3.259 \(\int (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=202 \[ \frac{\left (a^2 B+2 a A b-b^2 B\right ) (a+b \tan (c+d x))^2}{2 d}+\frac{b \left (3 a^2 A b+a^3 B-3 a b^2 B-A b^3\right ) \tan (c+d x)}{d}-\frac{\left (4 a^3 A b-6 a^2 b^2 B+a^4 B-4 a A b^3+b^4 B\right ) \log (\cos (c+d x))}{d}+x \left (-6 a^2 A b^2+a^4 A-4 a^3 b B+4 a b^3 B+A b^4\right )+\frac{(a B+A b) (a+b \tan (c+d x))^3}{3 d}+\frac{B (a+b \tan (c+d x))^4}{4 d} \]

[Out]

(a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*x - ((4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*
B)*Log[Cos[c + d*x]])/d + (b*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Tan[c + d*x])/d + ((2*a*A*b + a^2*B - b^2
*B)*(a + b*Tan[c + d*x])^2)/(2*d) + ((A*b + a*B)*(a + b*Tan[c + d*x])^3)/(3*d) + (B*(a + b*Tan[c + d*x])^4)/(4
*d)

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Rubi [A]  time = 0.230063, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3528, 3525, 3475} \[ \frac{\left (a^2 B+2 a A b-b^2 B\right ) (a+b \tan (c+d x))^2}{2 d}+\frac{b \left (3 a^2 A b+a^3 B-3 a b^2 B-A b^3\right ) \tan (c+d x)}{d}-\frac{\left (4 a^3 A b-6 a^2 b^2 B+a^4 B-4 a A b^3+b^4 B\right ) \log (\cos (c+d x))}{d}+x \left (-6 a^2 A b^2+a^4 A-4 a^3 b B+4 a b^3 B+A b^4\right )+\frac{(a B+A b) (a+b \tan (c+d x))^3}{3 d}+\frac{B (a+b \tan (c+d x))^4}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*A - 6*a^2*A*b^2 + A*b^4 - 4*a^3*b*B + 4*a*b^3*B)*x - ((4*a^3*A*b - 4*a*A*b^3 + a^4*B - 6*a^2*b^2*B + b^4*
B)*Log[Cos[c + d*x]])/d + (b*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Tan[c + d*x])/d + ((2*a*A*b + a^2*B - b^2
*B)*(a + b*Tan[c + d*x])^2)/(2*d) + ((A*b + a*B)*(a + b*Tan[c + d*x])^3)/(3*d) + (B*(a + b*Tan[c + d*x])^4)/(4
*d)

Rule 3528

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(d
*(a + b*Tan[e + f*x])^m)/(f*m), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*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] && GtQ[m, 0]

Rule 3525

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

Rubi steps

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

Mathematica [C]  time = 3.45446, size = 240, normalized size = 1.19 \[ \frac{B \left (-6 b^3 \left (b^2-10 a^2\right ) \tan ^2(c+d x)+60 a b^2 \left (2 a^2-b^2\right ) \tan (c+d x)+20 a b^4 \tan ^3(c+d x)+6 (b-i a)^5 \log (-\tan (c+d x)+i)+6 (b+i a)^5 \log (\tan (c+d x)+i)+3 b^5 \tan ^4(c+d x)\right )-2 (A b-a B) \left (6 b^2 \left (b^2-6 a^2\right ) \tan (c+d x)-12 a b^3 \tan ^2(c+d x)-3 i (a-i b)^4 \log (\tan (c+d x)+i)+3 i (a+i b)^4 \log (-\tan (c+d x)+i)-2 b^4 \tan ^3(c+d x)\right )}{12 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(A*b - a*B)*((3*I)*(a + I*b)^4*Log[I - Tan[c + d*x]] - (3*I)*(a - I*b)^4*Log[I + Tan[c + d*x]] + 6*b^2*(-6
*a^2 + b^2)*Tan[c + d*x] - 12*a*b^3*Tan[c + d*x]^2 - 2*b^4*Tan[c + d*x]^3) + B*(6*((-I)*a + b)^5*Log[I - Tan[c
+ d*x]] + 6*(I*a + b)^5*Log[I + Tan[c + d*x]] + 60*a*b^2*(2*a^2 - b^2)*Tan[c + d*x] - 6*b^3*(-10*a^2 + b^2)*T
an[c + d*x]^2 + 20*a*b^4*Tan[c + d*x]^3 + 3*b^5*Tan[c + d*x]^4))/(12*b*d)

