3.248 \(\int \tan ^2(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx\)
Optimal. Leaf size=201 \[ -\frac{b \left (a^2 B+2 a A b-b^2 B\right ) \tan (c+d x)}{d}+\frac{\left (3 a^2 A b+a^3 B-3 a b^2 B-A b^3\right ) \log (\cos (c+d x))}{d}-x \left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right )+\frac{(5 A b-a B) (a+b \tan (c+d x))^4}{20 b^2 d}-\frac{(a B+A b) (a+b \tan (c+d x))^2}{2 d}+\frac{B \tan (c+d x) (a+b \tan (c+d x))^4}{5 b d}-\frac{B (a+b \tan (c+d x))^3}{3 d} \]
[Out]
-((a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*x) + ((3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Log[Cos[c + d*x]])/d -
(b*(2*a*A*b + a^2*B - b^2*B)*Tan[c + d*x])/d - ((A*b + a*B)*(a + b*Tan[c + d*x])^2)/(2*d) - (B*(a + b*Tan[c +
d*x])^3)/(3*d) + ((5*A*b - a*B)*(a + b*Tan[c + d*x])^4)/(20*b^2*d) + (B*Tan[c + d*x]*(a + b*Tan[c + d*x])^4)/
(5*b*d)
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Rubi [A] time = 0.369349, antiderivative size = 201, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used =
{3607, 3630, 3528, 3525, 3475} \[ -\frac{b \left (a^2 B+2 a A b-b^2 B\right ) \tan (c+d x)}{d}+\frac{\left (3 a^2 A b+a^3 B-3 a b^2 B-A b^3\right ) \log (\cos (c+d x))}{d}-x \left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right )+\frac{(5 A b-a B) (a+b \tan (c+d x))^4}{20 b^2 d}-\frac{(a B+A b) (a+b \tan (c+d x))^2}{2 d}+\frac{B \tan (c+d x) (a+b \tan (c+d x))^4}{5 b d}-\frac{B (a+b \tan (c+d x))^3}{3 d} \]
Antiderivative was successfully verified.
[In]
Int[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]
[Out]
-((a^3*A - 3*a*A*b^2 - 3*a^2*b*B + b^3*B)*x) + ((3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Log[Cos[c + d*x]])/d -
(b*(2*a*A*b + a^2*B - b^2*B)*Tan[c + d*x])/d - ((A*b + a*B)*(a + b*Tan[c + d*x])^2)/(2*d) - (B*(a + b*Tan[c +
d*x])^3)/(3*d) + ((5*A*b - a*B)*(a + b*Tan[c + d*x])^4)/(20*b^2*d) + (B*Tan[c + d*x]*(a + b*Tan[c + d*x])^4)/
(5*b*d)
Rule 3607
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 - 2)*(c + d*Tan[e + f*x])^n*Simp[a^2*A*d*(m +
n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m
- 1) - b*(A*b + a*B)*d*(m + n))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&
!(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Rule 3630
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(C*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Int[(a + b*Tan[e + f*x])
^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0]
&& !LeQ[m, -1]
