3.698 \(\int \frac{\sec ^4(c+d x) (A+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^4} \, dx\)
Optimal. Leaf size=378 \[ -\frac{\left (5 A b^4-C \left (-23 a^2 b^2+12 a^4+6 b^4\right )\right ) \tan (c+d x)}{6 b^4 d \left (a^2-b^2\right )^2}-\frac{\left (a^2 b^6 (3 A+20 C)+28 a^6 b^2 C-35 a^4 b^4 C-8 a^8 C+2 A b^8\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac{\left (a^2 b^2 (2 A+9 C)-4 a^4 C+3 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{a \left (3 a^2 b^4 (A+4 C)-11 a^4 b^2 C+4 a^6 C+2 A b^6\right ) \tan (c+d x)}{2 b^4 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac{4 a C \tanh ^{-1}(\sin (c+d x))}{b^5 d} \]
[Out]
(-4*a*C*ArcTanh[Sin[c + d*x]])/(b^5*d) - ((2*A*b^8 - 8*a^8*C + 28*a^6*b^2*C - 35*a^4*b^4*C + a^2*b^6*(3*A + 20
*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) - ((5*A*b^4 - (1
2*a^4 - 23*a^2*b^2 + 6*b^4)*C)*Tan[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) - ((A*b^2 + a^2*C)*Sec[c + d*x]^3*Tan[c +
d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + ((3*A*b^4 - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Sec[c + d*x]^2*
Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (a*(2*A*b^6 + 4*a^6*C - 11*a^4*b^2*C + 3*a^2*b^
4*(A + 4*C))*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))
________________________________________________________________________________________
Rubi [A] time = 1.8112, antiderivative size = 378, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used =
{4099, 4098, 4090, 4082, 3998, 3770, 3831, 2659, 208} \[ -\frac{\left (5 A b^4-C \left (-23 a^2 b^2+12 a^4+6 b^4\right )\right ) \tan (c+d x)}{6 b^4 d \left (a^2-b^2\right )^2}-\frac{\left (a^2 b^6 (3 A+20 C)+28 a^6 b^2 C-35 a^4 b^4 C-8 a^8 C+2 A b^8\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left (a^2 C+A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac{\left (a^2 b^2 (2 A+9 C)-4 a^4 C+3 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{a \left (3 a^2 b^4 (A+4 C)-11 a^4 b^2 C+4 a^6 C+2 A b^6\right ) \tan (c+d x)}{2 b^4 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac{4 a C \tanh ^{-1}(\sin (c+d x))}{b^5 d} \]
Antiderivative was successfully verified.
[In]
Int[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]
[Out]
(-4*a*C*ArcTanh[Sin[c + d*x]])/(b^5*d) - ((2*A*b^8 - 8*a^8*C + 28*a^6*b^2*C - 35*a^4*b^4*C + a^2*b^6*(3*A + 20
*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) - ((5*A*b^4 - (1
2*a^4 - 23*a^2*b^2 + 6*b^4)*C)*Tan[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) - ((A*b^2 + a^2*C)*Sec[c + d*x]^3*Tan[c +
d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + ((3*A*b^4 - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Sec[c + d*x]^2*
Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (a*(2*A*b^6 + 4*a^6*C - 11*a^4*b^2*C + 3*a^2*b^
4*(A + 4*C))*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))
Rule 4099
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b
_.) + (a_))^(m_), x_Symbol] :> -Simp[(d*(A*b^2 + a^2*C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f
*x])^(n - 1))/(b*f*(a^2 - b^2)*(m + 1)), x] + Dist[d/(b*(a^2 - b^2)*(m + 1)), Int[(a + b*Csc[e + f*x])^(m + 1)
*(d*Csc[e + f*x])^(n - 1)*Simp[A*b^2*(n - 1) + a^2*C*(n - 1) + a*b*(A + C)*(m + 1)*Csc[e + f*x] - (A*b^2*(m +
n + 1) + C*(a^2*n + b^2*(m + 1)))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, C}, x] && NeQ[a^2 - b
^2, 0] && LtQ[m, -1] && GtQ[n, 0]
Rule 4098