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Maple [A]  time = 0.011, size = 362, normalized size = 1.8 \begin{align*}{\frac{B{b}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{A \left ( \tan \left ( dx+c \right ) \right ) ^{3}{b}^{4}}{3\,d}}+{\frac{4\,B \left ( \tan \left ( dx+c \right ) \right ) ^{3}a{b}^{3}}{3\,d}}+2\,{\frac{A \left ( \tan \left ( dx+c \right ) \right ) ^{2}a{b}^{3}}{d}}+3\,{\frac{B \left ( \tan \left ( dx+c \right ) \right ) ^{2}{a}^{2}{b}^{2}}{d}}-{\frac{B \left ( \tan \left ( dx+c \right ) \right ) ^{2}{b}^{4}}{2\,d}}+6\,{\frac{A{a}^{2}{b}^{2}\tan \left ( dx+c \right ) }{d}}-{\frac{A{b}^{4}\tan \left ( dx+c \right ) }{d}}+4\,{\frac{B{a}^{3}b\tan \left ( dx+c \right ) }{d}}-4\,{\frac{Ba{b}^{3}\tan \left ( dx+c \right ) }{d}}+2\,{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) A{a}^{3}b}{d}}-2\,{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) Aa{b}^{3}}{d}}+{\frac{B{a}^{4}\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}}-3\,{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) B{a}^{2}{b}^{2}}{d}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) B{b}^{4}}{2\,d}}+{\frac{A{a}^{4}\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}}-6\,{\frac{A\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{2}{b}^{2}}{d}}+{\frac{A\arctan \left ( \tan \left ( dx+c \right ) \right ){b}^{4}}{d}}-4\,{\frac{B\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{3}b}{d}}+4\,{\frac{B\arctan \left ( \tan \left ( dx+c \right ) \right ) a{b}^{3}}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/d*B*b^4*tan(d*x+c)^4+1/3/d*A*tan(d*x+c)^3*b^4+4/3/d*B*tan(d*x+c)^3*a*b^3+2/d*A*tan(d*x+c)^2*a*b^3+3/d*B*ta
n(d*x+c)^2*a^2*b^2-1/2/d*B*tan(d*x+c)^2*b^4+6/d*A*a^2*b^2*tan(d*x+c)-1/d*A*b^4*tan(d*x+c)+4/d*B*a^3*b*tan(d*x+
c)-4/d*B*a*b^3*tan(d*x+c)+2/d*ln(1+tan(d*x+c)^2)*A*a^3*b-2/d*ln(1+tan(d*x+c)^2)*A*a*b^3+1/2/d*a^4*B*ln(1+tan(d
*x+c)^2)-3/d*ln(1+tan(d*x+c)^2)*B*a^2*b^2+1/2/d*ln(1+tan(d*x+c)^2)*B*b^4+1/d*a^4*A*arctan(tan(d*x+c))-6/d*A*ar
ctan(tan(d*x+c))*a^2*b^2+1/d*A*arctan(tan(d*x+c))*b^4-4/d*B*arctan(tan(d*x+c))*a^3*b+4/d*B*arctan(tan(d*x+c))*
a*b^3

