3.455 \(\int \cot ^7(c+d x) (a+b \tan (c+d x))^4 \, dx\)

Optimal. Leaf size=198 \[ \frac{a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}+\frac{4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \cot ^2(c+d x)}{2 d}-\frac{4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \log (\sin (c+d x))}{d}-4 a b x \left (a^2-b^2\right )-\frac{7 a^3 b \cot ^5(c+d x)}{15 d}-\frac{a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d} \]

[Out]

-4*a*b*(a^2 - b^2)*x - (4*a*b*(a^2 - b^2)*Cot[c + d*x])/d - ((a^4 - 6*a^2*b^2 + b^4)*Cot[c + d*x]^2)/(2*d) + (
4*a*b*(a^2 - b^2)*Cot[c + d*x]^3)/(3*d) + (a^2*(3*a^2 - 16*b^2)*Cot[c + d*x]^4)/(12*d) - (7*a^3*b*Cot[c + d*x]
^5)/(15*d) - ((a^4 - 6*a^2*b^2 + b^4)*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]^6*(a + b*Tan[c + d*x])^2)/(6*d)

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Rubi [A]  time = 0.447365, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3565, 3635, 3628, 3529, 3531, 3475} \[ \frac{a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}+\frac{4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}-\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \cot ^2(c+d x)}{2 d}-\frac{4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac{\left (-6 a^2 b^2+a^4+b^4\right ) \log (\sin (c+d x))}{d}-4 a b x \left (a^2-b^2\right )-\frac{7 a^3 b \cot ^5(c+d x)}{15 d}-\frac{a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^7*(a + b*Tan[c + d*x])^4,x]

[Out]

-4*a*b*(a^2 - b^2)*x - (4*a*b*(a^2 - b^2)*Cot[c + d*x])/d - ((a^4 - 6*a^2*b^2 + b^4)*Cot[c + d*x]^2)/(2*d) + (
4*a*b*(a^2 - b^2)*Cot[c + d*x]^3)/(3*d) + (a^2*(3*a^2 - 16*b^2)*Cot[c + d*x]^4)/(12*d) - (7*a^3*b*Cot[c + d*x]
^5)/(15*d) - ((a^4 - 6*a^2*b^2 + b^4)*Log[Sin[c + d*x]])/d - (a^2*Cot[c + d*x]^6*(a + b*Tan[c + d*x])^2)/(6*d)

Rule 3565

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[((b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3635

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

Rule 3628

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

Rule 3529

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

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 ^7(c+d x) (a+b \tan (c+d x))^4 \, dx &=-\frac{a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\frac{1}{6} \int \cot ^6(c+d x) (a+b \tan (c+d x)) \left (14 a^2 b-6 a \left (a^2-3 b^2\right ) \tan (c+d x)-2 b \left (2 a^2-3 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=-\frac{7 a^3 b \cot ^5(c+d x)}{15 d}-\frac{a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\frac{1}{6} \int \cot ^5(c+d x) \left (-2 a^2 \left (3 a^2-16 b^2\right )-24 a b \left (a^2-b^2\right ) \tan (c+d x)-2 b^2 \left (2 a^2-3 b^2\right ) \tan ^2(c+d x)\right ) \, dx\\ &=\frac{a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac{7 a^3 b \cot ^5(c+d x)}{15 d}-\frac{a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\frac{1}{6} \int \cot ^4(c+d x) \left (-24 a b \left (a^2-b^2\right )+6 \left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right ) \, dx\\ &=\frac{4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}+\frac{a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac{7 a^3 b \cot ^5(c+d x)}{15 d}-\frac{a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\frac{1}{6} \int \cot ^3(c+d x) \left (6 \left (a^4-6 a^2 b^2+b^4\right )+24 a b \left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=-\frac{\left (a^4-6 a^2 b^2+b^4\right ) \cot ^2(c+d x)}{2 d}+\frac{4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}+\frac{a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac{7 a^3 b \cot ^5(c+d x)}{15 d}-\frac{a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\frac{1}{6} \int \cot ^2(c+d x) \left (24 a b \left (a^2-b^2\right )-6 \left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right ) \, dx\\ &=-\frac{4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac{\left (a^4-6 a^2 b^2+b^4\right ) \cot ^2(c+d x)}{2 d}+\frac{4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}+\frac{a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac{7 a^3 b \cot ^5(c+d x)}{15 d}-\frac{a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\frac{1}{6} \int \cot (c+d x) \left (-6 \left (a^4-6 a^2 b^2+b^4\right )-24 a b \left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=-4 a b \left (a^2-b^2\right ) x-\frac{4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac{\left (a^4-6 a^2 b^2+b^4\right ) \cot ^2(c+d x)}{2 d}+\frac{4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}+\frac{a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac{7 a^3 b \cot ^5(c+d x)}{15 d}-\frac{a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}+\left (-a^4+6 a^2 b^2-b^4\right ) \int \cot (c+d x) \, dx\\ &=-4 a b \left (a^2-b^2\right ) x-\frac{4 a b \left (a^2-b^2\right ) \cot (c+d x)}{d}-\frac{\left (a^4-6 a^2 b^2+b^4\right ) \cot ^2(c+d x)}{2 d}+\frac{4 a b \left (a^2-b^2\right ) \cot ^3(c+d x)}{3 d}+\frac{a^2 \left (3 a^2-16 b^2\right ) \cot ^4(c+d x)}{12 d}-\frac{7 a^3 b \cot ^5(c+d x)}{15 d}-\frac{\left (a^4-6 a^2 b^2+b^4\right ) \log (\sin (c+d x))}{d}-\frac{a^2 \cot ^6(c+d x) (a+b \tan (c+d x))^2}{6 d}\\ \end{align*}

