3.934 \(\int \frac{a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^5} \, dx\)

Optimal. Leaf size=336 \[ -\frac{b \left (-8 a^4 b^3 B+7 a^2 b^5 B+5 a^5 b^2 C-7 a^3 b^4 C+8 a^6 b B-10 a^7 C+2 a b^6 C-2 b^7 B\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b^2 \left (-17 a^2 b^3 B+13 a^3 b^2 C+26 a^4 b B-37 a^5 C-6 a b^4 C+6 b^5 B\right ) \tan (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}+\frac{b^2 \left (8 a^2 b B-13 a^3 C+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{b^2 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac{x (b B-a C)}{a^4} \]

[Out]

((b*B - a*C)*x)/a^4 - (b*(8*a^6*b*B - 8*a^4*b^3*B + 7*a^2*b^5*B - 2*b^7*B - 10*a^7*C + 5*a^5*b^2*C - 7*a^3*b^4
*C + 2*a*b^6*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d) + (b^
2*(b*B - 2*a*C)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (b^2*(8*a^2*b*B - 3*b^3*B - 13*a^3*
C + 3*a*b^2*C)*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b^2*(26*a^4*b*B - 17*a^2*b^3*B
+ 6*b^5*B - 37*a^5*C + 13*a^3*b^2*C - 6*a*b^4*C)*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 4.67127, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used = {24, 3923, 4060, 3919, 3831, 2659, 208} \[ -\frac{b \left (-8 a^4 b^3 B+7 a^2 b^5 B+5 a^5 b^2 C-7 a^3 b^4 C+8 a^6 b B-10 a^7 C+2 a b^6 C-2 b^7 B\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b^2 \left (-17 a^2 b^3 B+13 a^3 b^2 C+26 a^4 b B-37 a^5 C-6 a b^4 C+6 b^5 B\right ) \tan (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}+\frac{b^2 \left (8 a^2 b B-13 a^3 C+3 a b^2 C-3 b^3 B\right ) \tan (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{b^2 (b B-2 a C) \tan (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac{x (b B-a C)}{a^4} \]

Antiderivative was successfully verified.

[In]

Int[(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^5,x]

[Out]

((b*B - a*C)*x)/a^4 - (b*(8*a^6*b*B - 8*a^4*b^3*B + 7*a^2*b^5*B - 2*b^7*B - 10*a^7*C + 5*a^5*b^2*C - 7*a^3*b^4
*C + 2*a*b^6*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d) + (b^
2*(b*B - 2*a*C)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (b^2*(8*a^2*b*B - 3*b^3*B - 13*a^3*
C + 3*a*b^2*C)*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b^2*(26*a^4*b*B - 17*a^2*b^3*B
+ 6*b^5*B - 37*a^5*C + 13*a^3*b^2*C - 6*a*b^4*C)*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

