3.33 \(\int \cot ^5(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=177 \[ \frac{a^4 (64 B+67 i A) \cot (c+d x)}{12 d}+\frac{8 a^4 (A-i B) \log (\sin (c+d x))}{d}-\frac{(4 B+7 i A) \cot ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{12 d}+\frac{(19 A-16 i B) \cot ^2(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{12 d}+8 a^4 x (B+i A)-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^3}{4 d} \]

[Out]

8*a^4*(I*A + B)*x + (a^4*((67*I)*A + 64*B)*Cot[c + d*x])/(12*d) + (8*a^4*(A - I*B)*Log[Sin[c + d*x]])/d - (a*A
*Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^3)/(4*d) - (((7*I)*A + 4*B)*Cot[c + d*x]^3*(a^2 + I*a^2*Tan[c + d*x])^2
)/(12*d) + ((19*A - (16*I)*B)*Cot[c + d*x]^2*(a^4 + I*a^4*Tan[c + d*x]))/(12*d)

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Rubi [A]  time = 0.531504, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3593, 3591, 3531, 3475} \[ \frac{a^4 (64 B+67 i A) \cot (c+d x)}{12 d}+\frac{8 a^4 (A-i B) \log (\sin (c+d x))}{d}-\frac{(4 B+7 i A) \cot ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{12 d}+\frac{(19 A-16 i B) \cot ^2(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{12 d}+8 a^4 x (B+i A)-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^3}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^5*(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*((67*I)*A + 64*B)*Cot[c + d*x])/(12*d) + (8*a^4*(A - I*B)*Log[Sin[c + d*x]])/d - (a*A
*Cot[c + d*x]^4*(a + I*a*Tan[c + d*x])^3)/(4*d) - (((7*I)*A + 4*B)*Cot[c + d*x]^3*(a^2 + I*a^2*Tan[c + d*x])^2
)/(12*d) + ((19*A - (16*I)*B)*Cot[c + d*x]^2*(a^4 + I*a^4*Tan[c + d*x]))/(12*d)

Rule 3593

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] && EqQ
[a^2 + b^2, 0] && GtQ[m, 1] && LtQ[n, -1]

Rule 3591

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

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]

Rule 3475

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

Rubi steps

\begin{align*} \int \cot ^5(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx &=-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^3}{4 d}+\frac{1}{4} \int \cot ^4(c+d x) (a+i a \tan (c+d x))^3 (a (7 i A+4 B)-a (A-4 i B) \tan (c+d x)) \, dx\\ &=-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^3}{4 d}-\frac{(7 i A+4 B) \cot ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{12 d}+\frac{1}{12} \int \cot ^3(c+d x) (a+i a \tan (c+d x))^2 \left (-2 a^2 (19 A-16 i B)-2 a^2 (5 i A+8 B) \tan (c+d x)\right ) \, dx\\ &=-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^3}{4 d}-\frac{(7 i A+4 B) \cot ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{12 d}+\frac{(19 A-16 i B) \cot ^2(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{12 d}+\frac{1}{24} \int \cot ^2(c+d x) (a+i a \tan (c+d x)) \left (-2 a^3 (67 i A+64 B)+2 a^3 (29 A-32 i B) \tan (c+d x)\right ) \, dx\\ &=\frac{a^4 (67 i A+64 B) \cot (c+d x)}{12 d}-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^3}{4 d}-\frac{(7 i A+4 B) \cot ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{12 d}+\frac{(19 A-16 i B) \cot ^2(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{12 d}+\frac{1}{24} \int \cot (c+d x) \left (192 a^4 (A-i B)+192 a^4 (i A+B) \tan (c+d x)\right ) \, dx\\ &=8 a^4 (i A+B) x+\frac{a^4 (67 i A+64 B) \cot (c+d x)}{12 d}-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^3}{4 d}-\frac{(7 i A+4 B) \cot ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{12 d}+\frac{(19 A-16 i B) \cot ^2(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{12 d}+\left (8 a^4 (A-i B)\right ) \int \cot (c+d x) \, dx\\ &=8 a^4 (i A+B) x+\frac{a^4 (67 i A+64 B) \cot (c+d x)}{12 d}+\frac{8 a^4 (A-i B) \log (\sin (c+d x))}{d}-\frac{a A \cot ^4(c+d x) (a+i a \tan (c+d x))^3}{4 d}-\frac{(7 i A+4 B) \cot ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{12 d}+\frac{(19 A-16 i B) \cot ^2(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{12 d}\\ \end{align*}

