3.1.89 \(\int \cot ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx\) [89]

Optimal. Leaf size=261 \[ -\frac {3 a^{5/2} (121 A-120 i B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{64 d}+\frac {4 \sqrt {2} a^{5/2} (A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a^2 (149 i A+152 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}+\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{96 d}-\frac {a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d} \]

[Out]

-3/64*a^(5/2)*(121*A-120*I*B)*arctanh((a+I*a*tan(d*x+c))^(1/2)/a^(1/2))/d+4*a^(5/2)*(A-I*B)*arctanh(1/2*(a+I*a
*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*2^(1/2)/d+1/64*a^2*(149*I*A+152*B)*cot(d*x+c)*(a+I*a*tan(d*x+c))^(1/2)/d+1
/96*a^2*(107*A-104*I*B)*cot(d*x+c)^2*(a+I*a*tan(d*x+c))^(1/2)/d-1/24*a^2*(11*I*A+8*B)*cot(d*x+c)^3*(a+I*a*tan(
d*x+c))^(1/2)/d-1/4*a*A*cot(d*x+c)^4*(a+I*a*tan(d*x+c))^(3/2)/d

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Rubi [A]
time = 0.66, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 10, 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 {3 a^{5/2} (121 A-120 i B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{64 d}+\frac {4 \sqrt {2} a^{5/2} (A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {a^2 (8 B+11 i A) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}+\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{96 d}+\frac {a^2 (152 B+149 i A) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a^(5/2)*(121*A - (120*I)*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/Sqrt[a]])/(64*d) + (4*Sqrt[2]*a^(5/2)*(A -
I*B)*ArcTanh[Sqrt[a + I*a*Tan[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (a^2*((149*I)*A + 152*B)*Cot[c + d*x]*Sqrt[a +
I*a*Tan[c + d*x]])/(64*d) + (a^2*(107*A - (104*I)*B)*Cot[c + d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(96*d) - (a^2
*((11*I)*A + 8*B)*Cot[c + d*x]^3*Sqrt[a + I*a*Tan[c + d*x]])/(24*d) - (a*A*Cot[c + d*x]^4*(a + I*a*Tan[c + d*x
])^(3/2))/(4*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 ^5(c+d x) (a+i a \tan (c+d x))^{5/2} (A+B \tan (c+d x)) \, dx &=-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}+\frac {1}{4} \int \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2} \left (\frac {1}{2} a (11 i A+8 B)-\frac {1}{2} a (5 A-8 i B) \tan (c+d x)\right ) \, dx\\ &=-\frac {a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}+\frac {1}{12} \int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {1}{4} a^2 (107 A-104 i B)-\frac {1}{4} a^2 (85 i A+88 B) \tan (c+d x)\right ) \, dx\\ &=\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{96 d}-\frac {a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}+\frac {\int \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{8} a^3 (149 i A+152 B)+\frac {3}{8} a^3 (107 A-104 i B) \tan (c+d x)\right ) \, dx}{24 a}\\ &=\frac {a^2 (149 i A+152 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}+\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{96 d}-\frac {a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (\frac {9}{16} a^4 (121 A-120 i B)+\frac {3}{16} a^4 (149 i A+152 B) \tan (c+d x)\right ) \, dx}{24 a^2}\\ &=\frac {a^2 (149 i A+152 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}+\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{96 d}-\frac {a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}+\frac {1}{128} (3 a (121 A-120 i B)) \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \, dx+\left (4 a^2 (i A+B)\right ) \int \sqrt {a+i a \tan (c+d x)} \, dx\\ &=\frac {a^2 (149 i A+152 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}+\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{96 d}-\frac {a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}+\frac {\left (8 a^3 (A-i 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 (3 a^3 (121 A-120 i B)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{128 d}\\ &=\frac {4 \sqrt {2} a^{5/2} (A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a^2 (149 i A+152 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}+\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{96 d}-\frac {a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 d}-\frac {\left (3 a^2 (121 i A+120 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 )}{64 d}\\ &=-\frac {3 a^{5/2} (121 A-120 i B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{64 d}+\frac {4 \sqrt {2} a^{5/2} (A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {a^2 (149 i A+152 B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{64 d}+\frac {a^2 (107 A-104 i B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{96 d}-\frac {a^2 (11 i A+8 B) \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{24 d}-\frac {a A \cot ^4(c+d x) (a+i a \tan (c+d x))^{3/2}}{4 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. \(698\) vs. \(2(261)=522\).
time = 8.92, size = 698, normalized size = 2.67 \begin {gather*} \frac {e^{-2 i c} \sqrt {e^{i d x}} \left (2048 (A-i B) \sinh ^{-1}\left (e^{i (c+d x)}\right )+3 \sqrt {2} (121 A-120 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))}{256 \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) (583 i A \cos (c)+520 B \cos (c)-262 A \sin (c)+208 i B \sin (c)) \left (\frac {1}{192} \cos (2 c)-\frac {1}{192} i \sin (2 c)\right )+\csc ^4(c+d x) \left (-\frac {1}{4} A \cos (2 c)+\frac {1}{4} i A \sin (2 c)\right )+\csc (c) \csc ^2(c+d x) (87 i A+72 B-223 i A \cos (2 c)-136 B \cos (2 c)+223 A \sin (2 c)-136 i B \sin (2 c)) \left (\frac {1}{192} \cos (3 c)-\frac {1}{192} i \sin (3 c)\right )+\csc (c) \csc (c+d x) \left (\frac {1}{192} \cos (2 c)-\frac {1}{192} i \sin (2 c)\right ) (-583 i A \sin (d x)-520 B \sin (d x))+\csc (c) \csc ^3(c+d x) \left (\frac {1}{24} \cos (2 c)-\frac {1}{24} i \sin (2 c)\right ) (17 i A \sin (d x)+8 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]^5*(a + I*a*Tan[c + d*x])^(5/2)*(A + B*Tan[c + d*x]),x]