Rule 3923

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

Rule 4060

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

Rule 3919

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

Rule 3831

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^5} \, dx &=\frac{\int \frac{b^2 (b B-a C)+b^3 C \sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx}{b^2}\\ &=\frac{b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac{\int \frac{-3 b^2 \left (a^2-b^2\right ) (b B-a C)+3 a b^3 (b B-2 a C) \sec (c+d x)-2 b^4 (b B-2 a C) \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx}{3 a b^2 \left (a^2-b^2\right )}\\ &=\frac{b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b^2 \left (8 a^2 b B-3 b^3 B-13 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{\int \frac{6 b^2 \left (a^2-b^2\right )^2 (b B-a C)-2 a b^3 \left (6 a^2 b B-b^3 B-9 a^3 C-a b^2 C\right ) \sec (c+d x)+b^4 \left (8 a^2 b B-3 b^3 B-13 a^3 C+3 a b^2 C\right ) \sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx}{6 a^2 b^2 \left (a^2-b^2\right )^2}\\ &=\frac{b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b^2 \left (8 a^2 b B-3 b^3 B-13 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{b^2 \left (26 a^4 b B-17 a^2 b^3 B+6 b^5 B-37 a^5 C+13 a^3 b^2 C-6 a b^4 C\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac{\int \frac{-6 b^2 \left (a^2-b^2\right )^3 (b B-a C)+3 a b^3 \left (6 a^4 b B-2 a^2 b^3 B+b^5 B-8 a^5 C-a^3 b^2 C-a b^4 C\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{6 a^3 b^2 \left (a^2-b^2\right )^3}\\ &=\frac{(b B-a C) x}{a^4}+\frac{b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b^2 \left (8 a^2 b B-3 b^3 B-13 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{b^2 \left (26 a^4 b B-17 a^2 b^3 B+6 b^5 B-37 a^5 C+13 a^3 b^2 C-6 a b^4 C\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac{\left (b \left (8 a^6 b B-8 a^4 b^3 B+7 a^2 b^5 B-2 b^7 B-10 a^7 C+5 a^5 b^2 C-7 a^3 b^4 C+2 a b^6 C\right )\right ) \int \frac{\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^4 \left (a^2-b^2\right )^3}\\ &=\frac{(b B-a C) x}{a^4}+\frac{b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b^2 \left (8 a^2 b B-3 b^3 B-13 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{b^2 \left (26 a^4 b B-17 a^2 b^3 B+6 b^5 B-37 a^5 C+13 a^3 b^2 C-6 a b^4 C\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac{\left (8 a^6 b B-8 a^4 b^3 B+7 a^2 b^5 B-2 b^7 B-10 a^7 C+5 a^5 b^2 C-7 a^3 b^4 C+2 a b^6 C\right ) \int \frac{1}{1+\frac{a \cos (c+d x)}{b}} \, dx}{2 a^4 \left (a^2-b^2\right )^3}\\ &=\frac{(b B-a C) x}{a^4}+\frac{b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b^2 \left (8 a^2 b B-3 b^3 B-13 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{b^2 \left (26 a^4 b B-17 a^2 b^3 B+6 b^5 B-37 a^5 C+13 a^3 b^2 C-6 a b^4 C\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac{\left (8 a^6 b B-8 a^4 b^3 B+7 a^2 b^5 B-2 b^7 B-10 a^7 C+5 a^5 b^2 C-7 a^3 b^4 C+2 a b^6 C\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{a}{b}+\left (1-\frac{a}{b}\right ) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^4 \left (a^2-b^2\right )^3 d}\\ &=\frac{(b B-a C) x}{a^4}-\frac{b \left (8 a^6 b B-8 a^4 b^3 B+7 a^2 b^5 B-2 b^7 B-10 a^7 C+5 a^5 b^2 C-7 a^3 b^4 C+2 a b^6 C\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^4 (a-b)^{7/2} (a+b)^{7/2} d}+\frac{b^2 (b B-2 a C) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{b^2 \left (8 a^2 b B-3 b^3 B-13 a^3 C+3 a b^2 C\right ) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{b^2 \left (26 a^4 b B-17 a^2 b^3 B+6 b^5 B-37 a^5 C+13 a^3 b^2 C-6 a b^4 C\right ) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end{align*}