Mathematica [A]  time = 5.82022, size = 319, normalized size = 1.8 \[ \frac{a^4 \sin (c+d x) (\cot (c+d x)+i)^4 (A \cot (c+d x)+B) \left (192 d x (A-i B) (\sin (4 c)+i \cos (4 c)) \sin ^4(c+d x)+48 (A-i B) (\cos (4 c)-i \sin (4 c)) \sin ^4(c+d x) \log \left (\sin ^2(c+d x)\right )-96 i (A-i B) (\cos (4 c)-i \sin (4 c)) \sin ^4(c+d x) \tan ^{-1}(\tan (5 c+d x))+4 (\cos (4 c)-i \sin (4 c)) ((-B-4 i A) \cot (c)+12 A-6 i B) \sin ^2(c+d x)-8 i (14 A-11 i B) \csc (c) (\cos (4 c)-i \sin (4 c)) \sin (d x) \sin ^3(c+d x)+4 (B+4 i A) \csc (c) (\cos (4 c)-i \sin (4 c)) \sin (d x) \sin (c+d x)+3 i A \sin (4 c)-3 A \cos (4 c)\right )}{12 d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(I + Cot[c + d*x])^4*(B + A*Cot[c + d*x])*Sin[c + d*x]*(-3*A*Cos[4*c] + (3*I)*A*Sin[4*c] + 4*((4*I)*A + B
)*Csc[c]*(Cos[4*c] - I*Sin[4*c])*Sin[d*x]*Sin[c + d*x] + 4*(12*A - (6*I)*B + ((-4*I)*A - B)*Cot[c])*(Cos[4*c]
- I*Sin[4*c])*Sin[c + d*x]^2 - (8*I)*(14*A - (11*I)*B)*Csc[c]*(Cos[4*c] - I*Sin[4*c])*Sin[d*x]*Sin[c + d*x]^3
- (96*I)*(A - I*B)*ArcTan[Tan[5*c + d*x]]*(Cos[4*c] - I*Sin[4*c])*Sin[c + d*x]^4 + 48*(A - I*B)*Log[Sin[c + d*
x]^2]*(Cos[4*c] - I*Sin[4*c])*Sin[c + d*x]^4 + 192*(A - I*B)*d*x*(I*Cos[4*c] + Sin[4*c])*Sin[c + d*x]^4))/(12*
d*(Cos[d*x] + I*Sin[d*x])^4*(A*Cos[c + d*x] + B*Sin[c + d*x]))

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Maple [A]  time = 0.079, size = 189, normalized size = 1.1 \begin{align*} 8\,{\frac{A{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{A{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{B{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{7\,A{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+8\,{\frac{B{a}^{4}c}{d}}+7\,{\frac{\cot \left ( dx+c \right ) B{a}^{4}}{d}}-{\frac{2\,iB{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}+{\frac{8\,iA\cot \left ( dx+c \right ){a}^{4}}{d}}+{\frac{8\,iA{a}^{4}c}{d}}-{\frac{8\,iB{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+8\,iAx{a}^{4}+8\,B{a}^{4}x-{\frac{{\frac{4\,i}{3}}A{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.10027, size = 188, normalized size = 1.06 \begin{align*} -\frac{96 \,{\left (d x + c\right )}{\left (-i \, A - B\right )} a^{4} + 12 \,{\left (4 \, A - 4 i \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 12 \,{\left (8 \, A - 8 i \, B\right )} a^{4} \log \left (\tan \left (d x + c\right )\right ) - \frac{12 \,{\left (8 i \, A + 7 \, B\right )} a^{4} \tan \left (d x + c\right )^{3} +{\left (42 \, A - 24 i \, B\right )} a^{4} \tan \left (d x + c\right )^{2} + 4 \,{\left (-4 i \, A - B\right )} a^{4} \tan \left (d x + c\right ) - 3 \, A a^{4}}{\tan \left (d x + c\right )^{4}}}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(96*(d*x + c)*(-I*A - B)*a^4 + 12*(4*A - 4*I*B)*a^4*log(tan(d*x + c)^2 + 1) - 12*(8*A - 8*I*B)*a^4*log(t
an(d*x + c)) - (12*(8*I*A + 7*B)*a^4*tan(d*x + c)^3 + (42*A - 24*I*B)*a^4*tan(d*x + c)^2 + 4*(-4*I*A - B)*a^4*
tan(d*x + c) - 3*A*a^4)/tan(d*x + c)^4)/d