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Maxima [A]  time = 1.47115, size = 273, normalized size = 1.35 \begin{align*} \frac{3 \, B b^{4} \tan \left (d x + c\right )^{4} + 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )^{3} + 6 \,{\left (6 \, B a^{2} b^{2} + 4 \, A a b^{3} - B b^{4}\right )} \tan \left (d x + c\right )^{2} + 12 \,{\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )}{\left (d x + c\right )} + 6 \,{\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \,{\left (4 \, B a^{3} b + 6 \, A a^{2} b^{2} - 4 \, B a b^{3} - A b^{4}\right )} \tan \left (d x + c\right )}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.08274, size = 456, normalized size = 2.26 \begin{align*} \frac{3 \, B b^{4} \tan \left (d x + c\right )^{4} + 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )^{3} + 12 \,{\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} d x + 6 \,{\left (6 \, B a^{2} b^{2} + 4 \, A a b^{3} - B b^{4}\right )} \tan \left (d x + c\right )^{2} - 6 \,{\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 12 \,{\left (4 \, B a^{3} b + 6 \, A a^{2} b^{2} - 4 \, B a b^{3} - A b^{4}\right )} \tan \left (d x + c\right )}{12 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.885045, size = 347, normalized size = 1.72 \begin{align*} \begin{cases} A a^{4} x + \frac{2 A a^{3} b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - 6 A a^{2} b^{2} x + \frac{6 A a^{2} b^{2} \tan{\left (c + d x \right )}}{d} - \frac{2 A a b^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac{2 A a b^{3} \tan ^{2}{\left (c + d x \right )}}{d} + A b^{4} x + \frac{A b^{4} \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac{A b^{4} \tan{\left (c + d x \right )}}{d} + \frac{B a^{4} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - 4 B a^{3} b x + \frac{4 B a^{3} b \tan{\left (c + d x \right )}}{d} - \frac{3 B a^{2} b^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac{3 B a^{2} b^{2} \tan ^{2}{\left (c + d x \right )}}{d} + 4 B a b^{3} x + \frac{4 B a b^{3} \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac{4 B a b^{3} \tan{\left (c + d x \right )}}{d} + \frac{B b^{4} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{B b^{4} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac{B b^{4} \tan ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (A + B \tan{\left (c \right )}\right ) \left (a + b \tan{\left (c \right )}\right )^{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((A*a**4*x + 2*A*a**3*b*log(tan(c + d*x)**2 + 1)/d - 6*A*a**2*b**2*x + 6*A*a**2*b**2*tan(c + d*x)/d -
2*A*a*b**3*log(tan(c + d*x)**2 + 1)/d + 2*A*a*b**3*tan(c + d*x)**2/d + A*b**4*x + A*b**4*tan(c + d*x)**3/(3*d
) - A*b**4*tan(c + d*x)/d + B*a**4*log(tan(c + d*x)**2 + 1)/(2*d) - 4*B*a**3*b*x + 4*B*a**3*b*tan(c + d*x)/d -
3*B*a**2*b**2*log(tan(c + d*x)**2 + 1)/d + 3*B*a**2*b**2*tan(c + d*x)**2/d + 4*B*a*b**3*x + 4*B*a*b**3*tan(c
+ d*x)**3/(3*d) - 4*B*a*b**3*tan(c + d*x)/d + B*b**4*log(tan(c + d*x)**2 + 1)/(2*d) + B*b**4*tan(c + d*x)**4/(
4*d) - B*b**4*tan(c + d*x)**2/(2*d), Ne(d, 0)), (x*(A + B*tan(c))*(a + b*tan(c))**4, True))