Mathematica [B]  time = 5.59439, size = 1097, normalized size = 3.26 \[ \frac{(b+a \cos (c+d x)) \sec ^3(c+d x) (b B-a C+b C \sec (c+d x)) \left (\frac{24 b \left (-10 C a^7+8 b B a^6+5 b^2 C a^5-8 b^3 B a^4-7 b^4 C a^3+7 b^5 B a^2+2 b^6 C a-2 b^7 B\right ) \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right ) (b+a \cos (c+d x))^3}{\left (a^2-b^2\right )^{7/2}}+\frac{-6 c C \cos (3 (c+d x)) a^{10}-6 C d x \cos (3 (c+d x)) a^{10}-36 b c C a^9-36 b C d x a^9+6 b B c \cos (3 (c+d x)) a^9+6 b B d x \cos (3 (c+d x)) a^9+36 b^2 B c a^8+36 b^2 B d x a^8+18 b^2 c C \cos (3 (c+d x)) a^8+18 b^2 C d x \cos (3 (c+d x)) a^8-54 b^2 C \sin (c+d x) a^8-54 b^2 C \sin (3 (c+d x)) a^8+84 b^3 c C a^7+84 b^3 C d x a^7-18 b^3 B c \cos (3 (c+d x)) a^7-18 b^3 B d x \cos (3 (c+d x)) a^7+36 b^3 B \sin (c+d x) a^7-174 b^3 C \sin (2 (c+d x)) a^7+36 b^3 B \sin (3 (c+d x)) a^7-84 b^4 B c a^6-84 b^4 B d x a^6-18 b^4 c C \cos (3 (c+d x)) a^6-18 b^4 C d x \cos (3 (c+d x)) a^6-111 b^4 C \sin (c+d x) a^6+120 b^4 B \sin (2 (c+d x)) a^6+37 b^4 C \sin (3 (c+d x)) a^6-36 b^5 c C a^5-36 b^5 C d x a^5+18 b^5 B c \cos (3 (c+d x)) a^5+18 b^5 B d x \cos (3 (c+d x)) a^5+72 b^5 B \sin (c+d x) a^5+84 b^5 C \sin (2 (c+d x)) a^5-32 b^5 B \sin (3 (c+d x)) a^5+36 b^6 B c a^4+36 b^6 B d x a^4+6 b^6 c C \cos (3 (c+d x)) a^4+6 b^6 C d x \cos (3 (c+d x)) a^4+39 b^6 C \sin (c+d x) a^4-90 b^6 B \sin (2 (c+d x)) a^4-13 b^6 C \sin (3 (c+d x)) a^4-36 b^7 c C a^3-36 b^7 C d x a^3-6 b^7 B c \cos (3 (c+d x)) a^3-6 b^7 B d x \cos (3 (c+d x)) a^3-57 b^7 B \sin (c+d x) a^3-30 b^7 C \sin (2 (c+d x)) a^3+11 b^7 B \sin (3 (c+d x)) a^3+36 b^8 B c a^2+36 b^8 B d x a^2-36 b \left (a^2-b^2\right )^3 (a C-b B) (c+d x) \cos (2 (c+d x)) a^2-24 b^8 C \sin (c+d x) a^2+30 b^8 B \sin (2 (c+d x)) a^2+24 b^9 c C a+24 b^9 C d x a-18 \left (a^2-b^2\right )^3 \left (a^2+4 b^2\right ) (a C-b B) (c+d x) \cos (c+d x) a+24 b^9 B \sin (c+d x) a-24 b^{10} B c-24 b^{10} B d x}{\left (a^2-b^2\right )^3}\right )}{24 a^4 d (b C+(b B-a C) \cos (c+d x)) (a+b \sec (c+d x))^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^5,x]

[Out]