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 \tan ^2(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx &=\frac{B \tan (c+d x) (a+b \tan (c+d x))^4}{5 b d}+\frac{\int (a+b \tan (c+d x))^3 \left (-a B-5 b B \tan (c+d x)+(5 A b-a B) \tan ^2(c+d x)\right ) \, dx}{5 b}\\ &=\frac{(5 A b-a B) (a+b \tan (c+d x))^4}{20 b^2 d}+\frac{B \tan (c+d x) (a+b \tan (c+d x))^4}{5 b d}+\frac{\int (a+b \tan (c+d x))^3 (-5 A b-5 b B \tan (c+d x)) \, dx}{5 b}\\ &=-\frac{B (a+b \tan (c+d x))^3}{3 d}+\frac{(5 A b-a B) (a+b \tan (c+d x))^4}{20 b^2 d}+\frac{B \tan (c+d x) (a+b \tan (c+d x))^4}{5 b d}+\frac{\int (a+b \tan (c+d x))^2 (-5 b (a A-b B)-5 b (A b+a B) \tan (c+d x)) \, dx}{5 b}\\ &=-\frac{(A b+a B) (a+b \tan (c+d x))^2}{2 d}-\frac{B (a+b \tan (c+d x))^3}{3 d}+\frac{(5 A b-a B) (a+b \tan (c+d x))^4}{20 b^2 d}+\frac{B \tan (c+d x) (a+b \tan (c+d x))^4}{5 b d}+\frac{\int (a+b \tan (c+d x)) \left (-5 b \left (a^2 A-A b^2-2 a b B\right )-5 b \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)\right ) \, dx}{5 b}\\ &=-\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) x-\frac{b \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)}{d}-\frac{(A b+a B) (a+b \tan (c+d x))^2}{2 d}-\frac{B (a+b \tan (c+d x))^3}{3 d}+\frac{(5 A b-a B) (a+b \tan (c+d x))^4}{20 b^2 d}+\frac{B \tan (c+d x) (a+b \tan (c+d x))^4}{5 b d}+\left (-3 a^2 A b+A b^3-a^3 B+3 a b^2 B\right ) \int \tan (c+d x) \, dx\\ &=-\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) x+\frac{\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \log (\cos (c+d x))}{d}-\frac{b \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)}{d}-\frac{(A b+a B) (a+b \tan (c+d x))^2}{2 d}-\frac{B (a+b \tan (c+d x))^3}{3 d}+\frac{(5 A b-a B) (a+b \tan (c+d x))^4}{20 b^2 d}+\frac{B \tan (c+d x) (a+b \tan (c+d x))^4}{5 b d}\\ \end{align*}
Mathematica [C] time = 2.06747, size = 241, normalized size = 1.2 \[ \frac{10 B \left (6 b^2 \left (b^2-6 a^2\right ) \tan (c+d x)-12 a b^3 \tan ^2(c+d x)-3 i (a-i b)^4 \log (\tan (c+d x)+i)+3 i (a+i b)^4 \log (-\tan (c+d x)+i)-2 b^4 \tan ^3(c+d x)\right )-30 (A b-a B) \left (6 a b^2 \tan (c+d x)+(-b+i a)^3 \log (-\tan (c+d x)+i)-(b+i a)^3 \log (\tan (c+d x)+i)+b^3 \tan ^2(c+d x)\right )+\frac{3 (5 A b-a B) (a+b \tan (c+d x))^4}{b}+12 B \tan (c+d x) (a+b \tan (c+d x))^4}{60 b d} \]
Antiderivative was successfully verified.
[In]
Integrate[Tan[c + d*x]^2*(a + b*Tan[c + d*x])^3*(A + B*Tan[c + d*x]),x]
[Out]
((3*(5*A*b - a*B)*(a + b*Tan[c + d*x])^4)/b + 12*B*Tan[c + d*x]*(a + b*Tan[c + d*x])^4 - 30*(A*b - a*B)*((I*a
- b)^3*Log[I - Tan[c + d*x]] - (I*a + b)^3*Log[I + Tan[c + d*x]] + 6*a*b^2*Tan[c + d*x] + b^3*Tan[c + d*x]^2)
+ 10*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))/(60*b*d)
