3.29 \(\int \cot (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=142 \[ -\frac{(A-2 i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac{(3 A-4 i B) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+\frac{a^4 (7 A-8 i B) \log (\cos (c+d x))}{d}+8 a^4 x (B+i A)+\frac{a^4 A \log (\sin (c+d x))}{d}+\frac{i a B (a+i a \tan (c+d x))^3}{3 d} \]

[Out]

8*a^4*(I*A + B)*x + (a^4*(7*A - (8*I)*B)*Log[Cos[c + d*x]])/d + (a^4*A*Log[Sin[c + d*x]])/d + ((I/3)*a*B*(a +
I*a*Tan[c + d*x])^3)/d - ((A - (2*I)*B)*(a^2 + I*a^2*Tan[c + d*x])^2)/(2*d) - ((3*A - (4*I)*B)*(a^4 + I*a^4*Ta
n[c + d*x]))/d

________________________________________________________________________________________

Rubi [A]  time = 0.420767, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3594, 3589, 3475, 3531} \[ -\frac{(A-2 i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac{(3 A-4 i B) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+\frac{a^4 (7 A-8 i B) \log (\cos (c+d x))}{d}+8 a^4 x (B+i A)+\frac{a^4 A \log (\sin (c+d x))}{d}+\frac{i a B (a+i a \tan (c+d x))^3}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

8*a^4*(I*A + B)*x + (a^4*(7*A - (8*I)*B)*Log[Cos[c + d*x]])/d + (a^4*A*Log[Sin[c + d*x]])/d + ((I/3)*a*B*(a +
I*a*Tan[c + d*x])^3)/d - ((A - (2*I)*B)*(a^2 + I*a^2*Tan[c + d*x])^2)/(2*d) - ((3*A - (4*I)*B)*(a^4 + I*a^4*Ta
n[c + d*x]))/d

Rule 3594

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

Rule 3589

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

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3531

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

Rubi steps

\begin{align*} \int \cot (c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx &=\frac{i a B (a+i a \tan (c+d x))^3}{3 d}+\frac{1}{3} \int \cot (c+d x) (a+i a \tan (c+d x))^3 (3 a A+3 a (i A+2 B) \tan (c+d x)) \, dx\\ &=\frac{i a B (a+i a \tan (c+d x))^3}{3 d}-\frac{(A-2 i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}+\frac{1}{6} \int \cot (c+d x) (a+i a \tan (c+d x))^2 \left (6 a^2 A+6 a^2 (3 i A+4 B) \tan (c+d x)\right ) \, dx\\ &=\frac{i a B (a+i a \tan (c+d x))^3}{3 d}-\frac{(A-2 i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac{(3 A-4 i B) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+\frac{1}{6} \int \cot (c+d x) (a+i a \tan (c+d x)) \left (6 a^3 A+6 a^3 (7 i A+8 B) \tan (c+d x)\right ) \, dx\\ &=\frac{i a B (a+i a \tan (c+d x))^3}{3 d}-\frac{(A-2 i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac{(3 A-4 i B) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+\frac{1}{6} \int \cot (c+d x) \left (6 a^4 A+48 a^4 (i A+B) \tan (c+d x)\right ) \, dx-\left (a^4 (7 A-8 i B)\right ) \int \tan (c+d x) \, dx\\ &=8 a^4 (i A+B) x+\frac{a^4 (7 A-8 i B) \log (\cos (c+d x))}{d}+\frac{i a B (a+i a \tan (c+d x))^3}{3 d}-\frac{(A-2 i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac{(3 A-4 i B) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+\left (a^4 A\right ) \int \cot (c+d x) \, dx\\ &=8 a^4 (i A+B) x+\frac{a^4 (7 A-8 i B) \log (\cos (c+d x))}{d}+\frac{a^4 A \log (\sin (c+d x))}{d}+\frac{i a B (a+i a \tan (c+d x))^3}{3 d}-\frac{(A-2 i B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{2 d}-\frac{(3 A-4 i B) \left (a^4+i a^4 \tan (c+d x)\right )}{d}\\ \end{align*}