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Fricas [A]  time = 1.42994, size = 622, normalized size = 3.51 \begin{align*} -\frac{4 \,{\left (6 \,{\left (5 \, A - 3 i \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 9 \,{\left (7 \, A - 5 i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \,{\left (25 \, A - 19 i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} -{\left (14 \, A - 11 i \, B\right )} a^{4} - 6 \,{\left ({\left (A - i \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 4 \,{\left (A - i \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \,{\left (A - i \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 4 \,{\left (A - i \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (A - i \, B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )\right )}}{3 \,{\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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(6*(5*A - 3*I*B)*a^4*e^(6*I*d*x + 6*I*c) - 9*(7*A - 5*I*B)*a^4*e^(4*I*d*x + 4*I*c) + 2*(25*A - 19*I*B)*a^
4*e^(2*I*d*x + 2*I*c) - (14*A - 11*I*B)*a^4 - 6*((A - I*B)*a^4*e^(8*I*d*x + 8*I*c) - 4*(A - I*B)*a^4*e^(6*I*d*
x + 6*I*c) + 6*(A - I*B)*a^4*e^(4*I*d*x + 4*I*c) - 4*(A - I*B)*a^4*e^(2*I*d*x + 2*I*c) + (A - I*B)*a^4)*log(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 [A]  time = 20.1324, size = 221, normalized size = 1.25 \begin{align*} \frac{8 a^{4} \left (A - i B\right ) \log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac{- \frac{\left (40 A a^{4} - 24 i B a^{4}\right ) e^{- 2 i c} e^{6 i d x}}{d} + \frac{\left (56 A a^{4} - 44 i B a^{4}\right ) e^{- 8 i c}}{3 d} + \frac{\left (84 A a^{4} - 60 i B a^{4}\right ) e^{- 4 i c} e^{4 i d x}}{d} - \frac{\left (200 A a^{4} - 152 i B a^{4}\right ) e^{- 6 i c} e^{2 i d x}}{3 d}}{e^{8 i d x} - 4 e^{- 2 i c} e^{6 i d x} + 6 e^{- 4 i c} e^{4 i d x} - 4 e^{- 6 i c} e^{2 i d x} + e^{- 8 i c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8*a**4*(A - I*B)*log(exp(2*I*d*x) - exp(-2*I*c))/d + (-(40*A*a**4 - 24*I*B*a**4)*exp(-2*I*c)*exp(6*I*d*x)/d +
(56*A*a**4 - 44*I*B*a**4)*exp(-8*I*c)/(3*d) + (84*A*a**4 - 60*I*B*a**4)*exp(-4*I*c)*exp(4*I*d*x)/d - (200*A*a*
*4 - 152*I*B*a**4)*exp(-6*I*c)*exp(2*I*d*x)/(3*d))/(exp(8*I*d*x) - 4*exp(-2*I*c)*exp(6*I*d*x) + 6*exp(-4*I*c)*
exp(4*I*d*x) - 4*exp(-6*I*c)*exp(2*I*d*x) + exp(-8*I*c))

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Giac [B]  time = 1.89542, size = 439, normalized size = 2.48 \begin{align*} -\frac{3 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 32 i \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 8 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 180 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 96 i \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 864 i \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 696 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 384 \,{\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 ) - 384 \,{\left (4 \, A a^{4} - 4 i \, B a^{4}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + \frac{3200 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 3200 i \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 864 i \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 696 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 180 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 96 i \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 32 i \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 8 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, A a^{4}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4}}}{192 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(3*A*a^4*tan(1/2*d*x + 1/2*c)^4 - 32*I*A*a^4*tan(1/2*d*x + 1/2*c)^3 - 8*B*a^4*tan(1/2*d*x + 1/2*c)^3 -
180*A*a^4*tan(1/2*d*x + 1/2*c)^2 + 96*I*B*a^4*tan(1/2*d*x + 1/2*c)^2 + 864*I*A*a^4*tan(1/2*d*x + 1/2*c) + 696*
B*a^4*tan(1/2*d*x + 1/2*c) + 384*(8*A*a^4 - 8*I*B*a^4)*log(tan(1/2*d*x + 1/2*c) + I) - 384*(4*A*a^4 - 4*I*B*a^
4)*log(abs(tan(1/2*d*x + 1/2*c))) + (3200*A*a^4*tan(1/2*d*x + 1/2*c)^4 - 3200*I*B*a^4*tan(1/2*d*x + 1/2*c)^4 -
864*I*A*a^4*tan(1/2*d*x + 1/2*c)^3 - 696*B*a^4*tan(1/2*d*x + 1/2*c)^3 - 180*A*a^4*tan(1/2*d*x + 1/2*c)^2 + 96
*I*B*a^4*tan(1/2*d*x + 1/2*c)^2 + 32*I*A*a^4*tan(1/2*d*x + 1/2*c) + 8*B*a^4*tan(1/2*d*x + 1/2*c) + 3*A*a^4)/ta
n(1/2*d*x + 1/2*c)^4)/d