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Maple [A] time = 0.014, size = 383, normalized size = 1.9 \begin{align*}{\frac{B{b}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{A \left ( \tan \left ( dx+c \right ) \right ) ^{4}{b}^{3}}{4\,d}}+{\frac{3\,B \left ( \tan \left ( dx+c \right ) \right ) ^{4}a{b}^{2}}{4\,d}}+{\frac{A \left ( \tan \left ( dx+c \right ) \right ) ^{3}a{b}^{2}}{d}}+{\frac{B \left ( \tan \left ( dx+c \right ) \right ) ^{3}{a}^{2}b}{d}}-{\frac{B \left ( \tan \left ( dx+c \right ) \right ) ^{3}{b}^{3}}{3\,d}}+{\frac{3\,A \left ( \tan \left ( dx+c \right ) \right ) ^{2}{a}^{2}b}{2\,d}}-{\frac{A \left ( \tan \left ( dx+c \right ) \right ) ^{2}{b}^{3}}{2\,d}}+{\frac{B{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{3\,B \left ( \tan \left ( dx+c \right ) \right ) ^{2}a{b}^{2}}{2\,d}}+{\frac{A{a}^{3}\tan \left ( dx+c \right ) }{d}}-3\,{\frac{Aa{b}^{2}\tan \left ( dx+c \right ) }{d}}-3\,{\frac{B{a}^{2}b\tan \left ( dx+c \right ) }{d}}+{\frac{B{b}^{3}\tan \left ( dx+c \right ) }{d}}-{\frac{3\,\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) A{a}^{2}b}{2\,d}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) A{b}^{3}}{2\,d}}-{\frac{B{a}^{3}\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}}+{\frac{3\,\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) Ba{b}^{2}}{2\,d}}-{\frac{A{a}^{3}\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{A\arctan \left ( \tan \left ( dx+c \right ) \right ) a{b}^{2}}{d}}+3\,{\frac{B\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{2}b}{d}}-{\frac{B\arctan \left ( \tan \left ( dx+c \right ) \right ){b}^{3}}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tan(d*x+c)^2*(a+b*tan(d*x+c))^3*(A+B*tan(d*x+c)),x)
[Out]
1/5/d*B*b^3*tan(d*x+c)^5+1/4/d*A*tan(d*x+c)^4*b^3+3/4/d*B*tan(d*x+c)^4*a*b^2+1/d*A*tan(d*x+c)^3*a*b^2+1/d*B*ta
n(d*x+c)^3*a^2*b-1/3/d*B*tan(d*x+c)^3*b^3+3/2/d*A*tan(d*x+c)^2*a^2*b-1/2/d*A*tan(d*x+c)^2*b^3+1/2/d*a^3*B*tan(
d*x+c)^2-3/2/d*B*tan(d*x+c)^2*a*b^2+1/d*a^3*A*tan(d*x+c)-3/d*A*a*b^2*tan(d*x+c)-3/d*B*a^2*b*tan(d*x+c)+1/d*B*b
^3*tan(d*x+c)-3/2/d*ln(1+tan(d*x+c)^2)*A*a^2*b+1/2/d*ln(1+tan(d*x+c)^2)*A*b^3-1/2/d*a^3*B*ln(1+tan(d*x+c)^2)+3
/2/d*ln(1+tan(d*x+c)^2)*B*a*b^2-1/d*a^3*A*arctan(tan(d*x+c))+3/d*A*arctan(tan(d*x+c))*a*b^2+3/d*B*arctan(tan(d
*x+c))*a^2*b-1/d*B*arctan(tan(d*x+c))*b^3
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Maxima [A] time = 1.5374, size = 289, normalized size = 1.44 \begin{align*} \frac{12 \, B b^{3} \tan \left (d x + c\right )^{5} + 15 \,{\left (3 \, B a b^{2} + A b^{3}\right )} \tan \left (d x + c\right )^{4} + 20 \,{\left (3 \, B a^{2} b + 3 \, A a b^{2} - B b^{3}\right )} \tan \left (d x + c\right )^{3} + 30 \,{\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \tan \left (d x + c\right )^{2} - 60 \,{\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )}{\left (d x + c\right )} - 30 \,{\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 60 \,{\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} \tan \left (d x + c\right )}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)^2*(a+b*tan(d*x+c))^3*(A+B*tan(d*x+c)),x, algorithm="maxima")