Mathematica [B]  time = 6.97096, size = 429, normalized size = 3.02 \[ \frac{a^4 \sec (c) \sec ^3(c+d x) (\cos (4 d x)+i \sin (4 d x)) \left (3 \cos (d x) \left (3 (7 A-8 i B) \log \left (\cos ^2(c+d x)\right )+3 A \log \left (\sin ^2(c+d x)\right )+48 i A d x+4 A+48 B d x-16 i B\right )+3 \cos (2 c+d x) \left (3 (7 A-8 i B) \log \left (\cos ^2(c+d x)\right )+3 A \log \left (\sin ^2(c+d x)\right )+48 i A d x+4 A+48 B d x-16 i B\right )+48 i A \sin (2 c+d x)-48 i A \sin (2 c+3 d x)+48 i A d x \cos (2 c+3 d x)+48 i A d x \cos (4 c+3 d x)+21 A \cos (2 c+3 d x) \log \left (\cos ^2(c+d x)\right )+21 A \cos (4 c+3 d x) \log \left (\cos ^2(c+d x)\right )+3 A \cos (2 c+3 d x) \log \left (\sin ^2(c+d x)\right )+3 A \cos (4 c+3 d x) \log \left (\sin ^2(c+d x)\right )-96 i A \sin (d x)+96 B \sin (2 c+d x)-88 B \sin (2 c+3 d x)+48 B d x \cos (2 c+3 d x)+48 B d x \cos (4 c+3 d x)-24 i B \cos (2 c+3 d x) \log \left (\cos ^2(c+d x)\right )-24 i B \cos (4 c+3 d x) \log \left (\cos ^2(c+d x)\right )-168 B \sin (d x)\right )}{48 d (\cos (d x)+i \sin (d x))^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*Sec[c]*Sec[c + d*x]^3*(Cos[4*d*x] + I*Sin[4*d*x])*((48*I)*A*d*x*Cos[2*c + 3*d*x] + 48*B*d*x*Cos[2*c + 3*d
*x] + (48*I)*A*d*x*Cos[4*c + 3*d*x] + 48*B*d*x*Cos[4*c + 3*d*x] + 21*A*Cos[2*c + 3*d*x]*Log[Cos[c + d*x]^2] -
(24*I)*B*Cos[2*c + 3*d*x]*Log[Cos[c + d*x]^2] + 21*A*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] - (24*I)*B*Cos[4*c +
3*d*x]*Log[Cos[c + d*x]^2] + 3*A*Cos[2*c + 3*d*x]*Log[Sin[c + d*x]^2] + 3*A*Cos[4*c + 3*d*x]*Log[Sin[c + d*x]
^2] + 3*Cos[d*x]*(4*A - (16*I)*B + (48*I)*A*d*x + 48*B*d*x + 3*(7*A - (8*I)*B)*Log[Cos[c + d*x]^2] + 3*A*Log[S
in[c + d*x]^2]) + 3*Cos[2*c + d*x]*(4*A - (16*I)*B + (48*I)*A*d*x + 48*B*d*x + 3*(7*A - (8*I)*B)*Log[Cos[c + d
*x]^2] + 3*A*Log[Sin[c + d*x]^2]) - (96*I)*A*Sin[d*x] - 168*B*Sin[d*x] + (48*I)*A*Sin[2*c + d*x] + 96*B*Sin[2*
c + d*x] - (48*I)*A*Sin[2*c + 3*d*x] - 88*B*Sin[2*c + 3*d*x]))/(48*d*(Cos[d*x] + I*Sin[d*x])^4)