((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(b*B - a*C + b*C*Sec[c + d*x])*((24*b*(8*a^6*b*B - 8*a^4*b^3*B + 7*a^2*b^
5*B - 2*b^7*B - 10*a^7*C + 5*a^5*b^2*C - 7*a^3*b^4*C + 2*a*b^6*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2
- b^2]]*(b + a*Cos[c + d*x])^3)/(a^2 - b^2)^(7/2) + (36*a^8*b^2*B*c - 84*a^6*b^4*B*c + 36*a^4*b^6*B*c + 36*a^
2*b^8*B*c - 24*b^10*B*c - 36*a^9*b*c*C + 84*a^7*b^3*c*C - 36*a^5*b^5*c*C - 36*a^3*b^7*c*C + 24*a*b^9*c*C + 36*
a^8*b^2*B*d*x - 84*a^6*b^4*B*d*x + 36*a^4*b^6*B*d*x + 36*a^2*b^8*B*d*x - 24*b^10*B*d*x - 36*a^9*b*C*d*x + 84*a
^7*b^3*C*d*x - 36*a^5*b^5*C*d*x - 36*a^3*b^7*C*d*x + 24*a*b^9*C*d*x - 18*a*(a^2 - b^2)^3*(a^2 + 4*b^2)*(-(b*B)
+ a*C)*(c + d*x)*Cos[c + d*x] - 36*a^2*b*(a^2 - b^2)^3*(-(b*B) + a*C)*(c + d*x)*Cos[2*(c + d*x)] + 6*a^9*b*B*
c*Cos[3*(c + d*x)] - 18*a^7*b^3*B*c*Cos[3*(c + d*x)] + 18*a^5*b^5*B*c*Cos[3*(c + d*x)] - 6*a^3*b^7*B*c*Cos[3*(
c + d*x)] - 6*a^10*c*C*Cos[3*(c + d*x)] + 18*a^8*b^2*c*C*Cos[3*(c + d*x)] - 18*a^6*b^4*c*C*Cos[3*(c + d*x)] +
6*a^4*b^6*c*C*Cos[3*(c + d*x)] + 6*a^9*b*B*d*x*Cos[3*(c + d*x)] - 18*a^7*b^3*B*d*x*Cos[3*(c + d*x)] + 18*a^5*b
^5*B*d*x*Cos[3*(c + d*x)] - 6*a^3*b^7*B*d*x*Cos[3*(c + d*x)] - 6*a^10*C*d*x*Cos[3*(c + d*x)] + 18*a^8*b^2*C*d*
x*Cos[3*(c + d*x)] - 18*a^6*b^4*C*d*x*Cos[3*(c + d*x)] + 6*a^4*b^6*C*d*x*Cos[3*(c + d*x)] + 36*a^7*b^3*B*Sin[c
+ d*x] + 72*a^5*b^5*B*Sin[c + d*x] - 57*a^3*b^7*B*Sin[c + d*x] + 24*a*b^9*B*Sin[c + d*x] - 54*a^8*b^2*C*Sin[c
+ d*x] - 111*a^6*b^4*C*Sin[c + d*x] + 39*a^4*b^6*C*Sin[c + d*x] - 24*a^2*b^8*C*Sin[c + d*x] + 120*a^6*b^4*B*S
in[2*(c + d*x)] - 90*a^4*b^6*B*Sin[2*(c + d*x)] + 30*a^2*b^8*B*Sin[2*(c + d*x)] - 174*a^7*b^3*C*Sin[2*(c + d*x
)] + 84*a^5*b^5*C*Sin[2*(c + d*x)] - 30*a^3*b^7*C*Sin[2*(c + d*x)] + 36*a^7*b^3*B*Sin[3*(c + d*x)] - 32*a^5*b^
5*B*Sin[3*(c + d*x)] + 11*a^3*b^7*B*Sin[3*(c + d*x)] - 54*a^8*b^2*C*Sin[3*(c + d*x)] + 37*a^6*b^4*C*Sin[3*(c +
d*x)] - 13*a^4*b^6*C*Sin[3*(c + d*x)])/(a^2 - b^2)^3))/(24*a^4*d*(b*C + (b*B - a*C)*Cos[c + d*x])*(a + b*Sec[
c + d*x])^4)

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Maple [B]  time = 0.127, size = 2853, normalized size = 8.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*a*b-a^2*C+b^2*B*sec(d*x+c)+b^2*C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^5,x)

[Out]

