3.10.23 \(\int \frac {\sec ^4(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^4} \, dx\) [923]

Optimal. Leaf size=470 \[ \frac {(b B-4 a C) \tanh ^{-1}(\sin (c+d x))}{b^5 d}-\frac {\left (2 A b^8+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} b^5 (a+b)^{7/2} d}-\frac {\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \tan (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^3(c+d x) \tan (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]

[Out]

(B*b-4*C*a)*arctanh(sin(d*x+c))/b^5/d-(2*A*b^8+2*a^7*b*B-7*a^5*b^3*B+8*a^3*b^5*B-8*a*b^7*B-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)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(7/2)/b^5/(a+b)^(
7/2)/d-1/6*(5*A*b^4+3*B*a^3*b-8*B*a*b^3-12*C*a^4+23*C*a^2*b^2-6*C*b^4)*tan(d*x+c)/b^4/(a^2-b^2)^2/d-1/3*(A*b^2
-a*(B*b-C*a))*sec(d*x+c)^3*tan(d*x+c)/b/(a^2-b^2)/d/(a+b*sec(d*x+c))^3+1/6*(3*A*b^4+a^3*b*B-6*a*b^3*B-4*a^4*C+
a^2*b^2*(2*A+9*C))*sec(d*x+c)^2*tan(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*sec(d*x+c))^2+1/2*a*(2*A*b^6-a^5*b*B+2*a^3*b
^3*B-6*a*b^5*B+4*a^6*C-11*a^4*b^2*C+3*a^2*b^4*(A+4*C))*tan(d*x+c)/b^4/(a^2-b^2)^3/d/(a+b*sec(d*x+c))

________________________________________________________________________________________

Rubi [A]
time = 8.20, antiderivative size = 470, normalized size of antiderivative = 1.00, 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 = {4183, 4175, 4167, 4083, 3855, 3916, 2738, 214} \begin {gather*} -\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}-\frac {\tan (c+d x) \left (-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{6 b^4 d \left (a^2-b^2\right )^2}+\frac {\tan (c+d x) \sec ^2(c+d x) \left (-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right )}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac {a \tan (c+d x) \left (4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right )}{2 b^4 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac {\left (-8 a^8 C+2 a^7 b B+28 a^6 b^2 C-7 a^5 b^3 B-35 a^4 b^4 C+8 a^3 b^5 B+a^2 b^6 (3 A+20 C)-8 a b^7 B+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 {(b B-4 a C) \tanh ^{-1}(\sin (c+d x))}{b^5 d} \end {gather*}

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])^4,x]

[Out]

((b*B - 4*a*C)*ArcTanh[Sin[c + d*x]])/(b^5*d) - ((2*A*b^8 + 2*a^7*b*B - 7*a^5*b^3*B + 8*a^3*b^5*B - 8*a*b^7*B
- 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 + 3*a^3*b*B - 8*a*b^3*B - 12*a^4*C + 23*a^2*b^2*C - 6*b
^4*C)*Tan[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*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 + a^3*b*B - 6*a*b^3*B - 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 - a^5*b*B + 2*a^3*b^3*B - 6*a*
b^5*B + 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 214

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

Rule 2738

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 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3916

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

Rule 4083

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 4167

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 4175

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 4183

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]