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Giac [B]  time = 7.05837, size = 4567, normalized size = 22.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(12*A*a^4*d*x*tan(d*x)^4*tan(c)^4 - 48*B*a^3*b*d*x*tan(d*x)^4*tan(c)^4 - 72*A*a^2*b^2*d*x*tan(d*x)^4*tan(
c)^4 + 48*B*a*b^3*d*x*tan(d*x)^4*tan(c)^4 + 12*A*b^4*d*x*tan(d*x)^4*tan(c)^4 - 6*B*a^4*log(4*(tan(c)^2 + 1)/(t
an(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)
^4*tan(c)^4 - 24*A*a^3*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2
+ tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^4*tan(c)^4 + 36*B*a^2*b^2*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*ta
n(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^4*tan(c)^4
+ 24*A*a*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^
2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^4*tan(c)^4 - 6*B*b^4*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*
x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^4*tan(c)^4 - 48*A*a^4*d*x*ta
n(d*x)^3*tan(c)^3 + 192*B*a^3*b*d*x*tan(d*x)^3*tan(c)^3 + 288*A*a^2*b^2*d*x*tan(d*x)^3*tan(c)^3 - 192*B*a*b^3*
d*x*tan(d*x)^3*tan(c)^3 - 48*A*b^4*d*x*tan(d*x)^3*tan(c)^3 + 36*B*a^2*b^2*tan(d*x)^4*tan(c)^4 + 24*A*a*b^3*tan
(d*x)^4*tan(c)^4 - 9*B*b^4*tan(d*x)^4*tan(c)^4 + 24*B*a^4*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*
x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^3*tan(c)^3 + 96*A*a^3*b*log(
4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*ta
n(c) + 1))*tan(d*x)^3*tan(c)^3 - 144*B*a^2*b^2*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c)
+ tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^3*tan(c)^3 - 96*A*a*b^3*log(4*(tan(c)^2
+ 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*
tan(d*x)^3*tan(c)^3 + 24*B*b^4*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*ta
n(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^3*tan(c)^3 - 48*B*a^3*b*tan(d*x)^4*tan(c)^3 - 72*A*a^2*
b^2*tan(d*x)^4*tan(c)^3 + 48*B*a*b^3*tan(d*x)^4*tan(c)^3 + 12*A*b^4*tan(d*x)^4*tan(c)^3 - 48*B*a^3*b*tan(d*x)^
3*tan(c)^4 - 72*A*a^2*b^2*tan(d*x)^3*tan(c)^4 + 48*B*a*b^3*tan(d*x)^3*tan(c)^4 + 12*A*b^4*tan(d*x)^3*tan(c)^4
+ 72*A*a^4*d*x*tan(d*x)^2*tan(c)^2 - 288*B*a^3*b*d*x*tan(d*x)^2*tan(c)^2 - 432*A*a^2*b^2*d*x*tan(d*x)^2*tan(c)
^2 + 288*B*a*b^3*d*x*tan(d*x)^2*tan(c)^2 + 72*A*b^4*d*x*tan(d*x)^2*tan(c)^2 + 36*B*a^2*b^2*tan(d*x)^4*tan(c)^2
+ 24*A*a*b^3*tan(d*x)^4*tan(c)^2 - 6*B*b^4*tan(d*x)^4*tan(c)^2 - 72*B*a^2*b^2*tan(d*x)^3*tan(c)^3 - 48*A*a*b^
3*tan(d*x)^3*tan(c)^3 + 24*B*b^4*tan(d*x)^3*tan(c)^3 + 36*B*a^2*b^2*tan(d*x)^2*tan(c)^4 + 24*A*a*b^3*tan(d*x)^
2*tan(c)^4 - 6*B*b^4*tan(d*x)^2*tan(c)^4 - 16*B*a*b^3*tan(d*x)^4*tan(c) - 4*A*b^4*tan(d*x)^4*tan(c) - 36*B*a^4
*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*
x)*tan(c) + 1))*tan(d*x)^2*tan(c)^2 - 144*A*a^3*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan
(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^2*tan(c)^2 + 216*B*a^2*b^2*log(4*(ta
n(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c)
+ 1))*tan(d*x)^2*tan(c)^2 + 144*A*a*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(