[Out]
1/60*(12*B*b^3*tan(d*x + c)^5 + 15*(3*B*a*b^2 + A*b^3)*tan(d*x + c)^4 + 20*(3*B*a^2*b + 3*A*a*b^2 - B*b^3)*tan
(d*x + c)^3 + 30*(B*a^3 + 3*A*a^2*b - 3*B*a*b^2 - A*b^3)*tan(d*x + c)^2 - 60*(A*a^3 - 3*B*a^2*b - 3*A*a*b^2 +
B*b^3)*(d*x + c) - 30*(B*a^3 + 3*A*a^2*b - 3*B*a*b^2 - A*b^3)*log(tan(d*x + c)^2 + 1) + 60*(A*a^3 - 3*B*a^2*b
- 3*A*a*b^2 + B*b^3)*tan(d*x + c))/d
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Fricas [A] time = 2.08538, size = 494, normalized size = 2.46 \begin{align*} \frac{12 \, B b^{3} \tan \left (d x + c\right )^{5} + 15 \,{\left (3 \, B a b^{2} + A b^{3}\right )} \tan \left (d x + c\right )^{4} + 20 \,{\left (3 \, B a^{2} b + 3 \, A a b^{2} - B b^{3}\right )} \tan \left (d x + c\right )^{3} - 60 \,{\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} d x + 30 \,{\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \tan \left (d x + c\right )^{2} + 30 \,{\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 60 \,{\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} \tan \left (d x + c\right )}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)^2*(a+b*tan(d*x+c))^3*(A+B*tan(d*x+c)),x, algorithm="fricas")
[Out]
1/60*(12*B*b^3*tan(d*x + c)^5 + 15*(3*B*a*b^2 + A*b^3)*tan(d*x + c)^4 + 20*(3*B*a^2*b + 3*A*a*b^2 - B*b^3)*tan
(d*x + c)^3 - 60*(A*a^3 - 3*B*a^2*b - 3*A*a*b^2 + B*b^3)*d*x + 30*(B*a^3 + 3*A*a^2*b - 3*B*a*b^2 - A*b^3)*tan(
d*x + c)^2 + 30*(B*a^3 + 3*A*a^2*b - 3*B*a*b^2 - A*b^3)*log(1/(tan(d*x + c)^2 + 1)) + 60*(A*a^3 - 3*B*a^2*b -
3*A*a*b^2 + B*b^3)*tan(d*x + c))/d
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Sympy [A] time = 1.09906, size = 384, normalized size = 1.91 \begin{align*} \begin{cases} - A a^{3} x + \frac{A a^{3} \tan{\left (c + d x \right )}}{d} - \frac{3 A a^{2} b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{3 A a^{2} b \tan ^{2}{\left (c + d x \right )}}{2 d} + 3 A a b^{2} x + \frac{A a b^{2} \tan ^{3}{\left (c + d x \right )}}{d} - \frac{3 A a b^{2} \tan{\left (c + d x \right )}}{d} + \frac{A b^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{A b^{3} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac{A b^{3} \tan ^{2}{\left (c + d x \right )}}{2 d} - \frac{B a^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{B a^{3} \tan ^{2}{\left (c + d x \right )}}{2 d} + 3 B a^{2} b x + \frac{B a^{2} b \tan ^{3}{\left (c + d x \right )}}{d} - \frac{3 B a^{2} b \tan{\left (c + d x \right )}}{d} + \frac{3 B a b^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{3 B a b^{2} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac{3 B a b^{2} \tan ^{2}{\left (c + d x \right )}}{2 d} - B b^{3} x + \frac{B b^{3} \tan ^{5}{\left (c + d x \right )}}{5 d} - \frac{B b^{3} \tan ^{3}{\left (c + d x \right )}}{3 d} + \frac{B b^{3} \tan{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (A + B \tan{\left (c \right )}\right ) \left (a + b \tan{\left (c \right )}\right )^{3} \tan ^{2}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)**2*(a+b*tan(d*x+c))**3*(A+B*tan(d*x+c)),x)
[Out]
Piecewise((-A*a**3*x + A*a**3*tan(c + d*x)/d - 3*A*a**2*b*log(tan(c + d*x)**2 + 1)/(2*d) + 3*A*a**2*b*tan(c +
d*x)**2/(2*d) + 3*A*a*b**2*x + A*a*b**2*tan(c + d*x)**3/d - 3*A*a*b**2*tan(c + d*x)/d + A*b**3*log(tan(c + d*x
)**2 + 1)/(2*d) + A*b**3*tan(c + d*x)**4/(4*d) - A*b**3*tan(c + d*x)**2/(2*d) - B*a**3*log(tan(c + d*x)**2 + 1
)/(2*d) + B*a**3*tan(c + d*x)**2/(2*d) + 3*B*a**2*b*x + B*a**2*b*tan(c + d*x)**3/d - 3*B*a**2*b*tan(c + d*x)/d
+ 3*B*a*b**2*log(tan(c + d*x)**2 + 1)/(2*d) + 3*B*a*b**2*tan(c + d*x)**4/(4*d) - 3*B*a*b**2*tan(c + d*x)**2/(
2*d) - B*b**3*x + B*b**3*tan(c + d*x)**5/(5*d) - B*b**3*tan(c + d*x)**3/(3*d) + B*b**3*tan(c + d*x)/d, Ne(d, 0
)), (x*(A + B*tan(c))*(a + b*tan(c))**3*tan(c)**2, True))
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Giac [B] time = 9.77069, size = 5396, normalized size = 26.85 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)^2*(a+b*tan(d*x+c))^3*(A+B*tan(d*x+c)),x, algorithm="giac")
[Out]
-1/60*(60*A*a^3*d*x*tan(d*x)^5*tan(c)^5 - 180*B*a^2*b*d*x*tan(d*x)^5*tan(c)^5 - 180*A*a*b^2*d*x*tan(d*x)^5*tan
(c)^5 + 60*B*b^3*d*x*tan(d*x)^5*tan(c)^5 - 30*B*a^3*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*t
an(c) + tan(d*x)^2*tan(c)^2 + tan(d*x)^2 - 2*tan(d*x)*tan(c) + 1))*tan(d*x)^5*tan(c)^5 - 90*A*a^2*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)^5*tan(c)^5 + 90*B*a*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)^5*tan(c)^5 + 30*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)^5*
tan(c)^5 - 300*A*a^3*d*x*tan(d*x)^4*tan(c)^4 + 900*B*a^2*b*d*x*tan(d*x)^4*tan(c)^4 + 900*A*a*b^2*d*x*tan(d*x)^
4*tan(c)^4 - 300*B*b^3*d*x*tan(d*x)^4*tan(c)^4 - 30*B*a^3*tan(d*x)^5*tan(c)^5 - 90*A*a^2*b*tan(d*x)^5*tan(c)^5
+ 135*B*a*b^2*tan(d*x)^5*tan(c)^5 + 45*A*b^3*tan(d*x)^5*tan(c)^5 + 150*B*a^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 + 450*A*a^2*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 - 450*B*a*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)^4*tan(c)^4 - 150*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 + 60*A*a^3*tan(d*x)^5*tan(c)^4 - 180*B*a^2*b*tan(d*x)^5*tan(c)^4 - 180*A