[Out]

(Sqrt[E^(I*d*x)]*(2048*(A - I*B)*ArcSinh[E^(I*(c + d*x))] + 3*Sqrt[2]*(121*A - (120*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*Sqr
t[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]))/(256*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*Sin
[c + d*x])) + (Cos[c + d*x]^3*(Csc[c]*((583*I)*A*Cos[c] + 520*B*Cos[c] - 262*A*Sin[c] + (208*I)*B*Sin[c])*(Cos
[2*c]/192 - (I/192)*Sin[2*c]) + Csc[c + d*x]^4*(-1/4*(A*Cos[2*c]) + (I/4)*A*Sin[2*c]) + Csc[c]*Csc[c + d*x]^2*
((87*I)*A + 72*B - (223*I)*A*Cos[2*c] - 136*B*Cos[2*c] + 223*A*Sin[2*c] - (136*I)*B*Sin[2*c])*(Cos[3*c]/192 -
(I/192)*Sin[3*c]) + Csc[c]*Csc[c + d*x]*(Cos[2*c]/192 - (I/192)*Sin[2*c])*((-583*I)*A*Sin[d*x] - 520*B*Sin[d*x
]) + Csc[c]*Csc[c + d*x]^3*(Cos[2*c]/24 - (I/24)*Sin[2*c])*((17*I)*A*Sin[d*x] + 8*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. 3443 vs. \(2 (217 ) = 434\).
time = 0.53, size = 3444, normalized size = 13.20