d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^2*tan(c)^2 - 36*B*b^4*log(4*(tan(c)^2 + 1)/(ta
n(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^
2*tan(c)^2 + 144*B*a^3*b*tan(d*x)^3*tan(c)^2 + 216*A*a^2*b^2*tan(d*x)^3*tan(c)^2 - 192*B*a*b^3*tan(d*x)^3*tan(
c)^2 - 48*A*b^4*tan(d*x)^3*tan(c)^2 + 144*B*a^3*b*tan(d*x)^2*tan(c)^3 + 216*A*a^2*b^2*tan(d*x)^2*tan(c)^3 - 19
2*B*a*b^3*tan(d*x)^2*tan(c)^3 - 48*A*b^4*tan(d*x)^2*tan(c)^3 - 16*B*a*b^3*tan(d*x)*tan(c)^4 - 4*A*b^4*tan(d*x)
*tan(c)^4 + 3*B*b^4*tan(d*x)^4 - 48*A*a^4*d*x*tan(d*x)*tan(c) + 192*B*a^3*b*d*x*tan(d*x)*tan(c) + 288*A*a^2*b^
2*d*x*tan(d*x)*tan(c) - 192*B*a*b^3*d*x*tan(d*x)*tan(c) - 48*A*b^4*d*x*tan(d*x)*tan(c) - 72*B*a^2*b^2*tan(d*x)
^3*tan(c) - 48*A*a*b^3*tan(d*x)^3*tan(c) + 24*B*b^4*tan(d*x)^3*tan(c) + 72*B*a^2*b^2*tan(d*x)^2*tan(c)^2 + 48*
A*a*b^3*tan(d*x)^2*tan(c)^2 - 12*B*b^4*tan(d*x)^2*tan(c)^2 - 72*B*a^2*b^2*tan(d*x)*tan(c)^3 - 48*A*a*b^3*tan(d
*x)*tan(c)^3 + 24*B*b^4*tan(d*x)*tan(c)^3 + 3*B*b^4*tan(c)^4 + 16*B*a*b^3*tan(d*x)^3 + 4*A*b^4*tan(d*x)^3 + 24
*B*a^4*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*
tan(d*x)*tan(c) + 1))*tan(d*x)*tan(c) + 96*A*a^3*b*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*ta
n(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)*tan(c) - 144*B*a^2*b^2*log(4*(tan(c
)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1
))*tan(d*x)*tan(c) - 96*A*a*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*t
an(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)*tan(c) + 24*B*b^4*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan
(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)*tan(c) - 144
*B*a^3*b*tan(d*x)^2*tan(c) - 216*A*a^2*b^2*tan(d*x)^2*tan(c) + 192*B*a*b^3*tan(d*x)^2*tan(c) + 48*A*b^4*tan(d*
x)^2*tan(c) - 144*B*a^3*b*tan(d*x)*tan(c)^2 - 216*A*a^2*b^2*tan(d*x)*tan(c)^2 + 192*B*a*b^3*tan(d*x)*tan(c)^2
+ 48*A*b^4*tan(d*x)*tan(c)^2 + 16*B*a*b^3*tan(c)^3 + 4*A*b^4*tan(c)^3 + 12*A*a^4*d*x - 48*B*a^3*b*d*x - 72*A*a
^2*b^2*d*x + 48*B*a*b^3*d*x + 12*A*b^4*d*x + 36*B*a^2*b^2*tan(d*x)^2 + 24*A*a*b^3*tan(d*x)^2 - 6*B*b^4*tan(d*x
)^2 - 72*B*a^2*b^2*tan(d*x)*tan(c) - 48*A*a*b^3*tan(d*x)*tan(c) + 24*B*b^4*tan(d*x)*tan(c) + 36*B*a^2*b^2*tan(
c)^2 + 24*A*a*b^3*tan(c)^2 - 6*B*b^4*tan(c)^2 - 6*B*a^4*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)
^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)) - 24*A*a^3*b*log(4*(tan(c)^2 + 1)/(tan(
d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)) + 36*B*a^2*
b^2*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan
(d*x)*tan(c) + 1)) + 24*A*a*b^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*tan(c) + tan(d*x)^2*t
an(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)) - 6*B*b^4*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x
)^3*tan(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1)) + 48*B*a^3*b*tan(d*x) + 72*A*a^2*b^2*t
an(d*x) - 48*B*a*b^3*tan(d*x) - 12*A*b^4*tan(d*x) + 48*B*a^3*b*tan(c) + 72*A*a^2*b^2*tan(c) - 48*B*a*b^3*tan(c
) - 12*A*b^4*tan(c) + 36*B*a^2*b^2 + 24*A*a*b^3 - 9*B*b^4)/(d*tan(d*x)^4*tan(c)^4 - 4*d*tan(d*x)^3*tan(c)^3 +
6*d*tan(d*x)^2*tan(c)^2 - 4*d*tan(d*x)*tan(c) + d)