*a*b^2*tan(d*x)^5*tan(c)^4 + 60*B*b^3*tan(d*x)^5*tan(c)^4 + 60*A*a^3*tan(d*x)^4*tan(c)^5 - 180*B*a^2*b*tan(d*x
)^4*tan(c)^5 - 180*A*a*b^2*tan(d*x)^4*tan(c)^5 + 60*B*b^3*tan(d*x)^4*tan(c)^5 + 600*A*a^3*d*x*tan(d*x)^3*tan(c
)^3 - 1800*B*a^2*b*d*x*tan(d*x)^3*tan(c)^3 - 1800*A*a*b^2*d*x*tan(d*x)^3*tan(c)^3 + 600*B*b^3*d*x*tan(d*x)^3*t
an(c)^3 - 30*B*a^3*tan(d*x)^5*tan(c)^3 - 90*A*a^2*b*tan(d*x)^5*tan(c)^3 + 90*B*a*b^2*tan(d*x)^5*tan(c)^3 + 30*
A*b^3*tan(d*x)^5*tan(c)^3 + 90*B*a^3*tan(d*x)^4*tan(c)^4 + 270*A*a^2*b*tan(d*x)^4*tan(c)^4 - 495*B*a*b^2*tan(d
*x)^4*tan(c)^4 - 165*A*b^3*tan(d*x)^4*tan(c)^4 - 30*B*a^3*tan(d*x)^3*tan(c)^5 - 90*A*a^2*b*tan(d*x)^3*tan(c)^5
+ 90*B*a*b^2*tan(d*x)^3*tan(c)^5 + 30*A*b^3*tan(d*x)^3*tan(c)^5 + 60*B*a^2*b*tan(d*x)^5*tan(c)^2 + 60*A*a*b^2
*tan(d*x)^5*tan(c)^2 - 20*B*b^3*tan(d*x)^5*tan(c)^2 - 300*B*a^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 - 900*A*a^2
*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)^3*tan(c)^3 + 900*B*a*b^2*log(4*(tan(c)^2 + 1)/(tan(d*x)^4*tan(c)^2 - 2*tan(d*x)^3*t
an(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 + 300*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 - 240*A*a^3*tan(d*x)^4*tan(c)^3 + 900*B*a^2*b*tan(d*x)^4*tan(c)^3 + 900*A*a*b^2*tan(d*
x)^4*tan(c)^3 - 300*B*b^3*tan(d*x)^4*tan(c)^3 - 240*A*a^3*tan(d*x)^3*tan(c)^4 + 900*B*a^2*b*tan(d*x)^3*tan(c)^
4 + 900*A*a*b^2*tan(d*x)^3*tan(c)^4 - 300*B*b^3*tan(d*x)^3*tan(c)^4 + 60*B*a^2*b*tan(d*x)^2*tan(c)^5 + 60*A*a*
b^2*tan(d*x)^2*tan(c)^5 - 20*B*b^3*tan(d*x)^2*tan(c)^5 - 45*B*a*b^2*tan(d*x)^5*tan(c) - 15*A*b^3*tan(d*x)^5*ta
n(c) - 600*A*a^3*d*x*tan(d*x)^2*tan(c)^2 + 1800*B*a^2*b*d*x*tan(d*x)^2*tan(c)^2 + 1800*A*a*b^2*d*x*tan(d*x)^2*
tan(c)^2 - 600*B*b^3*d*x*tan(d*x)^2*tan(c)^2 + 90*B*a^3*tan(d*x)^4*tan(c)^2 + 270*A*a^2*b*tan(d*x)^4*tan(c)^2
- 450*B*a*b^2*tan(d*x)^4*tan(c)^2 - 150*A*b^3*tan(d*x)^4*tan(c)^2 - 120*B*a^3*tan(d*x)^3*tan(c)^3 - 360*A*a^2*
b*tan(d*x)^3*tan(c)^3 + 540*B*a*b^2*tan(d*x)^3*tan(c)^3 + 180*A*b^3*tan(d*x)^3*tan(c)^3 + 90*B*a^3*tan(d*x)^2*
tan(c)^4 + 270*A*a^2*b*tan(d*x)^2*tan(c)^4 - 450*B*a*b^2*tan(d*x)^2*tan(c)^4 - 150*A*b^3*tan(d*x)^2*tan(c)^4 -
45*B*a*b^2*tan(d*x)*tan(c)^5 - 15*A*b^3*tan(d*x)*tan(c)^5 + 12*B*b^3*tan(d*x)^5 - 120*B*a^2*b*tan(d*x)^4*tan(
c) - 120*A*a*b^2*tan(d*x)^4*tan(c) + 100*B*b^3*tan(d*x)^4*tan(c) + 300*B*a^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 + 900*A*a^2*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 - 900*B*a*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)^2*tan(c)^2 - 300*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 + 360*A*a^3*tan(d*x)^3*tan(c)^2 - 1440*B*a^2*b*tan(d*x)^3*tan(c)^2 - 1440