24/d*b^3*a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1
/2*c)^3*B+18/d*b^2*a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*t
an(1/2*d*x+1/2*c)^5*C-8/d*b^2*a^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+
1/2*c)/((a+b)*(a-b))^(1/2))*B+2/d*b^8/a^4/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(
1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*B+10/d*b*a^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-
b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C+7/d*b^5/a/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arcta
nh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C-2/d*b^7/a^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/
2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C-7/d*b^6/a^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a
-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*B-4/d*b^4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x
+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C-4/d*b^4/(tan(1/2*d*x+1/2*c)^2*a-tan(
1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C+40/3/d*b^4/(tan(1/2*d*x+1/2*c)^
2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C+4/d*b^4/(tan(1/2*d*x+
1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B-4/d*b^4/(tan(1/2
*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B-5/d*b^3*a
/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C+6/d
*b^5/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c
)*B+1/d*b^5/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*
x+1/2*c)*C+2/d*b^6/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*t
an(1/2*d*x+1/2*c)*C-2/d/a^3*arctan(tan(1/2*d*x+1/2*c))*C+6/d*b^5/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^
2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B-36/d*b^2*a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2
*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C+8/d*b^4/(a^6-3*a^4*b^2+3*a^2*b^4
-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*B-4/d*b^6/a^2/(tan(1/2*d*x+1/2
*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C-2/d*b^7/a^3/(tan(
1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B+2/d*b^
6/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)
^5*C-12/d*b^3*a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*
d*x+1/2*c)^5*B-1/d*b^5/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)
*tan(1/2*d*x+1/2*c)^5*C-1/d*b^6/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3
*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B+4/d*b^7/a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+
b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-2/d*b^7/a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/
(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B-7/d*b^3*a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-
a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C+18/d*b^2*a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1
/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C-44/3/d*b^5/a/(tan(1/2*d*x+1/2*c)^2*a-tan
(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-12/d*b^3*a/(tan(1/2*d*x+1/2*
c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B+1/d*b^6/a^2/(tan(1/2
*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B+7/d*b^3*a
/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C+
2/d/a^4*arctan(tan(1/2*d*x+1/2*c))*B*b

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*b*B-a^2*C+b^2*B*sec(d*x+c)+b^2*C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.12316, size = 5277, normalized size = 15.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*b*B-a^2*C+b^2*B*sec(d*x+c)+b^2*C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^5,x, algorithm="fricas")

[Out]