________________________________________________________________________________________

Maple [A]  time = 0.074, size = 169, normalized size = 1.2 \begin{align*}{\frac{A{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+7\,{\frac{A{a}^{4}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{B{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-7\,{\frac{B{a}^{4}\tan \left ( dx+c \right ) }{d}}+8\,B{a}^{4}x+8\,{\frac{B{a}^{4}c}{d}}-{\frac{4\,iA\tan \left ( dx+c \right ){a}^{4}}{d}}+8\,iAx{a}^{4}-{\frac{2\,iB{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{8\,iA{a}^{4}c}{d}}-{\frac{8\,iB{a}^{4}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{A{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)*(a+I*a*tan(d*x+c))^4*(A+B*tan(d*x+c)),x)

[Out]

1/2/d*a^4*A*tan(d*x+c)^2+7/d*A*a^4*ln(cos(d*x+c))+1/3/d*a^4*B*tan(d*x+c)^3-7/d*a^4*B*tan(d*x+c)+8*B*a^4*x+8/d*
B*a^4*c-4*I/d*A*tan(d*x+c)*a^4+8*I*A*x*a^4-2*I/d*B*a^4*tan(d*x+c)^2+8*I/d*A*a^4*c-8*I/d*B*a^4*ln(cos(d*x+c))+a
^4*A*ln(sin(d*x+c))/d

________________________________________________________________________________________

Maxima [A]  time = 2.18017, size = 149, normalized size = 1.05 \begin{align*} \frac{2 \, B a^{4} \tan \left (d x + c\right )^{3} +{\left (3 \, A - 12 i \, B\right )} a^{4} \tan \left (d x + c\right )^{2} - 48 \,{\left (d x + c\right )}{\left (-i \, A - B\right )} a^{4} - 6 \,{\left (4 \, A - 4 i \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 6 \, A a^{4} \log \left (\tan \left (d x + c\right )\right ) - 6 \,{\left (4 i \, A + 7 \, B\right )} a^{4} \tan \left (d x + c\right )}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*B*a^4*tan(d*x + c)^3 + (3*A - 12*I*B)*a^4*tan(d*x + c)^2 - 48*(d*x + c)*(-I*A - B)*a^4 - 6*(4*A - 4*I*B
)*a^4*log(tan(d*x + c)^2 + 1) + 6*A*a^4*log(tan(d*x + c)) - 6*(4*I*A + 7*B)*a^4*tan(d*x + c))/d

________________________________________________________________________________________

Fricas [B]  time = 1.48668, size = 672, normalized size = 4.73 \begin{align*} \frac{6 \,{\left (5 \, A - 12 i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 54 \,{\left (A - 2 i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 4 \,{\left (6 \, A - 11 i \, B\right )} a^{4} + 3 \,{\left ({\left (7 \, A - 8 i \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \,{\left (7 \, A - 8 i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \,{\left (7 \, A - 8 i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (7 \, A - 8 i \, B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 3 \,{\left (A a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, A a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, A a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + A a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{3 \,{\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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(6*(5*A - 12*I*B)*a^4*e^(4*I*d*x + 4*I*c) + 54*(A - 2*I*B)*a^4*e^(2*I*d*x + 2*I*c) + 4*(6*A - 11*I*B)*a^4
+ 3*((7*A - 8*I*B)*a^4*e^(6*I*d*x + 6*I*c) + 3*(7*A - 8*I*B)*a^4*e^(4*I*d*x + 4*I*c) + 3*(7*A - 8*I*B)*a^4*e^(
2*I*d*x + 2*I*c) + (7*A - 8*I*B)*a^4)*log(e^(2*I*d*x + 2*I*c) + 1) + 3*(A*a^4*e^(6*I*d*x + 6*I*c) + 3*A*a^4*e^
(4*I*d*x + 4*I*c) + 3*A*a^4*e^(2*I*d*x + 2*I*c) + A*a^4)*log(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)