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))^4} \, dx &=-\frac {\left (A b^2-a (b B-a 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 (b B-a C)\right )+3 b (b B-a (A+C)) \sec (c+d x)-\left (A b^2-a b B+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 (b B-a 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+a^3 b B-6 a b^3 B-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+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right )+2 b \left (2 a^2 b B+3 b^3 B+a^3 C-a b^2 (5 A+6 C)\right ) \sec (c+d x)-\left (5 A b^4+3 a^3 b B-8 a b^3 B-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))^2} \, dx}{6 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (A b^2-a (b B-a 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+a^3 b B-6 a b^3 B-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-a^5 b B+2 a^3 b^3 B-6 a b^5 B+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-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right )+\left (a^2-b^2\right ) \left (3 a^4 b B-4 a^2 b^3 B+6 b^5 B-12 a^5 C+25 a^3 b^2 C-a b^4 (5 A+18 C)\right ) \sec (c+d x)-b \left (a^2-b^2\right ) \left (5 A b^4+3 a^3 b B-8 a b^3 B-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+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \tan (c+d x)}{6 b^4 \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)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {\left (3 A b^4+a^3 b B-6 a b^3 B-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-a^5 b B+2 a^3 b^3 B-6 a b^5 B+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-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right )+6 b \left (a^2-b^2\right )^3 (b B-4 a 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+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \tan (c+d x)}{6 b^4 \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)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {\left (3 A b^4+a^3 b B-6 a b^3 B-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-a^5 b B+2 a^3 b^3 B-6 a b^5 B+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 {(b B-4 a C) \int \sec (c+d x) \, dx}{b^5}-\frac {\left (2 A b^8+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-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 {(b B-4 a C) \tanh ^{-1}(\sin (c+d x))}{b^5 d}-\frac {\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \tan (c+d x)}{6 b^4 \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)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {\left (3 A b^4+a^3 b B-6 a b^3 B-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-a^5 b B+2 a^3 b^3 B-6 a b^5 B+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+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-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 {(b B-4 a C) \tanh ^{-1}(\sin (c+d x))}{b^5 d}-\frac {\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \tan (c+d x)}{6 b^4 \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)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {\left (3 A b^4+a^3 b B-6 a b^3 B-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-a^5 b B+2 a^3 b^3 B-6 a b^5 B+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+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-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 ) \text {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 {(b B-4 a C) \tanh ^{-1}(\sin (c+d x))}{b^5 d}-\frac {\left (3 a^2 A b^6+2 A b^8+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-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+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \tan (c+d x)}{6 b^4 \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)}{3 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {\left (3 A b^4+a^3 b B-6 a b^3 B-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-a^5 b B+2 a^3 b^3 B-6 a b^5 B+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 [B] Leaf count is larger than twice the leaf count of optimal. \(1197\) vs. \(2(470)=940\).
time = 6.50, size = 1197, normalized size = 2.55 \begin {gather*} -\frac {2 \left (3 a^2 A b^6+2 A b^8+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-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 {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^4 \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{b^5 \sqrt {a^2-b^2} \left (-a^2+b^2\right )^3 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4}-\frac {2 (b B-4 a C) (b+a \cos (c+d x))^4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{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))^4}+\frac {2 (b B-4 a C) (b+a \cos (c+d x))^4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{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))^4}+\frac {(b+a \cos (c+d x)) \sec ^3(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-6 a^4 A b^5 \sin (c+d x)-54 a^2 A b^7 \sin (c+d x)+30 a^7 b^2 B \sin (c+d x)-90 a^5 b^4 B \sin (c+d x)+120 a^3 b^6 B \sin (c+d x)-120 a^8 b C \sin (c+d x)+294 a^6 b^3 C \sin (c+d x)-174 a^4 b^5 C \sin (c+d x)-108 a^2 b^7 C \sin (c+d x)+48 b^9 C \sin (c+d x)-16 a^5 A b^4 \sin (2 (c+d x))-2 a^3 A b^6 \sin (2 (c+d x))-72 a A b^8 \sin (2 (c+d x))+12 a^8 b B \sin (2 (c+d x))+10 a^6 b^3 B \sin (2 (c+d x))-76 a^4 b^5 B \sin (2 (c+d x))+144 a^2 b^7 B \sin (2 (c+d x))-48 a^9 C \sin (2 (c+d x))-40 a^7 b^2 C \sin (2 (c+d x))+370 a^5 b^4 C \sin (2 (c+d x))-444 a^3 b^6 C \sin (2 (c+d x))+72 a b^8 C \sin (2 (c+d x))-6 a^4 A b^5 \sin (3 (c+d x))-54 a^2 A b^7 \sin (3 (c+d x))+30 a^7 b^2 B \sin (3 (c+d x))-90 a^5 b^4 B \sin (3 (c+d x))+120 a^3 b^6 B \sin (3 (c+d x))-120 a^8 b C \sin (3 (c+d x))+342 a^6 b^3 C \sin (3 (c+d x))-318 a^4 b^5 C \sin (3 (c+d x))+36 a^2 b^7 C \sin (3 (c+d x))-4 a^5 A b^4 \sin (4 (c+d x))-11 a^3 A b^6 \sin (4 (c+d x))+6 a^8 b B \sin (4 (c+d x))-17 a^6 b^3 B \sin (4 (c+d x))+26 a^4 b^5 B \sin (4 (c+d x))-24 a^9 C \sin (4 (c+d x))+68 a^7 b^2 C \sin (4 (c+d x))-65 a^5 b^4 C \sin (4 (c+d x))+6 a^3 b^6 C \sin (4 (c+d x))\right )}{24 b^4 \left (-a^2+b^2\right )^3 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+b \sec (c+d x))^4} \end {gather*}

Warning: Unable to verify antiderivative.

