3.916 \(\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=465 \[ \frac{\tan (c+d x) \left (-a^3 b^2 (2 A-21 C)-11 a^2 b^3 B+6 a^4 b B-12 a^5 C+a b^4 (5 A-6 C)+2 b^5 B\right )}{2 b^4 d \left (a^2-b^2\right )^2}+\frac{\left (12 a^2 C-6 a b B+2 A b^2+b^2 C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}-\frac{a \left (a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+15 a^3 b^3 B-6 a^5 b B+12 a^6 C-12 a b^5 B+6 A b^6\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac{\tan (c+d x) \sec ^2(c+d x) \left (a \left (2 a^2 b B-4 a^3 C+7 a b^2 C-5 b^3 B\right )+3 A b^4\right )}{2 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac{\tan (c+d x) \sec (c+d x) \left (-a^2 b^2 (A-10 C)+3 a^3 b B-6 a^4 C-6 a b^3 B+b^4 (4 A-C)\right )}{2 b^3 d \left (a^2-b^2\right )^2} \]

[Out]

((2*A*b^2 - 6*a*b*B + 12*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*b^5*d) - (a*(6*A*b^6 - 6*a^5*b*B + 15*a^3*b^
3*B - 12*a*b^5*B + a^4*b^2*(2*A - 29*C) - 5*a^2*b^4*(A - 4*C) + 12*a^6*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2
])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) + ((6*a^4*b*B - 11*a^2*b^3*B + 2*b^5*B - a^3*b^2*(2*A - 2
1*C) + a*b^4*(5*A - 6*C) - 12*a^5*C)*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*b*B - 6*a*b^3*B - a^2*b^2
*(A - 10*C) + b^4*(4*A - C) - 6*a^4*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B -
a*C))*Sec[c + d*x]^3*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((3*A*b^4 + a*(2*a^2*b*B - 5*
b^3*B - 4*a^3*C + 7*a*b^2*C))*Sec[c + d*x]^2*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))

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Rubi [A]  time = 4.74502, antiderivative size = 465, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.195, Rules used = {4098, 4092, 4082, 3998, 3770, 3831, 2659, 208} \[ \frac{\tan (c+d x) \left (-a^3 b^2 (2 A-21 C)-11 a^2 b^3 B+6 a^4 b B-12 a^5 C+a b^4 (5 A-6 C)+2 b^5 B\right )}{2 b^4 d \left (a^2-b^2\right )^2}+\frac{\left (12 a^2 C-6 a b B+2 A b^2+b^2 C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}-\frac{a \left (a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+15 a^3 b^3 B-6 a^5 b B+12 a^6 C-12 a b^5 B+6 A b^6\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac{\tan (c+d x) \sec ^2(c+d x) \left (a \left (2 a^2 b B-4 a^3 C+7 a b^2 C-5 b^3 B\right )+3 A b^4\right )}{2 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac{\tan (c+d x) \sec (c+d x) \left (-a^2 b^2 (A-10 C)+3 a^3 b B-6 a^4 C-6 a b^3 B+b^4 (4 A-C)\right )}{2 b^3 d \left (a^2-b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]

[Out]

((2*A*b^2 - 6*a*b*B + 12*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*b^5*d) - (a*(6*A*b^6 - 6*a^5*b*B + 15*a^3*b^
3*B - 12*a*b^5*B + a^4*b^2*(2*A - 29*C) - 5*a^2*b^4*(A - 4*C) + 12*a^6*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2
])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) + ((6*a^4*b*B - 11*a^2*b^3*B + 2*b^5*B - a^3*b^2*(2*A - 2
1*C) + a*b^4*(5*A - 6*C) - 12*a^5*C)*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*b*B - 6*a*b^3*B - a^2*b^2
*(A - 10*C) + b^4*(4*A - C) - 6*a^4*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B -
a*C))*Sec[c + d*x]^3*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((3*A*b^4 + a*(2*a^2*b*B - 5*
b^3*B - 4*a^3*C + 7*a*b^2*C))*Sec[c + d*x]^2*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))