Mathematica [C]  time = 0.481572, size = 178, normalized size = 0.9 \[ -\frac{-\frac{1}{4} a^2 \left (a^2-6 b^2\right ) \cot ^4(c+d x)+\frac{1}{2} \left (-6 a^2 b^2+a^4+b^4\right ) \cot ^2(c+d x)+\frac{4}{5} a^3 b \cot ^5(c+d x)+\frac{1}{6} a^4 \cot ^6(c+d x)-\frac{4}{3} a b (a-b) (a+b) \cot ^3(c+d x)+4 a b (a-b) (a+b) \cot (c+d x)-\frac{1}{2} (a-i b)^4 \log (-\cot (c+d x)+i)-\frac{1}{2} (a+i b)^4 \log (\cot (c+d x)+i)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^7*(a + b*Tan[c + d*x])^4,x]

[Out]

-((4*a*(a - b)*b*(a + b)*Cot[c + d*x] + ((a^4 - 6*a^2*b^2 + b^4)*Cot[c + d*x]^2)/2 - (4*a*(a - b)*b*(a + b)*Co
t[c + d*x]^3)/3 - (a^2*(a^2 - 6*b^2)*Cot[c + d*x]^4)/4 + (4*a^3*b*Cot[c + d*x]^5)/5 + (a^4*Cot[c + d*x]^6)/6 -
((a - I*b)^4*Log[I - Cot[c + d*x]])/2 - ((a + I*b)^4*Log[I + Cot[c + d*x]])/2)/d)

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Maple [A]  time = 0.066, size = 267, normalized size = 1.4 \begin{align*} -{\frac{{b}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{{b}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{4\,{b}^{3}a \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+4\,{\frac{\cot \left ( dx+c \right ) a{b}^{3}}{d}}+4\,{b}^{3}ax+4\,{\frac{{b}^{3}ac}{d}}-{\frac{3\,{a}^{2}{b}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{2\,d}}+3\,{\frac{{a}^{2}{b}^{2} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{d}}+6\,{\frac{{a}^{2}{b}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{4\,b{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{4\,b{a}^{3} \left ( \cot \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-4\,{\frac{b{a}^{3}\cot \left ( dx+c \right ) }{d}}-4\,x{a}^{3}b-4\,{\frac{b{a}^{3}c}{d}}-{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{6}}{6\,d}}+{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{{a}^{4} \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{{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)^7*(a+b*tan(d*x+c))^4,x)

[Out]

-1/2/d*b^4*cot(d*x+c)^2-1/d*b^4*ln(sin(d*x+c))-4/3/d*b^3*a*cot(d*x+c)^3+4/d*cot(d*x+c)*a*b^3+4*b^3*a*x+4/d*a*b
^3*c-3/2/d*a^2*b^2*cot(d*x+c)^4+3/d*a^2*b^2*cot(d*x+c)^2+6/d*a^2*b^2*ln(sin(d*x+c))-4/5*a^3*b*cot(d*x+c)^5/d+4
/3*a^3*b*cot(d*x+c)^3/d-4*a^3*b*cot(d*x+c)/d-4*x*a^3*b-4/d*a^3*b*c-1/6/d*a^4*cot(d*x+c)^6+1/4*a^4*cot(d*x+c)^4
/d-1/2*a^4*cot(d*x+c)^2/d-a^4*ln(sin(d*x+c))/d