[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])^4,x]

[Out]

(-2*(3*a^2*A*b^6 + 2*A*b^8 + 2*a^7*b*B - 7*a^5*b^3*B + 8*a^3*b^5*B - 8*a*b^7*B - 8*a^8*C + 28*a^6*b^2*C - 35*a
^4*b^4*C + 20*a^2*b^6*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^4*Sec[c + d
*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b^5*Sqrt[a^2 - b^2]*(-a^2 + b^2)^3*d*(A + 2*C + 2*B*Cos[c + d*
x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) - (2*(b*B - 4*a*C)*(b + a*Cos[c + d*x])^4*Log[Cos[(c + d*x)/2
] - Sin[(c + d*x)/2]]*Sec[c + d*x]^2*(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])^4) + (2*(b*B - 4*a*C)*(b + a*Cos[c + d*x])^4*Log[Cos[(c + d*x)/2
] + Sin[(c + d*x)/2]]*Sec[c + d*x]^2*(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])^4) + ((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^5*Sin[c + d*x] - 54*a^2*A*b^7*Sin[c + d*x] + 30*a^7*b^2*B*Sin[c + d*x] - 90*a^5*
b^4*B*Sin[c + d*x] + 120*a^3*b^6*B*Sin[c + d*x] - 120*a^8*b*C*Sin[c + d*x] + 294*a^6*b^3*C*Sin[c + d*x] - 174*
a^4*b^5*C*Sin[c + d*x] - 108*a^2*b^7*C*Sin[c + d*x] + 48*b^9*C*Sin[c + d*x] - 16*a^5*A*b^4*Sin[2*(c + d*x)] -
2*a^3*A*b^6*Sin[2*(c + d*x)] - 72*a*A*b^8*Sin[2*(c + d*x)] + 12*a^8*b*B*Sin[2*(c + d*x)] + 10*a^6*b^3*B*Sin[2*
(c + d*x)] - 76*a^4*b^5*B*Sin[2*(c + d*x)] + 144*a^2*b^7*B*Sin[2*(c + d*x)] - 48*a^9*C*Sin[2*(c + d*x)] - 40*a
^7*b^2*C*Sin[2*(c + d*x)] + 370*a^5*b^4*C*Sin[2*(c + d*x)] - 444*a^3*b^6*C*Sin[2*(c + d*x)] + 72*a*b^8*C*Sin[2
*(c + d*x)] - 6*a^4*A*b^5*Sin[3*(c + d*x)] - 54*a^2*A*b^7*Sin[3*(c + d*x)] + 30*a^7*b^2*B*Sin[3*(c + d*x)] - 9
0*a^5*b^4*B*Sin[3*(c + d*x)] + 120*a^3*b^6*B*Sin[3*(c + d*x)] - 120*a^8*b*C*Sin[3*(c + d*x)] + 342*a^6*b^3*C*S
in[3*(c + d*x)] - 318*a^4*b^5*C*Sin[3*(c + d*x)] + 36*a^2*b^7*C*Sin[3*(c + d*x)] - 4*a^5*A*b^4*Sin[4*(c + d*x)
] - 11*a^3*A*b^6*Sin[4*(c + d*x)] + 6*a^8*b*B*Sin[4*(c + d*x)] - 17*a^6*b^3*B*Sin[4*(c + d*x)] + 26*a^4*b^5*B*
Sin[4*(c + d*x)] - 24*a^9*C*Sin[4*(c + d*x)] + 68*a^7*b^2*C*Sin[4*(c + d*x)] - 65*a^5*b^4*C*Sin[4*(c + d*x)] +
6*a^3*b^6*C*Sin[4*(c + d*x)]))/(24*b^4*(-a^2 + b^2)^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a
+ b*Sec[c + d*x])^4)