method	result	size
default	\(\text {Expression too large to display}\)	\(3444\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384/d*a^2*((I*sin(d*x+c)+cos(d*x+c))*a/cos(d*x+c))^(1/2)*(1322*A*cos(d*x+c)^2*sin(d*x+c)+894*A*cos(d*x+c)*si
n(d*x+c)+1536*I*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)^5+1456*B*cos(d*x+c)^5-2368*B*cos(d*x+c)^3+416*B*cos(d*x+c)^4-416*B*cos(d*x+c)^2+912*B*cos
(d*x+c)+1456*I*B*cos(d*x+c)^4*sin(d*x+c)-1089*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)^5+1080*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)^5-1089*A*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ln((si
n(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)^4+1080*B*(-2*cos(d*x+c)/(co
s(d*x+c)+1))^(1/2)*arctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^4+1536*I*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)^5+1536*I*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)^4+1536*I*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)^4-3072*I*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)^3-3072*I*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)^3-30
72*I*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))*c
os(d*x+c)^2-3072*I*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+1536*I*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)+1536*I*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)-1089*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))+10
80*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))+3072*A*(-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))*2^(1/2)
*cos(d*x+c)^3-3072*B*(-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))*2^(1/2)*cos(d*x+c)^3+3072*A*(-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))*2^(1/2)*cos(d*x+c)^2-3072*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*arcta
n(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*2^(1/2)*cos(d*x+c)^2-1536*A*(-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))*2^(1/2)*cos(d*x+c)+1536
*B*(-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))*2^(1/2)*cos(d
*x+c)+1089*I*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))+1080*I*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
))-912*I*B*cos(d*x+c)*sin(d*x+c)+1040*I*B*cos(d*x+c)^3*sin(d*x+c)-1328*I*B*cos(d*x+c)^2*sin(d*x+c)-1166*A*cos(
d*x+c)^3*sin(d*x+c)+2178*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)^3-2160*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+2178*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)^2-2160*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*ar
ctan(1/(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*cos(d*x+c)^2-1536*A*(-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))*2^(1/2)-1089*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)+1536*B*
(-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))*2^(1/2)+1080*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)-1690*A*cos(d*x+c
)^4*sin(d*x+c)+894*I*A*cos(d*x+c)+1690*I*A*cos(d*x+c)^5+524*I*A*cos(d*x+c)^4-2488*I*A*cos(d*x+c)^3-428*I*A*cos
(d*x+c)^2+1089*I*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)+1080*I*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)-1536*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)^5+1536*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*...