[-1/12*(12*(C*a^12 - B*a^11*b - 4*C*a^10*b^2 + 4*B*a^9*b^3 + 6*C*a^8*b^4 - 6*B*a^7*b^5 - 4*C*a^6*b^6 + 4*B*a^5
*b^7 + C*a^4*b^8 - B*a^3*b^9)*d*x*cos(d*x + c)^3 + 36*(C*a^11*b - B*a^10*b^2 - 4*C*a^9*b^3 + 4*B*a^8*b^4 + 6*C
*a^7*b^5 - 6*B*a^6*b^6 - 4*C*a^5*b^7 + 4*B*a^4*b^8 + C*a^3*b^9 - B*a^2*b^10)*d*x*cos(d*x + c)^2 + 36*(C*a^10*b
^2 - B*a^9*b^3 - 4*C*a^8*b^4 + 4*B*a^7*b^5 + 6*C*a^6*b^6 - 6*B*a^5*b^7 - 4*C*a^4*b^8 + 4*B*a^3*b^9 + C*a^2*b^1
0 - B*a*b^11)*d*x*cos(d*x + c) + 12*(C*a^9*b^3 - B*a^8*b^4 - 4*C*a^7*b^5 + 4*B*a^6*b^6 + 6*C*a^5*b^7 - 6*B*a^4
*b^8 - 4*C*a^3*b^9 + 4*B*a^2*b^10 + C*a*b^11 - B*b^12)*d*x + 3*(10*C*a^7*b^4 - 8*B*a^6*b^5 - 5*C*a^5*b^6 + 8*B
*a^4*b^7 + 7*C*a^3*b^8 - 7*B*a^2*b^9 - 2*C*a*b^10 + 2*B*b^11 + (10*C*a^10*b - 8*B*a^9*b^2 - 5*C*a^8*b^3 + 8*B*
a^7*b^4 + 7*C*a^6*b^5 - 7*B*a^5*b^6 - 2*C*a^4*b^7 + 2*B*a^3*b^8)*cos(d*x + c)^3 + 3*(10*C*a^9*b^2 - 8*B*a^8*b^
3 - 5*C*a^7*b^4 + 8*B*a^6*b^5 + 7*C*a^5*b^6 - 7*B*a^4*b^7 - 2*C*a^3*b^8 + 2*B*a^2*b^9)*cos(d*x + c)^2 + 3*(10*
C*a^8*b^3 - 8*B*a^7*b^4 - 5*C*a^6*b^5 + 8*B*a^5*b^6 + 7*C*a^4*b^7 - 7*B*a^3*b^8 - 2*C*a^2*b^9 + 2*B*a*b^10)*co
s(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(
d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) + 2*(37*C*a^8*b^4 -
26*B*a^7*b^5 - 50*C*a^6*b^6 + 43*B*a^5*b^7 + 19*C*a^4*b^8 - 23*B*a^3*b^9 - 6*C*a^2*b^10 + 6*B*a*b^11 + (54*C*
a^10*b^2 - 36*B*a^9*b^3 - 91*C*a^8*b^4 + 68*B*a^7*b^5 + 50*C*a^6*b^6 - 43*B*a^5*b^7 - 13*C*a^4*b^8 + 11*B*a^3*
b^9)*cos(d*x + c)^2 + 3*(29*C*a^9*b^3 - 20*B*a^8*b^4 - 43*C*a^7*b^5 + 35*B*a^6*b^6 + 19*C*a^5*b^7 - 20*B*a^4*b
^8 - 5*C*a^3*b^9 + 5*B*a^2*b^10)*cos(d*x + c))*sin(d*x + c))/((a^15 - 4*a^13*b^2 + 6*a^11*b^4 - 4*a^9*b^6 + a^
7*b^8)*d*cos(d*x + c)^3 + 3*(a^14*b - 4*a^12*b^3 + 6*a^10*b^5 - 4*a^8*b^7 + a^6*b^9)*d*cos(d*x + c)^2 + 3*(a^1
3*b^2 - 4*a^11*b^4 + 6*a^9*b^6 - 4*a^7*b^8 + a^5*b^10)*d*cos(d*x + c) + (a^12*b^3 - 4*a^10*b^5 + 6*a^8*b^7 - 4
*a^6*b^9 + a^4*b^11)*d), -1/6*(6*(C*a^12 - B*a^11*b - 4*C*a^10*b^2 + 4*B*a^9*b^3 + 6*C*a^8*b^4 - 6*B*a^7*b^5 -
4*C*a^6*b^6 + 4*B*a^5*b^7 + C*a^4*b^8 - B*a^3*b^9)*d*x*cos(d*x + c)^3 + 18*(C*a^11*b - B*a^10*b^2 - 4*C*a^9*b
^3 + 4*B*a^8*b^4 + 6*C*a^7*b^5 - 6*B*a^6*b^6 - 4*C*a^5*b^7 + 4*B*a^4*b^8 + C*a^3*b^9 - B*a^2*b^10)*d*x*cos(d*x
+ c)^2 + 18*(C*a^10*b^2 - B*a^9*b^3 - 4*C*a^8*b^4 + 4*B*a^7*b^5 + 6*C*a^6*b^6 - 6*B*a^5*b^7 - 4*C*a^4*b^8 + 4