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 4092

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[(C*Csc[e + f*x]*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m
+ 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[a*C + b*(C*(m + 2)
+ A*(m + 3))*Csc[e + f*x] - (2*a*C - b*B*(m + 3))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m
}, 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+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx &=-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac{\int \frac{\sec ^3(c+d x) \left (3 \left (A b^2-a (b B-a C)\right )+2 b (b B-a (A+C)) \sec (c+d x)-2 \left (A b^2-a b B+2 a^2 C-b^2 C\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{\int \frac{\sec ^2(c+d x) \left (2 \left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right )+b \left (a^2 b B+2 b^3 B+a^3 C-a b^2 (3 A+4 C)\right ) \sec (c+d x)-2 \left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{\int \frac{\sec (c+d x) \left (-2 a \left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right )+2 b \left (a^3 b B-4 a b^3 B-2 a^4 C+b^4 (2 A+C)+a^2 b^2 (A+4 C)\right ) \sec (c+d x)+2 \left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 b^3 \left (a^2-b^2\right )^2}\\ &=\frac{\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac{\int \frac{\sec (c+d x) \left (-2 a b \left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right )+2 \left (a^2-b^2\right )^2 \left (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 b^4 \left (a^2-b^2\right )^2}\\ &=\frac{\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\left (a \left (6 A b^6-6 a^5 b B+15 a^3 b^3 B-12 a b^5 B+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\right )\right ) \int \frac{\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 b^5 \left (a^2-b^2\right )^2}+\frac{\left (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) \int \sec (c+d x) \, dx}{2 b^5}\\ &=\frac{\left (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac{\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\left (a \left (6 A b^6-6 a^5 b B+15 a^3 b^3 B-12 a b^5 B+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\right )\right ) \int \frac{1}{1+\frac{a \cos (c+d x)}{b}} \, dx}{2 b^6 \left (a^2-b^2\right )^2}\\ &=\frac{\left (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac{\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac{\left (a \left (6 A b^6-6 a^5 b B+15 a^3 b^3 B-12 a b^5 B+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\right )\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 )^2 d}\\ &=\frac{\left (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}-\frac{a \left (2 a^4 A b^2-5 a^2 A b^4+6 A b^6-6 a^5 b B+15 a^3 b^3 B-12 a b^5 B+12 a^6 C-29 a^4 b^2 C+20 a^2 b^4 C\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{(a-b)^{5/2} b^5 (a+b)^{5/2} d}+\frac{\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac{\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end{align*}

Mathematica [B]  time = 6.47949, size = 1124, normalized size = 2.42 \[ \frac{\left (-12 C a^2+6 b B a-2 A b^2-b^2 C\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sec (c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) (b+a \cos (c+d x))^3}{b^5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{\left (12 C a^2-6 b B a+2 A b^2+b^2 C\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sec (c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) (b+a \cos (c+d x))^3}{b^5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{2 a \left (12 C a^6-6 b B a^5+2 A b^2 a^4-29 b^2 C a^4+15 b^3 B a^3-5 A b^4 a^2+20 b^4 C a^2-12 b^5 B a+6 A b^6\right ) \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right ) \sec (c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) (b+a \cos (c+d x))^3}{b^5 \sqrt{a^2-b^2} \left (b^2-a^2\right )^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{\sec ^3(c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (-24 C \sin (2 (c+d x)) a^7-12 C \sin (4 (c+d x)) a^7-36 b C \sin (c+d x) a^6+12 b B \sin (2 (c+d x)) a^6-36 b C \sin (3 (c+d x)) a^6+6 b B \sin (4 (c+d x)) a^6+18 b^2 B \sin (c+d x) a^5-4 A b^2 \sin (2 (c+d x)) a^5+26 b^2 C \sin (2 (c+d x)) a^5+18 b^2 B \sin (3 (c+d x)) a^5-2 A b^2 \sin (4 (c+d x)) a^5+21 b^2 C \sin (4 (c+d x)) a^5-6 A b^3 \sin (c+d x) a^4+72 b^3 C \sin (c+d x) a^4-14 b^3 B \sin (2 (c+d x)) a^4-6 A b^3 \sin (3 (c+d x)) a^4+64 b^3 C \sin (3 (c+d x)) a^4-11 b^3 B \sin (4 (c+d x)) a^4-32 b^4 B \sin (c+d x) a^3+10 A b^4 \sin (2 (c+d x)) a^3+20 b^4 C \sin (2 (c+d x)) a^3-32 b^4 B \sin (3 (c+d x)) a^3+5 A b^4 \sin (4 (c+d x)) a^3-6 b^4 C \sin (4 (c+d x)) a^3+12 A b^5 \sin (c+d x) a^2-38 b^5 C \sin (c+d x) a^2-12 b^5 B \sin (2 (c+d x)) a^2+12 A b^5 \sin (3 (c+d x)) a^2-22 b^5 C \sin (3 (c+d x)) a^2+2 b^5 B \sin (4 (c+d x)) a^2+8 b^6 B \sin (c+d x) a-16 b^6 C \sin (2 (c+d x)) a+8 b^6 B \sin (3 (c+d x)) a+8 b^7 C \sin (c+d x)+8 b^7 B \sin (2 (c+d x))\right ) (b+a \cos (c+d x))}{8 b^4 \left (b^2-a^2\right )^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]