________________________________________________________________________________________

Maple [A]
time = 1.16, size = 679, normalized size = 1.44

method	result	size
derivativedivides	\(\frac {-\frac {C}{b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-b B +4 a C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{5}}-\frac {C}{b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (b B -4 a C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{5}}+\frac {\frac {2 \left (-\frac {\left (2 a^{2} A \,b^{4}+3 A a \,b^{5}+6 A \,b^{6}-2 a^{5} b B +B \,a^{4} b^{2}+6 a^{3} b^{3} B -4 B \,a^{2} b^{4}-12 a \,b^{5} B +6 a^{6} C -2 C \,a^{5} b -18 a^{4} b^{2} C +5 C \,a^{3} b^{3}+20 C \,a^{2} b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {2 \left (a^{2} A \,b^{4}+9 A \,b^{6}-3 a^{5} b B +11 a^{3} b^{3} B -18 a \,b^{5} B +9 a^{6} C -29 a^{4} b^{2} C +30 C \,a^{2} b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (2 a^{2} A \,b^{4}-3 A a \,b^{5}+6 A \,b^{6}-2 a^{5} b B -B \,a^{4} b^{2}+6 a^{3} b^{3} B +4 B \,a^{2} b^{4}-12 a \,b^{5} B +6 a^{6} C +2 C \,a^{5} b -18 a^{4} b^{2} C -5 C \,a^{3} b^{3}+20 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}\right )}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{3}}-\frac {\left (3 a^{2} A \,b^{6}+2 A \,b^{8}+2 a^{7} b B -7 a^{5} b^{3} B +8 a^{3} b^{5} B -8 a \,b^{7} B -8 a^{8} C +28 a^{6} b^{2} C -35 a^{4} b^{4} C +20 C \,a^{2} b^{6}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{b^{5}}}{d}\)	\(679\)
default	\(\frac {-\frac {C}{b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-b B +4 a C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{5}}-\frac {C}{b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (b B -4 a C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{5}}+\frac {\frac {2 \left (-\frac {\left (2 a^{2} A \,b^{4}+3 A a \,b^{5}+6 A \,b^{6}-2 a^{5} b B +B \,a^{4} b^{2}+6 a^{3} b^{3} B -4 B \,a^{2} b^{4}-12 a \,b^{5} B +6 a^{6} C -2 C \,a^{5} b -18 a^{4} b^{2} C +5 C \,a^{3} b^{3}+20 C \,a^{2} b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {2 \left (a^{2} A \,b^{4}+9 A \,b^{6}-3 a^{5} b B +11 a^{3} b^{3} B -18 a \,b^{5} B +9 a^{6} C -29 a^{4} b^{2} C +30 C \,a^{2} b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (2 a^{2} A \,b^{4}-3 A a \,b^{5}+6 A \,b^{6}-2 a^{5} b B -B \,a^{4} b^{2}+6 a^{3} b^{3} B +4 B \,a^{2} b^{4}-12 a \,b^{5} B +6 a^{6} C +2 C \,a^{5} b -18 a^{4} b^{2} C -5 C \,a^{3} b^{3}+20 C \,a^{2} b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}\right )}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{3}}-\frac {\left (3 a^{2} A \,b^{6}+2 A \,b^{8}+2 a^{7} b B -7 a^{5} b^{3} B +8 a^{3} b^{5} B -8 a \,b^{7} B -8 a^{8} C +28 a^{6} b^{2} C -35 a^{4} b^{4} C +20 C \,a^{2} b^{6}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{b^{5}}}{d}\)	\(679\)
risch	\(\text {Expression too large to display}\)	\(3228\)