*B*a^3*b^9 + C*a^2*b^10 - B*a*b^11)*d*x*cos(d*x + c) + 6*(C*a^9*b^3 - B*a^8*b^4 - 4*C*a^7*b^5 + 4*B*a^6*b^6 +
6*C*a^5*b^7 - 6*B*a^4*b^8 - 4*C*a^3*b^9 + 4*B*a^2*b^10 + C*a*b^11 - B*b^12)*d*x - 3*(10*C*a^7*b^4 - 8*B*a^6*b^
5 - 5*C*a^5*b^6 + 8*B*a^4*b^7 + 7*C*a^3*b^8 - 7*B*a^2*b^9 - 2*C*a*b^10 + 2*B*b^11 + (10*C*a^10*b - 8*B*a^9*b^2
- 5*C*a^8*b^3 + 8*B*a^7*b^4 + 7*C*a^6*b^5 - 7*B*a^5*b^6 - 2*C*a^4*b^7 + 2*B*a^3*b^8)*cos(d*x + c)^3 + 3*(10*C
*a^9*b^2 - 8*B*a^8*b^3 - 5*C*a^7*b^4 + 8*B*a^6*b^5 + 7*C*a^5*b^6 - 7*B*a^4*b^7 - 2*C*a^3*b^8 + 2*B*a^2*b^9)*co
s(d*x + c)^2 + 3*(10*C*a^8*b^3 - 8*B*a^7*b^4 - 5*C*a^6*b^5 + 8*B*a^5*b^6 + 7*C*a^4*b^7 - 7*B*a^3*b^8 - 2*C*a^2
*b^9 + 2*B*a*b^10)*cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*s
in(d*x + c))) + (37*C*a^8*b^4 - 26*B*a^7*b^5 - 50*C*a^6*b^6 + 43*B*a^5*b^7 + 19*C*a^4*b^8 - 23*B*a^3*b^9 - 6*C
*a^2*b^10 + 6*B*a*b^11 + (54*C*a^10*b^2 - 36*B*a^9*b^3 - 91*C*a^8*b^4 + 68*B*a^7*b^5 + 50*C*a^6*b^6 - 43*B*a^5
*b^7 - 13*C*a^4*b^8 + 11*B*a^3*b^9)*cos(d*x + c)^2 + 3*(29*C*a^9*b^3 - 20*B*a^8*b^4 - 43*C*a^7*b^5 + 35*B*a^6*
b^6 + 19*C*a^5*b^7 - 20*B*a^4*b^8 - 5*C*a^3*b^9 + 5*B*a^2*b^10)*cos(d*x + c))*sin(d*x + c))/((a^15 - 4*a^13*b^
2 + 6*a^11*b^4 - 4*a^9*b^6 + a^7*b^8)*d*cos(d*x + c)^3 + 3*(a^14*b - 4*a^12*b^3 + 6*a^10*b^5 - 4*a^8*b^7 + a^6
*b^9)*d*cos(d*x + c)^2 + 3*(a^13*b^2 - 4*a^11*b^4 + 6*a^9*b^6 - 4*a^7*b^8 + a^5*b^10)*d*cos(d*x + c) + (a^12*b
^3 - 4*a^10*b^5 + 6*a^8*b^7 - 4*a^6*b^9 + a^4*b^11)*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{B b}{a^{4} + 4 a^{3} b \sec{\left (c + d x \right )} + 6 a^{2} b^{2} \sec ^{2}{\left (c + d x \right )} + 4 a b^{3} \sec ^{3}{\left (c + d x \right )} + b^{4} \sec ^{4}{\left (c + d x \right )}}\, dx - \int \frac{C a}{a^{4} + 4 a^{3} b \sec{\left (c + d x \right )} + 6 a^{2} b^{2} \sec ^{2}{\left (c + d x \right )} + 4 a b^{3} \sec ^{3}{\left (c + d x \right )} + b^{4} \sec ^{4}{\left (c + d x \right )}}\, dx - \int - \frac{C b \sec{\left (c + d x \right )}}{a^{4} + 4 a^{3} b \sec{\left (c + d x \right )} + 6 a^{2} b^{2} \sec ^{2}{\left (c + d x \right )} + 4 a b^{3} \sec ^{3}{\left (c + d x \right )} + b^{4} \sec ^{4}{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*b*B-a**2*C+b**2*B*sec(d*x+c)+b**2*C*sec(d*x+c)**2)/(a+b*sec(d*x+c))**5,x)