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Simp[(d*(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(
a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1))/(b*f*(a^2 - b^2)*(m + 1)), x] + Dist[d/(b*(a^2 - b^2)*(m
+ 1)), Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1)*Simp[A*b^2*(n - 1) - a*(b*B - a*C)*(n - 1) +
b*(a*A - b*B + a*C)*(m + 1)*Csc[e + f*x] - (b*(A*b - a*B)*(m + n + 1) + C*(a^2*n + b^2*(m + 1)))*Csc[e + f*x]
^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 0]
Rule 4090
Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(
e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(a*(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc[e
+ f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)), Int[Csc[e + f*x]*(a + b*C
sc[e + f*x])^(m + 1)*Simp[b*(m + 1)*(-(a*(b*B - a*C)) + A*b^2) + (b*B*(a^2 + b^2*(m + 1)) - a*(A*b^2*(m + 2) +
C*(a^2 + b^2*(m + 1))))*Csc[e + f*x] - b*C*(m + 1)*(a^2 - b^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 4082
Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_
.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Simp[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1))/(b*f*(m
+ 2)), x] + Dist[1/(b*(m + 2)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[b*A*(m + 2) + b*C*(m + 1) + (b*B*
(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Rule 3998
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x
_Symbol] :> Dist[B/b, Int[Csc[e + f*x], x], x] + Dist[(A*b - a*B)/b, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x]
, x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0]
Rule 3770
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
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{\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx &=-\frac{\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac{\int \frac{\sec ^3(c+d x) \left (3 \left (A b^2+a^2 C\right )-3 a b (A+C) \sec (c+d x)-\left (A b^2+4 a^2 C-3 b^2 C\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx}{3 b \left (a^2-b^2\right )}\\ &=-\frac{\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{\int \frac{\sec ^2(c+d x) \left (2 \left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right )-2 a b \left (5 A b^2-\left (a^2-6 b^2\right ) C\right ) \sec (c+d x)-\left (5 A b^4-\left (12 a^4-23 a^2 b^2+6 b^4\right ) C\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{6 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{a \left (2 A b^6+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac{\int \frac{\sec (c+d x) \left (-3 b \left (2 A b^6+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right )-a \left (a^2-b^2\right ) \left (5 A b^4+12 a^4 C-25 a^2 b^2 C+18 b^4 C\right ) \sec (c+d x)-b \left (a^2-b^2\right ) \left (5 A b^4-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 b^4 \left (a^2-b^2\right )^3}\\ &=-\frac{\left (5 A b^4-\left (12 a^4-23 a^2 b^2+6 b^4\right ) C\right ) \tan (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{a \left (2 A b^6+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac{\int \frac{\sec (c+d x) \left (-3 b^2 \left (2 A b^6+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right )-24 a b \left (a^2-b^2\right )^3 C \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 b^5 \left (a^2-b^2\right )^3}\\ &=-\frac{\left (5 A b^4-\left (12 a^4-23 a^2 b^2+6 b^4\right ) C\right ) \tan (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{a \left (2 A