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Maxima [A]
time = 0.55, size = 293, normalized size = 1.12 \begin {gather*} -\frac {a^{4} {\left (\frac {768 \, \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 )}{a^{\frac {3}{2}}} - \frac {9 \, {\left (121 \, A - 120 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 )}{a^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} {\left (149 \, A - 152 i \, B\right )} - {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} {\left (1127 \, A - 1160 i \, B\right )} a + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} {\left (1049 \, A - 1016 i \, B\right )} a^{2} - 3 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} {\left (107 \, A - 104 i \, B\right )} a^{3}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4} a - 4 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a^{2} + 6 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{3} - 4 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{4} + a^{5}}\right )}}{384 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*a^4*(768*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*tan(d*x + c) + a)))/a^(3/2) - 9*(121*A - 120*I*B)*log((sqrt(I*a*tan(d*x + c) + a) - sqrt(a))/(sqrt(I*a*tan
(d*x + c) + a) + sqrt(a)))/a^(3/2) + 2*(3*(I*a*tan(d*x + c) + a)^(7/2)*(149*A - 152*I*B) - (I*a*tan(d*x + c) +
a)^(5/2)*(1127*A - 1160*I*B)*a + (I*a*tan(d*x + c) + a)^(3/2)*(1049*A - 1016*I*B)*a^2 - 3*sqrt(I*a*tan(d*x +
c) + a)*(107*A - 104*I*B)*a^3)/((I*a*tan(d*x + c) + a)^4*a - 4*(I*a*tan(d*x + c) + a)^3*a^2 + 6*(I*a*tan(d*x +
c) + a)^2*a^3 - 4*(I*a*tan(d*x + c) + a)*a^4 + a^5))/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. 944 vs. \(2 (206) = 412\).
time = 1.98, size = 944, normalized size = 3.62 \begin {gather*} \frac {1536 \, \sqrt {2} \sqrt {\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, 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 (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, 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 ) - 1536 \, \sqrt {2} \sqrt {\frac {{\left (A^{2} - 2 i \, A B - B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, 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 (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, 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 ) + 9 \, \sqrt {\frac {{\left (14641 \, A^{2} - 29040 i \, A B - 14400 \, B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {16 \, {\left (3 \, {\left (-121 i \, A - 120 \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-121 i \, A - 120 \, B\right )} a^{3} + 2 \, \sqrt {2} \sqrt {\frac {{\left (14641 \, A^{2} - 29040 i \, A B - 14400 \, B^{2}\right )} a^{5}}{d^{2}}} {\left (i \, d e^{\left (3 i \, d x + 3 i \, c\right )} + i \, 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 (-121 i \, A - 120 \, B\right )} a}\right ) - 9 \, \sqrt {\frac {{\left (14641 \, A^{2} - 29040 i \, A B - 14400 \, B^{2}\right )} a^{5}}{d^{2}}} {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {16 \, {\left (3 \, {\left (-121 i \, A - 120 \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-121 i \, A - 120 \, B\right )} a^{3} + 2 \, \sqrt {2} \sqrt {\frac {{\left (14641 \, A^{2} - 29040 i \, A B - 14400 \, B^{2}\right )} a^{5}}{d^{2}}} {\left (-i \, d e^{\left (3 i \, d x + 3 i \, c\right )} - i \, 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 (-121 i \, A - 120 \, B\right )} a}\right ) - 4 \, \sqrt {2} {\left (13 \, {\left (65 \, A - 56 i \, B\right )} a^{2} e^{\left (9 i \, d x + 9 i \, c\right )} - 2 \, {\left (215 \, A - 392 i \, B\right )} a^{2} e^{\left (7 i \, d x + 7 i \, c\right )} - 4 \, {\left (35 \, A - 104 i \, B\right )} a^{2} e^{\left (5 i \, d x + 5 i \, c\right )} + 2 \, {\left (407 \, A - 392 i \, B\right )} a^{2} e^{\left (3 i \, d x + 3 i \, c\right )} - 3 \, {\left (107 \, A - 104 i \, 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}}}{768 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \, 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)^5*(a+I*a*tan(d*x+c))^(5/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/768*(1536*sqrt(2)*sqrt((A^2 - 2*I*A*B - B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d
*e^(4*I*d*x + 4*I*c) - 4*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)*(I*d*e^(2*I*d*x + 2*I*c) + I*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-I*d*x - I*c)/((-I*A -
B)*a^2)) - 1536*sqrt(2)*sqrt((A^2 - 2*I*A*B - B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c)
+ 6*d*e^(4*I*d*x + 4*I*c) - 4*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)*(-I*d*e^(2*I*d*x + 2*I*c) - I*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-I*d*x - I*c)/((
-I*A - B)*a^2)) + 9*sqrt((14641*A^2 - 29040*I*A*B - 14400*B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*
x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)*log(16*(3*(-121*I*A - 120*B)*a^3*e^(2*I*d*
x + 2*I*c) + (-121*I*A - 120*B)*a^3 + 2*sqrt(2)*sqrt((14641*A^2 - 29040*I*A*B - 14400*B^2)*a^5/d^2)*(I*d*e^(3*
I*d*x + 3*I*c) + I*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)/((-121*I*A - 120
*B)*a)) - 9*sqrt((14641*A^2 - 29040*I*A*B - 14400*B^2)*a^5/d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*
c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)*log(16*(3*(-121*I*A - 120*B)*a^3*e^(2*I*d*x + 2*I*
c) + (-121*I*A - 120*B)*a^3 + 2*sqrt(2)*sqrt((14641*A^2 - 29040*I*A*B - 14400*B^2)*a^5/d^2)*(-I*d*e^(3*I*d*x +
3*I*c) - I*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)/((-121*I*A - 120*B)*a))
- 4*sqrt(2)*(13*(65*A - 56*I*B)*a^2*e^(9*I*d*x + 9*I*c) - 2*(215*A - 392*I*B)*a^2*e^(7*I*d*x + 7*I*c) - 4*(35
*A - 104*I*B)*a^2*e^(5*I*d*x + 5*I*c) + 2*(407*A - 392*I*B)*a^2*e^(3*I*d*x + 3*I*c) - 3*(107*A - 104*I*B)*a^2*
e^(I*d*x + I*c))*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1)))/(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(
4*I*d*x + 4*I*c) - 4*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)**5*(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 4847 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)^5*(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)^5, x)

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Mupad [B]
time = 8.59, size = 2500, normalized size = 9.58 \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)^5*(A + B*tan(c + d*x))*(a + a*tan(c + d*x)*1i)^(5/2),x)

[Out]