[Out]

-Integral(-B*b/(a**4 + 4*a**3*b*sec(c + d*x) + 6*a**2*b**2*sec(c + d*x)**2 + 4*a*b**3*sec(c + d*x)**3 + b**4*s
ec(c + d*x)**4), x) - Integral(C*a/(a**4 + 4*a**3*b*sec(c + d*x) + 6*a**2*b**2*sec(c + d*x)**2 + 4*a*b**3*sec(
c + d*x)**3 + b**4*sec(c + d*x)**4), x) - Integral(-C*b*sec(c + d*x)/(a**4 + 4*a**3*b*sec(c + d*x) + 6*a**2*b*
*2*sec(c + d*x)**2 + 4*a*b**3*sec(c + d*x)**3 + b**4*sec(c + d*x)**4), x)

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Giac [B]  time = 1.61446, size = 1161, normalized size = 3.46 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*b*B-a^2*C+b^2*B*sec(d*x+c)+b^2*C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^5,x, algorithm="giac")

[Out]

1/3*(3*(10*C*a^7*b - 8*B*a^6*b^2 - 5*C*a^5*b^3 + 8*B*a^4*b^4 + 7*C*a^3*b^5 - 7*B*a^2*b^6 - 2*C*a*b^7 + 2*B*b^8
)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c)
)/sqrt(-a^2 + b^2)))/((a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*sqrt(-a^2 + b^2)) - 3*(C*a - B*b)*(d*x + c)/a^4
+ (54*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^5 - 36*B*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 - 87*C*a^6*b^3*tan(1/2*d*x + 1/2
*c)^5 + 60*B*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 42*C*a^4*b^5*tan(1/2*d*x +
1/2*c)^5 - 45*B*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^2*b^7*tan(1/2*d*x + 1/2*c)^5 - 15*C*a^2*b^7*tan(1/2*d*x
+ 1/2*c)^5 + 15*B*a*b^8*tan(1/2*d*x + 1/2*c)^5 + 6*C*a*b^8*tan(1/2*d*x + 1/2*c)^5 - 6*B*b^9*tan(1/2*d*x + 1/2
*c)^5 - 108*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^3 + 72*B*a^6*b^3*tan(1/2*d*x + 1/2*c)^3 + 148*C*a^5*b^4*tan(1/2*d*x
+ 1/2*c)^3 - 116*B*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 - 52*C*a^3*b^6*tan(1/2*d*x + 1/2*c)^3 + 56*B*a^2*b^7*tan(1/
2*d*x + 1/2*c)^3 + 12*C*a*b^8*tan(1/2*d*x + 1/2*c)^3 - 12*B*b^9*tan(1/2*d*x + 1/2*c)^3 + 54*C*a^7*b^2*tan(1/2*
d*x + 1/2*c) - 36*B*a^6*b^3*tan(1/2*d*x + 1/2*c) + 87*C*a^6*b^3*tan(1/2*d*x + 1/2*c) - 60*B*a^5*b^4*tan(1/2*d*
x + 1/2*c) + 6*B*a^4*b^5*tan(1/2*d*x + 1/2*c) - 42*C*a^4*b^5*tan(1/2*d*x + 1/2*c) + 45*B*a^3*b^6*tan(1/2*d*x +
1/2*c) + 6*B*a^2*b^7*tan(1/2*d*x + 1/2*c) + 15*C*a^2*b^7*tan(1/2*d*x + 1/2*c) - 15*B*a*b^8*tan(1/2*d*x + 1/2*
c) + 6*C*a*b^8*tan(1/2*d*x + 1/2*c) - 6*B*b^9*tan(1/2*d*x + 1/2*c))/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*(
a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)^3))/d