt(-
(A^2 - 2*I*A*B - B^2)*a^5/d^2)*(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) - d)
*log(4*((-I*A - B)*a^3*e^(I*d*x + I*c) - sqrt(-(A^2 - 2*I*A*B - B^2)*a^5/d^2)*(d*e^(2*I*d*x + 2*I*c) + d)*sqrt
(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-I*d*x - I*c)/((-I*A - B)*a^2)) - 3*sqrt(-(2025*A^2 - 4140*I*A*B - 2116*B^2)
*a^5/d^2)*(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) - d)*log(-16*(3*(-45*I*A
- 46*B)*a^3*e^(2*I*d*x + 2*I*c) + (-45*I*A - 46*B)*a^3 + 2*sqrt(2)*sqrt(-(2025*A^2 - 4140*I*A*B - 2116*B^2)*a^
5/d^2)*(d*e^(3*I*d*x + 3*I*c) + d*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-2*I*d*x - 2*I*c)/((4
5*I*A + 46*B)*a)) + 3*sqrt(-(2025*A^2 - 4140*I*A*B - 2116*B^2)*a^5/d^2)*(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d*
x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) - d)*log(-16*(3*(-45*I*A - 46*B)*a^3*e^(2*I*d*x + 2*I*c) + (-45*I*A - 46*
B)*a^3 - 2*sqrt(2)*sqrt(-(2025*A^2 - 4140*I*A*B - 2116*B^2)*a^5/d^2)*(d*e^(3*I*d*x + 3*I*c) + d*e^(I*d*x + I*c
))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-2*I*d*x - 2*I*c)/((45*I*A + 46*B)*a)) + 4*sqrt(2)*((91*I*A + 66*B)*a
^2*e^(7*I*d*x + 7*I*c) - 7*(I*A + 6*B)*a^2*e^(5*I*d*x + 5*I*c) + (-59*I*A - 66*B)*a^2*e^(3*I*d*x + 3*I*c) - 3*
(-13*I*A - 14*B)*a^2*e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))/(d*e^(6*I*d*x + 6*I*c) - 3*d*e^(4*I*d
*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) - d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**4*(a+I*a*tan(d*x+c))**(5/2)*(A+B*tan(d*x+c)),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3878 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)*(I*a*tan(d*x + c) + a)^(5/2)*cot(d*x + c)^4, x)

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Mupad [B]
time = 8.47, size = 2500, normalized size = 11.52 \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)^4*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^(5/2),x)

[Out]

2*atanh((23*A^2*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((1041*B^2*a^5)/(128*d^2) - (4073*A^2*a^5)/(512*d^2) - (
(529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A
^3*B*a^22*253i)/(8*d^4))^(1/2)/(64*a^6) + (A*B*a^5*2059i)/(128*d^2))^(1/2))/(4*((A^3*a^11*d*1771i)/32 + (663*B
^3*a^11*d)/4 - (A*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*
a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)*13i)/4 - (7*B*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^
22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2))/2 + (
A*B^2*a^11*d*2167i)/8 - (797*A^2*B*a^11*d)/16)) - (6*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*((1041*B^2*a^5)/(128*d^
2) - (4073*A^2*a^5)/(512*d^2) - ((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4)
+ (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)/(64*a^6) + (A*B*a^5*2059i)/(128*d^2))^(1/2)*((
529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^
3*B*a^22*253i)/(8*d^4))^(1/2))/((A^3*a^14*d*1771i)/32 + (663*B^3*a^14*d)/4 + (A*B^2*a^14*d*2167i)/8 - (797*A^2
*B*a^14*d)/16 - (A*a^3*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A
*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)*13i)/4 - (7*B*a^3*d^3*((529*A^4*a^22)/(64*d^4) + (2
89*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/
2))/2) + (17*B^2*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((1041*B^2*a^5)/(128*d^2) - (4073*A^2*a^5)/(512*d^2) -
((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (
A^3*B*a^22*253i)/(8*d^4))^(1/2)/(64*a^6) + (A*B*a^5*2059i)/(128*d^2))^(1/2))/((A^3*a^11*d*1771i)/32 + (663*B^3
*a^11*d)/4 - (A*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^
22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)*13i)/4 - (7*B*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22
)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2))/2 + (A*
B^2*a^11*d*2167i)/8 - (797*A^2*B*a^11*d)/16) + (A*B*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*((1041*B^2*a^5)/(128
*d^2) - (4073*A^2*a^5)/(512*d^2) - ((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d
^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)/(64*a^6) + (A*B*a^5*2059i)/(128*d^2))^(1/2)
*11i)/((A^3*a^11*d*1771i)/32 + (663*B^3*a^11*d)/4 - (A*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) +
(149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)*13i)/4 - (7*B*d^3*(
(529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A
^3*B*a^22*253i)/(8*d^4))^(1/2))/2 + (A*B^2*a^11*d*2167i)/8 - (797*A^2*B*a^11*d)/16))*((1041*B^2*a^5)/(128*d^2)
- (4073*A^2*a^5)/(512*d^2) - ((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) +
(A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)/(64*a^6) + (A*B*a^5*2059i)/(128*d^2))^(1/2) + 2*
atanh((6*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*(((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a
^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)/(64*a^6) - (4073*A^2*a^5)/(512*d^2
) + (1041*B^2*a^5)/(128*d^2) + (A*B*a^5*2059i)/(128*d^2))^(1/2)*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d
^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2))/((A^3*a^14*d*
1771i)/32 + (663*B^3*a^14*d)/4 + (A*B^2*a^14*d*2167i)/8 - (797*A^2*B*a^14*d)/16 + (A*a^3*d^3*((529*A^4*a^22)/(
64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/
(8*d^4))^(1/2)*13i)/4 + (7*B*a^3*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8
*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2))/2) + (23*A^2*a^8*d^2*(a + a*tan(c + d*x)
*1i)^(1/2)*(((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)
/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)/(64*a^6) - (4073*A^2*a^5)/(512*d^2) + (1041*B^2*a^5)/(128*d^2) + (
A*B*a^5*2059i)/(128*d^2))^(1/2))/(4*((A^3*a^11*d*1771i)/32 + (663*B^3*a^11*d)/4 + (A*d^3*((529*A^4*a^22)/(64*d
^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d
^4))^(1/2)*13i)/4 + (7*B*d^3*((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149*A^2*B^2*a^22)/(8*d^4) +
(A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2))/2 + (A*B^2*a^11*d*2167i)/8 - (797*A^2*B*a^11*d)/
16)) + (17*B^2*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((529*A^4*a^22)/(64*d^4) + (289*B^4*a^22)/(4*d^4) + (149
*A^2*B^2*a^22)/(8*d^4) + (A*B^3*a^22*187i)/(2*d^4) + (A^3*B*a^22*253i)/(8*d^4))^(1/2)/(64*a^6) - (4073*A^2*a^5
)/(512*d^2) + (1041*B^2*a^5)/(128*d^2) + (A*B*a...

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