b^6+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac{(4 a C) \int \sec (c+d x) \, dx}{b^5}-\frac{\left (2 A b^8-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)\right ) \int \frac{\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 b^5 \left (a^2-b^2\right )^3}\\ &=-\frac{4 a C \tanh ^{-1}(\sin (c+d x))}{b^5 d}-\frac{\left (5 A b^4-\left (12 a^4-23 a^2 b^2+6 b^4\right ) C\right ) \tan (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{a \left (2 A b^6+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac{\left (2 A b^8-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)\right ) \int \frac{1}{1+\frac{a \cos (c+d x)}{b}} \, dx}{2 b^6 \left (a^2-b^2\right )^3}\\ &=-\frac{4 a C \tanh ^{-1}(\sin (c+d x))}{b^5 d}-\frac{\left (5 A b^4-\left (12 a^4-23 a^2 b^2+6 b^4\right ) C\right ) \tan (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{a \left (2 A b^6+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac{\left (2 A b^8-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 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 )}{b^6 \left (a^2-b^2\right )^3 d}\\ &=-\frac{4 a C \tanh ^{-1}(\sin (c+d x))}{b^5 d}-\frac{\left (3 a^2 A b^6+2 A b^8-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+20 a^2 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-b)^{7/2} b^5 (a+b)^{7/2} d}-\frac{\left (5 A b^4-\left (12 a^4-23 a^2 b^2+6 b^4\right ) C\right ) \tan (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2+a^2 C\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac{\left (3 A b^4-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{a \left (2 A b^6+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 4.4066, size = 564, normalized size = 1.49 \[ \frac{\sec ^3(c+d x) (a \cos (c+d x)+b) \left (A+C \sec ^2(c+d x)\right ) \left (-\frac{2 b \sin (c+d x) \left (6 a^2 b \left (a^2 b^4 (A+53 C)-57 a^4 b^2 C+20 a^6 C+3 b^6 (3 A-2 C)\right ) \cos (2 (c+d x))+a \left (5 a^4 b^4 (4 A-61 C)+a^2 b^6 (13 A+438 C)-28 a^6 b^2 C+72 a^8 C+72 b^8 (A-C)\right ) \cos (c+d x)+4 a^5 A b^4 \cos (3 (c+d x))+11 a^3 A b^6 \cos (3 (c+d x))+6 a^4 A b^5+54 a^2 A b^7-68 a^7 b^2 C \cos (3 (c+d x))+65 a^5 b^4 C \cos (3 (c+d x))-6 a^3 b^6 C \cos (3 (c+d x))-318 a^6 b^3 C+246 a^4 b^5 C+36 a^2 b^7 C+120 a^8 b C+24 a^9 C \cos (3 (c+d x))-24 b^9 C\right )}{\left (b^2-a^2\right )^3}-\frac{48 \left (-a^2 b^6 (3 A+20 C)-28 a^6 b^2 C+35 a^4 b^4 C+8 a^8 C-2 A b^8\right ) \cos (c+d x) (a \cos (c+d x)+b)^3 \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}+192 a C \cos (c+d x) (a \cos (c+d x)+b)^3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-192 a C \cos (c+d x) (a \cos (c+d x)+b)^3 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{24 b^5 d (a+b \sec (c+d x))^4 (A \cos (2 (c+d x))+A+2 C)} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]
[Out]
((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2)*((-48*(-2*A*b^8 + 8*a^8*C - 28*a^6*b^2*C + 35*a^4*
b^4*C - a^2*b^6*(3*A + 20*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*Cos[c + d*x]*(b + a*Cos[c +
d*x])^3)/(a^2 - b^2)^(7/2) + 192*a*C*Cos[c + d*x]*(b + a*Cos[c + d*x])^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)
/2]] - 192*a*C*Cos[c + d*x]*(b + a*Cos[c + d*x])^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - (2*b*(6*a^4*A*b^
5 + 54*a^2*A*b^7 + 120*a^8*b*C - 318*a^6*b^3*C + 246*a^4*b^5*C + 36*a^2*b^7*C - 24*b^9*C + a*(5*a^4*b^4*(4*A -
61*C) + 72*b^8*(A - C) + 72*a^8*C - 28*a^6*b^2*C + a^2*b^6*(13*A + 438*C))*Cos[c + d*x] + 6*a^2*b*(3*b^6*(3*A