[Out]

(2*a*(2*a^4*A*b^2 - 5*a^2*A*b^4 + 6*A*b^6 - 6*a^5*b*B + 15*a^3*b^3*B - 12*a*b^5*B + 12*a^6*C - 29*a^4*b^2*C +
20*a^2*b^4*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*
Sec[c + d*x] + C*Sec[c + d*x]^2))/(b^5*Sqrt[a^2 - b^2]*(-a^2 + b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*
c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((-2*A*b^2 + 6*a*b*B - 12*a^2*C - b^2*C)*(b + a*Cos[c + d*x])^3*Log[Cos[
(c + d*x)/2] - Sin[(c + d*x)/2]]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b^5*d*(A + 2*C + 2*B*C
os[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((2*A*b^2 - 6*a*b*B + 12*a^2*C + b^2*C)*(b + a*Cos
[c + d*x])^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b
^5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])*Sec[c +
d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-6*a^4*A*b^3*Sin[c + d*x] + 12*a^2*A*b^5*Sin[c + d*x] + 18*a^
5*b^2*B*Sin[c + d*x] - 32*a^3*b^4*B*Sin[c + d*x] + 8*a*b^6*B*Sin[c + d*x] - 36*a^6*b*C*Sin[c + d*x] + 72*a^4*b
^3*C*Sin[c + d*x] - 38*a^2*b^5*C*Sin[c + d*x] + 8*b^7*C*Sin[c + d*x] - 4*a^5*A*b^2*Sin[2*(c + d*x)] + 10*a^3*A
*b^4*Sin[2*(c + d*x)] + 12*a^6*b*B*Sin[2*(c + d*x)] - 14*a^4*b^3*B*Sin[2*(c + d*x)] - 12*a^2*b^5*B*Sin[2*(c +
d*x)] + 8*b^7*B*Sin[2*(c + d*x)] - 24*a^7*C*Sin[2*(c + d*x)] + 26*a^5*b^2*C*Sin[2*(c + d*x)] + 20*a^3*b^4*C*Si
n[2*(c + d*x)] - 16*a*b^6*C*Sin[2*(c + d*x)] - 6*a^4*A*b^3*Sin[3*(c + d*x)] + 12*a^2*A*b^5*Sin[3*(c + d*x)] +
18*a^5*b^2*B*Sin[3*(c + d*x)] - 32*a^3*b^4*B*Sin[3*(c + d*x)] + 8*a*b^6*B*Sin[3*(c + d*x)] - 36*a^6*b*C*Sin[3*
(c + d*x)] + 64*a^4*b^3*C*Sin[3*(c + d*x)] - 22*a^2*b^5*C*Sin[3*(c + d*x)] - 2*a^5*A*b^2*Sin[4*(c + d*x)] + 5*
a^3*A*b^4*Sin[4*(c + d*x)] + 6*a^6*b*B*Sin[4*(c + d*x)] - 11*a^4*b^3*B*Sin[4*(c + d*x)] + 2*a^2*b^5*B*Sin[4*(c
+ d*x)] - 12*a^7*C*Sin[4*(c + d*x)] + 21*a^5*b^2*C*Sin[4*(c + d*x)] - 6*a^3*b^4*C*Sin[4*(c + d*x)]))/(8*b^4*(
-a^2 + b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3)

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Maple [B]  time = 0.123, size = 2275, normalized size = 4.9 \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+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x)

[Out]