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Maxima [A]  time = 1.62936, size = 263, normalized size = 1.33 \begin{align*} -\frac{240 \,{\left (a^{3} b - a b^{3}\right )}{\left (d x + c\right )} - 30 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 60 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac{240 \,{\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{5} + 48 \, a^{3} b \tan \left (d x + c\right ) + 30 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{4} + 10 \, a^{4} - 80 \,{\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} - 15 \,{\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{6}}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7*(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/60*(240*(a^3*b - a*b^3)*(d*x + c) - 30*(a^4 - 6*a^2*b^2 + b^4)*log(tan(d*x + c)^2 + 1) + 60*(a^4 - 6*a^2*b^
2 + b^4)*log(tan(d*x + c)) + (240*(a^3*b - a*b^3)*tan(d*x + c)^5 + 48*a^3*b*tan(d*x + c) + 30*(a^4 - 6*a^2*b^2
+ b^4)*tan(d*x + c)^4 + 10*a^4 - 80*(a^3*b - a*b^3)*tan(d*x + c)^3 - 15*(a^4 - 6*a^2*b^2)*tan(d*x + c)^2)/tan
(d*x + c)^6)/d

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Fricas [A]  time = 2.01114, size = 504, normalized size = 2.55 \begin{align*} -\frac{30 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\frac{\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{6} + 5 \,{\left (11 \, a^{4} - 54 \, a^{2} b^{2} + 6 \, b^{4} + 48 \,{\left (a^{3} b - a b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{6} + 240 \,{\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{5} + 48 \, a^{3} b \tan \left (d x + c\right ) + 30 \,{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{4} + 10 \, a^{4} - 80 \,{\left (a^{3} b - a b^{3}\right )} \tan \left (d x + c\right )^{3} - 15 \,{\left (a^{4} - 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{60 \, d \tan \left (d x + c\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7*(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/60*(30*(a^4 - 6*a^2*b^2 + b^4)*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c)^6 + 5*(11*a^4 - 54*a^2
*b^2 + 6*b^4 + 48*(a^3*b - a*b^3)*d*x)*tan(d*x + c)^6 + 240*(a^3*b - a*b^3)*tan(d*x + c)^5 + 48*a^3*b*tan(d*x
+ c) + 30*(a^4 - 6*a^2*b^2 + b^4)*tan(d*x + c)^4 + 10*a^4 - 80*(a^3*b - a*b^3)*tan(d*x + c)^3 - 15*(a^4 - 6*a^
2*b^2)*tan(d*x + c)^2)/(d*tan(d*x + c)^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**7*(a+b*tan(d*x+c))**4,x)

[Out]

Timed out

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Giac [B]  time = 2.84815, size = 683, normalized size = 3.45 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^7*(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

-1/1920*(5*a^4*tan(1/2*d*x + 1/2*c)^6 - 48*a^3*b*tan(1/2*d*x + 1/2*c)^5 - 60*a^4*tan(1/2*d*x + 1/2*c)^4 + 180*
a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 560*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 320*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 435*a^4
*tan(1/2*d*x + 1/2*c)^2 - 2160*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 240*b^4*tan(1/2*d*x + 1/2*c)^2 - 5280*a^3*b*ta
n(1/2*d*x + 1/2*c) + 4800*a*b^3*tan(1/2*d*x + 1/2*c) + 7680*(a^3*b - a*b^3)*(d*x + c) - 1920*(a^4 - 6*a^2*b^2
+ b^4)*log(tan(1/2*d*x + 1/2*c)^2 + 1) + 1920*(a^4 - 6*a^2*b^2 + b^4)*log(abs(tan(1/2*d*x + 1/2*c))) - (4704*a
^4*tan(1/2*d*x + 1/2*c)^6 - 28224*a^2*b^2*tan(1/2*d*x + 1/2*c)^6 + 4704*b^4*tan(1/2*d*x + 1/2*c)^6 - 5280*a^3*
b*tan(1/2*d*x + 1/2*c)^5 + 4800*a*b^3*tan(1/2*d*x + 1/2*c)^5 - 435*a^4*tan(1/2*d*x + 1/2*c)^4 + 2160*a^2*b^2*t
an(1/2*d*x + 1/2*c)^4 - 240*b^4*tan(1/2*d*x + 1/2*c)^4 + 560*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 320*a*b^3*tan(1/2*
d*x + 1/2*c)^3 + 60*a^4*tan(1/2*d*x + 1/2*c)^2 - 180*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 - 48*a^3*b*tan(1/2*d*x + 1
/2*c) - 5*a^4)/tan(1/2*d*x + 1/2*c)^6)/d