________________________________________________________________________________________

Sympy [B]  time = 29.7044, size = 262, normalized size = 1.85 \begin{align*} \frac{\frac{\left (10 A a^{4} - 24 i B a^{4}\right ) e^{- 2 i c} e^{4 i d x}}{d} + \frac{\left (18 A a^{4} - 36 i B a^{4}\right ) e^{- 4 i c} e^{2 i d x}}{d} + \frac{\left (24 A a^{4} - 44 i B a^{4}\right ) e^{- 6 i c}}{3 d}}{e^{6 i d x} + 3 e^{- 2 i c} e^{4 i d x} + 3 e^{- 4 i c} e^{2 i d x} + e^{- 6 i c}} + \operatorname{RootSum}{\left (z^{2} d^{2} + z \left (- 8 A a^{4} d + 8 i B a^{4} d\right ) + 7 A^{2} a^{8} - 8 i A B a^{8}, \left ( i \mapsto i \log{\left (\frac{i i d}{3 i A a^{4} e^{2 i c} + 4 B a^{4} e^{2 i c}} - \frac{4 i A + 4 B}{3 i A e^{2 i c} + 4 B e^{2 i c}} + e^{2 i d x} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((10*A*a**4 - 24*I*B*a**4)*exp(-2*I*c)*exp(4*I*d*x)/d + (18*A*a**4 - 36*I*B*a**4)*exp(-4*I*c)*exp(2*I*d*x)/d +
(24*A*a**4 - 44*I*B*a**4)*exp(-6*I*c)/(3*d))/(exp(6*I*d*x) + 3*exp(-2*I*c)*exp(4*I*d*x) + 3*exp(-4*I*c)*exp(2
*I*d*x) + exp(-6*I*c)) + RootSum(_z**2*d**2 + _z*(-8*A*a**4*d + 8*I*B*a**4*d) + 7*A**2*a**8 - 8*I*A*B*a**8, La
mbda(_i, _i*log(_i*I*d/(3*I*A*a**4*exp(2*I*c) + 4*B*a**4*exp(2*I*c)) - (4*I*A + 4*B)/(3*I*A*exp(2*I*c) + 4*B*e
xp(2*I*c)) + exp(2*I*d*x))))

________________________________________________________________________________________

Giac [B]  time = 1.64026, size = 454, normalized size = 3.2 \begin{align*} \frac{6 \, A a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - 12 \,{\left (8 \, A a^{4} - 8 i \, B a^{4}\right )} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right ) + 6 \,{\left (7 \, A a^{4} - 8 i \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) + 6 \,{\left (7 \, A a^{4} - 8 i \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{77 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 88 i \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 48 i \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 84 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 243 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 312 i \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 96 i \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 184 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 243 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 312 i \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 48 i \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 84 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 77 \, A a^{4} + 88 i \, B a^{4}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*A*a^4*log(abs(tan(1/2*d*x + 1/2*c))) - 12*(8*A*a^4 - 8*I*B*a^4)*log(tan(1/2*d*x + 1/2*c) + I) + 6*(7*A*
a^4 - 8*I*B*a^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) + 6*(7*A*a^4 - 8*I*B*a^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1
)) - (77*A*a^4*tan(1/2*d*x + 1/2*c)^6 - 88*I*B*a^4*tan(1/2*d*x + 1/2*c)^6 - 48*I*A*a^4*tan(1/2*d*x + 1/2*c)^5
- 84*B*a^4*tan(1/2*d*x + 1/2*c)^5 - 243*A*a^4*tan(1/2*d*x + 1/2*c)^4 + 312*I*B*a^4*tan(1/2*d*x + 1/2*c)^4 + 96
*I*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 184*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 243*A*a^4*tan(1/2*d*x + 1/2*c)^2 - 312*I*
B*a^4*tan(1/2*d*x + 1/2*c)^2 - 48*I*A*a^4*tan(1/2*d*x + 1/2*c) - 84*B*a^4*tan(1/2*d*x + 1/2*c) - 77*A*a^4 + 88
*I*B*a^4)/(tan(1/2*d*x + 1/2*c)^2 - 1)^3)/d