12/d/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*B*a^2+1/d/b
^3*ln(tan(1/2*d*x+1/2*c)+1)*A+1/2/d*C/b^3/(tan(1/2*d*x+1/2*c)-1)^2-1/d/b^3/(tan(1/2*d*x+1/2*c)-1)*B+1/2/d/b^3/
(tan(1/2*d*x+1/2*c)-1)*C-1/d/b^3*ln(tan(1/2*d*x+1/2*c)-1)*A-1/2/d*C/b^3/(tan(1/2*d*x+1/2*c)+1)^2-1/d/b^3/(tan(
1/2*d*x+1/2*c)+1)*B+1/2/d/b^3/(tan(1/2*d*x+1/2*c)+1)*C-1/2/d/b^3*ln(tan(1/2*d*x+1/2*c)-1)*C+1/2/d/b^3*ln(tan(1
/2*d*x+1/2*c)+1)*C+29/d*a^5/b^3/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b
)*(a-b))^(1/2))*C-20/d*a^3/b/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(
a-b))^(1/2))*C+1/d*a^4/b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2
*d*x+1/2*c)^3*B-8/d*a^3/b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*
c)*B-6/d*a^6/b^4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*C+4/d*
a^5/b^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B+1/d*a^4/b^2/(
tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B+8/d*a^3/b/(tan(1/2*d*x
+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+2/d*a^4/b^2/(tan(1/2*d*
x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-2/d*a^4/b^2/(tan(1/2*d
*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A-1/d*a^3/b/(tan(1/2*d*x+1/2*c)^2
*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A+10/d*a^4/b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1
/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*C-10/d*a^4/b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1
/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C+3/d/b^4/(tan(1/2*d*x+1/2*c)-1)*a*C+3/d/b^4*ln(ta
n(1/2*d*x+1/2*c)-1)*B*a-6/d/b^5*ln(tan(1/2*d*x+1/2*c)-1)*a^2*C-1/d*a^3/b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1
/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-4/d*a^5/b^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+
1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+6/d*a^6/b^4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x
+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C-1/d*a^5/b^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*
x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C-1/d*a^5/b^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d
*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*C+3/d/b^4/(tan(1/2*d*x+1/2*c)+1)*a*C-12/d*a^7/b^5/(a^4-2
*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C+5/d*a^3/b/(a^4-2*a^2
*b^2+b^4)/((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/d*a*b/(a^4-2*a^2*b^2+b
^4)/((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/d*a^2/(tan(1/2*d*x+1/2*c)^2*
a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A-6/d*a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x
+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-2/d*a^5/b^3/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^
(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A-3/d/b^4*ln(tan(1/2*d*x+1/2*c)+1)*B*a+6/d/b^5*ln(
tan(1/2*d*x+1/2*c)+1)*a^2*C+6/d*a^6/b^4/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*
c)/((a+b)*(a-b))^(1/2))*B-15/d*a^4/b^2/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c
)/((a+b)*(a-b))^(1/2))*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(sec(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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(sec(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**4*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/(a+b*sec(d*x+c))**3,x)

[Out]

Integral((A + B*sec(c + d*x) + C*sec(c + d*x)**2)*sec(c + d*x)**4/(a + b*sec(c + d*x))**3, x)

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Giac [B]  time = 1.47309, size = 2349, normalized size = 5.05 \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+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, algorithm="giac")

[Out]