- 2*C) + 20*a^6*C - 57*a^4*b^2*C + a^2*b^4*(A + 53*C))*Cos[2*(c + d*x)] + 4*a^5*A*b^4*Cos[3*(c + d*x)] + 11*a
^3*A*b^6*Cos[3*(c + d*x)] + 24*a^9*C*Cos[3*(c + d*x)] - 68*a^7*b^2*C*Cos[3*(c + d*x)] + 65*a^5*b^4*C*Cos[3*(c
+ d*x)] - 6*a^3*b^6*C*Cos[3*(c + d*x)])*Sin[c + d*x])/(-a^2 + b^2)^3))/(24*b^5*d*(A + 2*C + A*Cos[2*(c + d*x)]
)*(a + b*Sec[c + d*x])^4)
________________________________________________________________________________________
Maple [B] time = 0.11, size = 2318, normalized size = 6.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sec(d*x+c)^4*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x)
[Out]
8/d/b^5/(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)
)*a^8*C-3/d*b/(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))*A*a^2-28/d/b^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))*a^6*C+35/d/b/(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))*a^4*C-20/d/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^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+4/3/d/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^3/(a^2-2*a
*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-4/d*a*C/b^5*ln(tan(1/2*d*x+1/2*c)+1)+40/d/(tan(1/2*d*x+1/2*c)^2
*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^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/(tan(1/2*d*x+1
/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A-20/d/(tan(1/2
*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^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/(ta
n(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A-
20/d*b/(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*a^2+5/d/b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^4/(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^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^6/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3
)*tan(1/2*d*x+1/2*c)*C-6/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^7/(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/b^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^6/(a-b)/(a^3
+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C-6/d*b^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/(
a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A-6/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^7/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C-116/3/d/b^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^5/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C+18/d/b^2/(tan(1/2*d*x+1/2*c)^2*a-t
an(1/2*d*x+1/2*c)^2*b-a-b)^3*a^5/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C-5/d/b/(tan(1/2*d*x+1/2
*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*a^4/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C+3/d*b/(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/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A-3/d*b/
(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/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5
*A-6/d*b^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/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x
+1/2*c)^5*A+12/d*b^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/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*t
an(1/2*d*x+1/2*c)^3*A+12/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^7/(a^2-2*a*b+b^2)/(a^2+