(((107*A*a^6 - B*a^6*104i)*(a + a*tan(c + d*x)*1i)^(1/2))/(64*d) - ((149*A*a^3 - B*a^3*152i)*(a + a*tan(c + d*
x)*1i)^(7/2))/(64*d) - ((1049*A*a^5 - B*a^5*1016i)*(a + a*tan(c + d*x)*1i)^(3/2))/(192*d) + ((1127*A*a^4 - B*a
^4*1160i)*(a + a*tan(c + d*x)*1i)^(5/2))/(192*d))/((a + a*tan(c + d*x)*1i)^4 - 4*a^3*(a + a*tan(c + d*x)*1i) -
4*a*(a + a*tan(c + d*x)*1i)^3 + 6*a^2*(a + a*tan(c + d*x)*1i)^2 + a^4) - 2*atanh((384*d^4*(a + a*tan(c + d*x)
*1i)^(1/2)*(((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B
^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)/(64*a^6) + (262841*A^2*a^5)/(32768*d^2) - (40
73*B^2*a^5)/(512*d^2) - (A*B*a^5*32719i)/(2048*d^2))^(1/2)*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(6
4*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2
))/((431443*A^3*a^14*d)/256 - B^3*a^14*d*3542i + (21783*A*B^2*a^14*d)/4 + (A^2*B*a^14*d*6993i)/32 + 214*A*a^3*
d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*
1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2) - B*a^3*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^
4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*
d^4))^(1/2)*208i) + (697*A^2*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((485809*A^4*a^22)/(262144*d^4) + (529*B^4
*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d
^4))^(1/2)/(64*a^6) + (262841*A^2*a^5)/(32768*d^2) - (4073*B^2*a^5)/(512*d^2) - (A*B*a^5*32719i)/(2048*d^2))^(
1/2))/(4*((431443*A^3*a^11*d)/256 - B^3*a^11*d*3542i + 214*A*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^
22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4)
)^(1/2) - B*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) +
(A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)*208i + (21783*A*B^2*a^11*d)/4 + (A^2*B*a^
11*d*6993i)/32)) + (368*B^2*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((485809*A^4*a^22)/(262144*d^4) + (529*B^4*
a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^
4))^(1/2)/(64*a^6) + (262841*A^2*a^5)/(32768*d^2) - (4073*B^2*a^5)/(512*d^2) - (A*B*a^5*32719i)/(2048*d^2))^(1
/2))/((431443*A^3*a^11*d)/256 - B^3*a^11*d*3542i + 214*A*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/
(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1
/2) - B*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B
^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)*208i + (21783*A*B^2*a^11*d)/4 + (A^2*B*a^11*d
*6993i)/32) + (A*B*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1/2)*(((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64
*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)
/(64*a^6) + (262841*A^2*a^5)/(32768*d^2) - (4073*B^2*a^5)/(512*d^2) - (A*B*a^5*32719i)/(2048*d^2))^(1/2)*196i)
/((431443*A^3*a^11*d)/256 - B^3*a^11*d*3542i + 214*A*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*
d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)
- B*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a
^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)*208i + (21783*A*B^2*a^11*d)/4 + (A^2*B*a^11*d*699
3i)/32))*(((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3
*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)/(64*a^6) + (262841*A^2*a^5)/(32768*d^2) - (4073
*B^2*a^5)/(512*d^2) - (A*B*a^5*32719i)/(2048*d^2))^(1/2) - 2*atanh((697*A^2*a^8*d^2*(a + a*tan(c + d*x)*1i)^(1
/2)*((262841*A^2*a^5)/(32768*d^2) - ((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2
*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)/(64*a^6) - (4073*B^2*
a^5)/(512*d^2) - (A*B*a^5*32719i)/(2048*d^2))^(1/2))/(4*((431443*A^3*a^11*d)/256 - B^3*a^11*d*3542i - 214*A*d^
3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*11
27i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2) + B*d^3*((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22
)/(64*d^4) + (11229*A^2*B^2*a^22)/(2048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^
(1/2)*208i + (21783*A*B^2*a^11*d)/4 + (A^2*B*a^11*d*6993i)/32)) - (384*d^4*(a + a*tan(c + d*x)*1i)^(1/2)*((262
841*A^2*a^5)/(32768*d^2) - ((485809*A^4*a^22)/(262144*d^4) + (529*B^4*a^22)/(64*d^4) + (11229*A^2*B^2*a^22)/(2
048*d^4) + (A*B^3*a^22*1127i)/(128*d^4) + (A^3*B*a^22*34153i)/(8192*d^4))^(1/2)/(64*a^6) - (4073*B^2*a^5)/(512
*d^2) - (A*B*a^5*32719i)/(2048*d^2))^(1/2)*((48...

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