-1/2*(2*(12*C*a^7 - 6*B*a^6*b + 2*A*a^5*b^2 - 29*C*a^5*b^2 + 15*B*a^4*b^3 - 5*A*a^3*b^4 + 20*C*a^3*b^4 - 12*B*
a^2*b^5 + 6*A*a*b^6)*(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*t
an(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/((a^4*b^5 - 2*a^2*b^7 + b^9)*sqrt(-a^2 + b^2)) - 2*(12*C*a^7*tan(1/2*d
*x + 1/2*c)^7 - 6*B*a^6*b*tan(1/2*d*x + 1/2*c)^7 - 18*C*a^6*b*tan(1/2*d*x + 1/2*c)^7 + 2*A*a^5*b^2*tan(1/2*d*x
+ 1/2*c)^7 + 9*B*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 - 17*C*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 - 3*A*a^4*b^3*tan(1/2*d
*x + 1/2*c)^7 + 9*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 + 33*C*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 - 5*A*a^3*b^4*tan(1/2
*d*x + 1/2*c)^7 - 16*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 - 2*C*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 + 6*A*a^2*b^5*tan(1
/2*d*x + 1/2*c)^7 + 2*B*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 - 13*C*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 + 4*B*a*b^6*tan(1
/2*d*x + 1/2*c)^7 + 4*C*a*b^6*tan(1/2*d*x + 1/2*c)^7 - 2*B*b^7*tan(1/2*d*x + 1/2*c)^7 + C*b^7*tan(1/2*d*x + 1/
2*c)^7 - 36*C*a^7*tan(1/2*d*x + 1/2*c)^5 + 18*B*a^6*b*tan(1/2*d*x + 1/2*c)^5 + 18*C*a^6*b*tan(1/2*d*x + 1/2*c)
^5 - 6*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 - 9*B*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 + 67*C*a^5*b^2*tan(1/2*d*x + 1/2*
c)^5 + 3*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 - 35*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 - 29*C*a^4*b^3*tan(1/2*d*x + 1
/2*c)^5 + 15*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 + 16*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 - 26*C*a^3*b^4*tan(1/2*d*x
+ 1/2*c)^5 - 6*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 + 10*B*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 + 5*C*a^2*b^5*tan(1/2*d
*x + 1/2*c)^5 - 4*B*a*b^6*tan(1/2*d*x + 1/2*c)^5 + 4*C*a*b^6*tan(1/2*d*x + 1/2*c)^5 - 2*B*b^7*tan(1/2*d*x + 1/
2*c)^5 + 3*C*b^7*tan(1/2*d*x + 1/2*c)^5 + 36*C*a^7*tan(1/2*d*x + 1/2*c)^3 - 18*B*a^6*b*tan(1/2*d*x + 1/2*c)^3
+ 18*C*a^6*b*tan(1/2*d*x + 1/2*c)^3 + 6*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 - 9*B*a^5*b^2*tan(1/2*d*x + 1/2*c)^3
- 67*C*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 + 3*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 + 35*B*a^4*b^3*tan(1/2*d*x + 1/2*c)
^3 - 29*C*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 - 15*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 + 16*B*a^3*b^4*tan(1/2*d*x + 1/
2*c)^3 + 26*C*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 - 6*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 - 10*B*a^2*b^5*tan(1/2*d*x +
1/2*c)^3 + 5*C*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 - 4*B*a*b^6*tan(1/2*d*x + 1/2*c)^3 - 4*C*a*b^6*tan(1/2*d*x + 1/
2*c)^3 + 2*B*b^7*tan(1/2*d*x + 1/2*c)^3 + 3*C*b^7*tan(1/2*d*x + 1/2*c)^3 - 12*C*a^7*tan(1/2*d*x + 1/2*c) + 6*B
*a^6*b*tan(1/2*d*x + 1/2*c) - 18*C*a^6*b*tan(1/2*d*x + 1/2*c) - 2*A*a^5*b^2*tan(1/2*d*x + 1/2*c) + 9*B*a^5*b^2
*tan(1/2*d*x + 1/2*c) + 17*C*a^5*b^2*tan(1/2*d*x + 1/2*c) - 3*A*a^4*b^3*tan(1/2*d*x + 1/2*c) - 9*B*a^4*b^3*tan
(1/2*d*x + 1/2*c) + 33*C*a^4*b^3*tan(1/2*d*x + 1/2*c) + 5*A*a^3*b^4*tan(1/2*d*x + 1/2*c) - 16*B*a^3*b^4*tan(1/
2*d*x + 1/2*c) + 2*C*a^3*b^4*tan(1/2*d*x + 1/2*c) + 6*A*a^2*b^5*tan(1/2*d*x + 1/2*c) - 2*B*a^2*b^5*tan(1/2*d*x
+ 1/2*c) - 13*C*a^2*b^5*tan(1/2*d*x + 1/2*c) + 4*B*a*b^6*tan(1/2*d*x + 1/2*c) - 4*C*a*b^6*tan(1/2*d*x + 1/2*c
) + 2*B*b^7*tan(1/2*d*x + 1/2*c) + C*b^7*tan(1/2*d*x + 1/2*c))/((a^4*b^4 - 2*a^2*b^6 + b^8)*(a*tan(1/2*d*x + 1
/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*a*tan(1/2*d*x + 1/2*c)^2 + a + b)^2) - (12*C*a^2 - 6*B*a*b + 2*A*b^2 +
C*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^5 + (12*C*a^2 - 6*B*a*b + 2*A*b^2 + C*b^2)*log(abs(tan(1/2*d*x + 1
/2*c) - 1))/b^5)/d