2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C+18/d/b^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^5/(a+b)/(a^
3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C-1/d*C/b^4/(tan(1/2*d*x+1/2*c)+1)-1/d*C/b^4/(tan(1/2*d*x+1/2*c)-1)-
2/d*b^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)
)*A+4/d*a*C/b^5*ln(tan(1/2*d*x+1/2*c)-1)
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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(sec(d*x+c)^4*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 65.1669, size = 5434, normalized size = 14.38 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(d*x+c)^4*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x, algorithm="fricas")
[Out]
[1/12*(3*((8*C*a^11 - 28*C*a^9*b^2 + 35*C*a^7*b^4 - (3*A + 20*C)*a^5*b^6 - 2*A*a^3*b^8)*cos(d*x + c)^4 + 3*(8*
C*a^10*b - 28*C*a^8*b^3 + 35*C*a^6*b^5 - (3*A + 20*C)*a^4*b^7 - 2*A*a^2*b^9)*cos(d*x + c)^3 + 3*(8*C*a^9*b^2 -
28*C*a^7*b^4 + 35*C*a^5*b^6 - (3*A + 20*C)*a^3*b^8 - 2*A*a*b^10)*cos(d*x + c)^2 + (8*C*a^8*b^3 - 28*C*a^6*b^5
+ 35*C*a^4*b^7 - (3*A + 20*C)*a^2*b^9 - 2*A*b^11)*cos(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)) - 24*((C*a^12 - 4*C*a^10*b^2 + 6*C*a^8*b^4 - 4*C*a^6*b^6 + C*a^4*b^8)*cos(d*
x + c)^4 + 3*(C*a^11*b - 4*C*a^9*b^3 + 6*C*a^7*b^5 - 4*C*a^5*b^7 + C*a^3*b^9)*cos(d*x + c)^3 + 3*(C*a^10*b^2 -
4*C*a^8*b^4 + 6*C*a^6*b^6 - 4*C*a^4*b^8 + C*a^2*b^10)*cos(d*x + c)^2 + (C*a^9*b^3 - 4*C*a^7*b^5 + 6*C*a^5*b^7
- 4*C*a^3*b^9 + C*a*b^11)*cos(d*x + c))*log(sin(d*x + c) + 1) + 24*((C*a^12 - 4*C*a^10*b^2 + 6*C*a^8*b^4 - 4*
C*a^6*b^6 + C*a^4*b^8)*cos(d*x + c)^4 + 3*(C*a^11*b - 4*C*a^9*b^3 + 6*C*a^7*b^5 - 4*C*a^5*b^7 + C*a^3*b^9)*cos
(d*x + c)^3 + 3*(C*a^10*b^2 - 4*C*a^8*b^4 + 6*C*a^6*b^6 - 4*C*a^4*b^8 + C*a^2*b^10)*cos(d*x + c)^2 + (C*a^9*b^
3 - 4*C*a^7*b^5 + 6*C*a^5*b^7 - 4*C*a^3*b^9 + C*a*b^11)*cos(d*x + c))*log(-sin(d*x + c) + 1) + 2*(6*C*a^8*b^4
- 24*C*a^6*b^6 + 36*C*a^4*b^8 - 24*C*a^2*b^10 + 6*C*b^12 + (24*C*a^11*b - 92*C*a^9*b^3 + (4*A + 133*C)*a^7*b^5
+ (7*A - 71*C)*a^5*b^7 - (11*A - 6*C)*a^3*b^9)*cos(d*x + c)^3 + 3*(20*C*a^10*b^2 - 77*C*a^8*b^4 + (A + 110*C)
*a^6*b^6 + (8*A - 59*C)*a^4*b^8 - 3*(3*A - 2*C)*a^2*b^10)*cos(d*x + c)^2 + (44*C*a^9*b^3 + (2*A - 169*C)*a^7*b
^5 - (7*A - 239*C)*a^5*b^7 + (23*A - 132*C)*a^3*b^9 - 18*(A - C)*a*b^11)*cos(d*x + c))*sin(d*x + c))/((a^11*b^
5 - 4*a^9*b^7 + 6*a^7*b^9 - 4*a^5*b^11 + a^3*b^13)*d*cos(d*x + c)^4 + 3*(a^10*b^6 - 4*a^8*b^8 + 6*a^6*b^10 - 4
*a^4*b^12 + a^2*b^14)*d*cos(d*x + c)^3 + 3*(a^9*b^7 - 4*a^7*b^9 + 6*a^5*b^11 - 4*a^3*b^13 + a*b^15)*d*cos(d*x
+ c)^2 + (a^8*b^8 - 4*a^6*b^10 + 6*a^4*b^12 - 4*a^2*b^14 + b^16)*d*cos(d*x + c)), 1/6*(3*((8*C*a^11 - 28*C*a^9
*b^2 + 35*C*a^7*b^4 - (3*A + 20*C)*a^5*b^6 - 2*A*a^3*b^8)*cos(d*x + c)^4 + 3*(8*C*a^10*b - 28*C*a^8*b^3 + 35*C
*a^6*b^5 - (3*A + 20*C)*a^4*b^7 - 2*A*a^2*b^9)*cos(d*x + c)^3 + 3*(8*C*a^9*b^2 - 28*C*a^7*b^4 + 35*C*a^5*b^6 -
(3*A + 20*C)*a^3*b^8 - 2*A*a*b^10)*cos(d*x + c)^2 + (8*C*a^8*b^3 - 28*C*a^6*b^5 + 35*C*a^4*b^7 - (3*A + 20*C)
*a^2*b^9 - 2*A*b^11)*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)
*sin(d*x + c))) - 12*((C*a^12 - 4*C*a^10*b^2 + 6*C*a^8*b^4 - 4*C*a^6*b^6 + C*a^4*b^8)*cos(d*x + c)^4 + 3*(C*a^
11*b - 4*C*a^9*b^3 + 6*C*a^7*b^5 - 4*C*a^5*b^7 + C*a^3*b^9)*cos(d*x + c)^3 + 3*(C*a^10*b^2 - 4*C*a^8*b^4 + 6*C
*a^6*b^6 - 4*C*a^4*b^8 + C*a^2*b^10)*cos(d*x + c)^2 + (C*a^9*b^3 - 4*C*a^7*b^5 + 6*C*a^5*b^7 - 4*C*a^3*b^9 + C
*a*b^11)*cos(d*x + c))*log(sin(d*x + c) + 1) + 12*((C*a^12 - 4*C*a^10*b^2 + 6*C*a^8*b^4 - 4*C*a^6*b^6 + C*a^4*