*A*a*b^2*tan(d*x)^3*tan(c)^2 + 600*B*b^3*tan(d*x)^3*tan(c)^2 + 360*A*a^3*tan(d*x)^2*tan(c)^3 - 1440*B*a^2*b*ta
n(d*x)^2*tan(c)^3 - 1440*A*a*b^2*tan(d*x)^2*tan(c)^3 + 600*B*b^3*tan(d*x)^2*tan(c)^3 - 120*B*a^2*b*tan(d*x)*ta
n(c)^4 - 120*A*a*b^2*tan(d*x)*tan(c)^4 + 100*B*b^3*tan(d*x)*tan(c)^4 + 12*B*b^3*tan(c)^5 + 45*B*a*b^2*tan(d*x)
^4 + 15*A*b^3*tan(d*x)^4 + 300*A*a^3*d*x*tan(d*x)*tan(c) - 900*B*a^2*b*d*x*tan(d*x)*tan(c) - 900*A*a*b^2*d*x*t
an(d*x)*tan(c) + 300*B*b^3*d*x*tan(d*x)*tan(c) - 90*B*a^3*tan(d*x)^3*tan(c) - 270*A*a^2*b*tan(d*x)^3*tan(c) +
450*B*a*b^2*tan(d*x)^3*tan(c) + 150*A*b^3*tan(d*x)^3*tan(c) + 120*B*a^3*tan(d*x)^2*tan(c)^2 + 360*A*a^2*b*tan(
d*x)^2*tan(c)^2 - 540*B*a*b^2*tan(d*x)^2*tan(c)^2 - 180*A*b^3*tan(d*x)^2*tan(c)^2 - 90*B*a^3*tan(d*x)*tan(c)^3
- 270*A*a^2*b*tan(d*x)*tan(c)^3 + 450*B*a*b^2*tan(d*x)*tan(c)^3 + 150*A*b^3*tan(d*x)*tan(c)^3 + 45*B*a*b^2*ta
n(c)^4 + 15*A*b^3*tan(c)^4 + 60*B*a^2*b*tan(d*x)^3 + 60*A*a*b^2*tan(d*x)^3 - 20*B*b^3*tan(d*x)^3 - 150*B*a^3*l
og(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) - 450*A*a^2*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)*tan(c) + 450*B*a*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) + 150*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)*tan(c) - 240*A*a^3*tan(d*x)^2*tan(c) + 900*B*a^2*b*tan(d*x)^2*ta
n(c) + 900*A*a*b^2*tan(d*x)^2*tan(c) - 300*B*b^3*tan(d*x)^2*tan(c) - 240*A*a^3*tan(d*x)*tan(c)^2 + 900*B*a^2*b
*tan(d*x)*tan(c)^2 + 900*A*a*b^2*tan(d*x)*tan(c)^2 - 300*B*b^3*tan(d*x)*tan(c)^2 + 60*B*a^2*b*tan(c)^3 + 60*A*
a*b^2*tan(c)^3 - 20*B*b^3*tan(c)^3 - 60*A*a^3*d*x + 180*B*a^2*b*d*x + 180*A*a*b^2*d*x - 60*B*b^3*d*x + 30*B*a^
3*tan(d*x)^2 + 90*A*a^2*b*tan(d*x)^2 - 90*B*a*b^2*tan(d*x)^2 - 30*A*b^3*tan(d*x)^2 - 90*B*a^3*tan(d*x)*tan(c)
- 270*A*a^2*b*tan(d*x)*tan(c) + 495*B*a*b^2*tan(d*x)*tan(c) + 165*A*b^3*tan(d*x)*tan(c) + 30*B*a^3*tan(c)^2 +
90*A*a^2*b*tan(c)^2 - 90*B*a*b^2*tan(c)^2 - 30*A*b^3*tan(c)^2 + 30*B*a^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)) + 90*A*a^2*b*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)) - 90*B*a*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)) - 30*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)) + 60*A*a^3*tan(d*x) - 180*B*a^2*b*tan(d*x) - 180*A
*a*b^2*tan(d*x) + 60*B*b^3*tan(d*x) + 60*A*a^3*tan(c) - 180*B*a^2*b*tan(c) - 180*A*a*b^2*tan(c) + 60*B*b^3*tan
(c) + 30*B*a^3 + 90*A*a^2*b - 135*B*a*b^2 - 45*A*b^3)/(d*tan(d*x)^5*tan(c)^5 - 5*d*tan(d*x)^4*tan(c)^4 + 10*d*
tan(d*x)^3*tan(c)^3 - 10*d*tan(d*x)^2*tan(c)^2 + 5*d*tan(d*x)*tan(c) - d)