b^8)*cos(d*x + c)^4 + 3*(C*a^11*b - 4*C*a^9*b^3 + 6*C*a^7*b^5 - 4*C*a^5*b^7 + C*a^3*b^9)*cos(d*x + c)^3 + 3*(C
*a^10*b^2 - 4*C*a^8*b^4 + 6*C*a^6*b^6 - 4*C*a^4*b^8 + C*a^2*b^10)*cos(d*x + c)^2 + (C*a^9*b^3 - 4*C*a^7*b^5 +
6*C*a^5*b^7 - 4*C*a^3*b^9 + C*a*b^11)*cos(d*x + c))*log(-sin(d*x + c) + 1) + (6*C*a^8*b^4 - 24*C*a^6*b^6 + 36*
C*a^4*b^8 - 24*C*a^2*b^10 + 6*C*b^12 + (24*C*a^11*b - 92*C*a^9*b^3 + (4*A + 133*C)*a^7*b^5 + (7*A - 71*C)*a^5*
b^7 - (11*A - 6*C)*a^3*b^9)*cos(d*x + c)^3 + 3*(20*C*a^10*b^2 - 77*C*a^8*b^4 + (A + 110*C)*a^6*b^6 + (8*A - 59
*C)*a^4*b^8 - 3*(3*A - 2*C)*a^2*b^10)*cos(d*x + c)^2 + (44*C*a^9*b^3 + (2*A - 169*C)*a^7*b^5 - (7*A - 239*C)*a
^5*b^7 + (23*A - 132*C)*a^3*b^9 - 18*(A - C)*a*b^11)*cos(d*x + c))*sin(d*x + c))/((a^11*b^5 - 4*a^9*b^7 + 6*a^
7*b^9 - 4*a^5*b^11 + a^3*b^13)*d*cos(d*x + c)^4 + 3*(a^10*b^6 - 4*a^8*b^8 + 6*a^6*b^10 - 4*a^4*b^12 + a^2*b^14
)*d*cos(d*x + c)^3 + 3*(a^9*b^7 - 4*a^7*b^9 + 6*a^5*b^11 - 4*a^3*b^13 + a*b^15)*d*cos(d*x + c)^2 + (a^8*b^8 -
4*a^6*b^10 + 6*a^4*b^12 - 4*a^2*b^14 + b^16)*d*cos(d*x + c))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}}{\left (a + b \sec{\left (c + d x \right )}\right )^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(d*x+c)**4*(A+C*sec(d*x+c)**2)/(a+b*sec(d*x+c))**4,x)
[Out]
Integral((A + C*sec(c + d*x)**2)*sec(c + d*x)**4/(a + b*sec(c + d*x))**4, x)
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Giac [B] time = 1.41093, size = 1185, normalized size = 3.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(d*x+c)^4*(A+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x, algorithm="giac")
[Out]
1/3*(3*(8*C*a^8 - 28*C*a^6*b^2 + 35*C*a^4*b^4 - 3*A*a^2*b^6 - 20*C*a^2*b^6 - 2*A*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^6
*b^5 - 3*a^4*b^7 + 3*a^2*b^9 - b^11)*sqrt(-a^2 + b^2)) - 12*C*a*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^5 + 12*C*
a*log(abs(tan(1/2*d*x + 1/2*c) - 1))/b^5 - (18*C*a^9*tan(1/2*d*x + 1/2*c)^5 - 42*C*a^8*b*tan(1/2*d*x + 1/2*c)^
5 - 24*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^5 + 117*C*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 6*A*a^5*b^4*tan(1/2*d*x + 1/2
*c)^5 - 24*C*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 3*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 - 105*C*a^4*b^5*tan(1/2*d*x +
1/2*c)^5 + 6*A*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 + 60*C*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 - 27*A*a^2*b^7*tan(1/2*d*
x + 1/2*c)^5 + 18*A*a*b^8*tan(1/2*d*x + 1/2*c)^5 - 36*C*a^9*tan(1/2*d*x + 1/2*c)^3 + 152*C*a^7*b^2*tan(1/2*d*x
+ 1/2*c)^3 - 4*A*a^5*b^4*tan(1/2*d*x + 1/2*c)^3 - 236*C*a^5*b^4*tan(1/2*d*x + 1/2*c)^3 - 32*A*a^3*b^6*tan(1/2
*d*x + 1/2*c)^3 + 120*C*a^3*b^6*tan(1/2*d*x + 1/2*c)^3 + 36*A*a*b^8*tan(1/2*d*x + 1/2*c)^3 + 18*C*a^9*tan(1/2*
d*x + 1/2*c) + 42*C*a^8*b*tan(1/2*d*x + 1/2*c) - 24*C*a^7*b^2*tan(1/2*d*x + 1/2*c) - 117*C*a^6*b^3*tan(1/2*d*x
+ 1/2*c) + 6*A*a^5*b^4*tan(1/2*d*x + 1/2*c) - 24*C*a^5*b^4*tan(1/2*d*x + 1/2*c) + 3*A*a^4*b^5*tan(1/2*d*x + 1
/2*c) + 105*C*a^4*b^5*tan(1/2*d*x + 1/2*c) + 6*A*a^3*b^6*tan(1/2*d*x + 1/2*c) + 60*C*a^3*b^6*tan(1/2*d*x + 1/2
*c) + 27*A*a^2*b^7*tan(1/2*d*x + 1/2*c) + 18*A*a*b^8*tan(1/2*d*x + 1/2*c))/((a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 -
b^10)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)^3) - 6*C*tan(1/2*d*x + 1/2*c)/((tan(1/2*d
*x + 1/2*c)^2 - 1)*b^4))/d