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))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(-C/b^4/(tan(1/2*d*x+1/2*c)-1)+1/b^5*(-B*b+4*C*a)*ln(tan(1/2*d*x+1/2*c)-1)-C/b^4/(tan(1/2*d*x+1/2*c)+1)+(B
*b-4*C*a)/b^5*ln(tan(1/2*d*x+1/2*c)+1)+2/b^5*((-1/2*(2*A*a^2*b^4+3*A*a*b^5+6*A*b^6-2*B*a^5*b+B*a^4*b^2+6*B*a^3
*b^3-4*B*a^2*b^4-12*B*a*b^5+6*C*a^6-2*C*a^5*b-18*C*a^4*b^2+5*C*a^3*b^3+20*C*a^2*b^4)*a*b/(a-b)/(a^3+3*a^2*b+3*
a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5+2/3*(A*a^2*b^4+9*A*b^6-3*B*a^5*b+11*B*a^3*b^3-18*B*a*b^5+9*C*a^6-29*C*a^4*b^2+
30*C*a^2*b^4)*a*b/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-1/2*(2*A*a^2*b^4-3*A*a*b^5+6*A*b^6-2*B*
a^5*b-B*a^4*b^2+6*B*a^3*b^3+4*B*a^2*b^4-12*B*a*b^5+6*C*a^6+2*C*a^5*b-18*C*a^4*b^2-5*C*a^3*b^3+20*C*a^2*b^4)*a*
b/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c))/(a*tan(1/2*d*x+1/2*c)^2-b*tan(1/2*d*x+1/2*c)^2-a-b)^3-1/
2*(3*A*a^2*b^6+2*A*b^8+2*B*a^7*b-7*B*a^5*b^3+8*B*a^3*b^5-8*B*a*b^7-8*C*a^8+28*C*a^6*b^2-35*C*a^4*b^4+20*C*a^2*
b^6)/(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))))

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more de

________________________________________________________________________________________

Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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))^4,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 )^{4}}\, dx \end {gather*}

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))**4,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))**4, x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1264 vs. \(2 (456) = 912\).
time = 0.58, size = 1264, normalized size = 2.69 \begin {gather*} \text {Too large to display} \end {gather*}

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))^4,x, algorithm="giac")

[Out]

1/3*(3*(8*C*a^8 - 2*B*a^7*b - 28*C*a^6*b^2 + 7*B*a^5*b^3 + 35*C*a^4*b^4 - 8*B*a^3*b^5 - 3*A*a^2*b^6 - 20*C*a^2
*b^6 + 8*B*a*b^7 - 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)) - (
18*C*a^9*tan(1/2*d*x + 1/2*c)^5 - 6*B*a^8*b*tan(1/2*d*x + 1/2*c)^5 - 42*C*a^8*b*tan(1/2*d*x + 1/2*c)^5 + 15*B*
a^7*b^2*tan(1/2*d*x + 1/2*c)^5 - 24*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 11
7*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 - 45*B*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 + 6*B*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*B*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 - 36*B*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 + 12*B*a^8*b*tan(1/2*d*x + 1/2
*c)^3 + 152*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^3 - 56*B*a^6*b^3*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 + 116*B*a^4*b^5*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 - 72*B*a^2*b^7*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) - 6*B*a^8*b*tan(1/2*d*x + 1/2*c) + 42*C*a^8*b*tan(1/2*d*x +
1/2*c) - 15*B*a^7*b^2*tan(1/2*d*x + 1/2*c) - 24*C*a^7*b^2*tan(1/2*d*x + 1/2*c) + 6*B*a^6*b^3*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) + 45*B*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) + 6*B*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*B*a^3*b^6*tan(1/2*d*x + 1/2*c) + 6
0*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) - 36*B*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) - 3*(4*C*a - B*b)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^5 + 3*(4*C*a - B*b)*log(abs(
tan(1/2*d*x + 1/2*c) - 1))/b^5 - 6*C*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 - 1)*b^4))/d

________________________________________________________________________________________

Mupad [B]
time = 20.92, size = 2500, normalized size = 5.32 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B/cos(c + d*x) + C/cos(c + d*x)^2)/(cos(c + d*x)^4*(a + b/cos(c + d*x))^4),x)

[Out]

(atan(((((8*tan(c/2 + (d*x)/2)*(4*A^2*b^16 + 4*B^2*b^16 + 128*C^2*a^16 - 8*B^2*a*b^15 - 128*C^2*a^15*b + 12*A^
2*a^2*b^14 + 9*A^2*a^4*b^12 + 44*B^2*a^2*b^14 + 48*B^2*a^3*b^13 - 92*B^2*a^4*b^12 - 120*B^2*a^5*b^11 + 156*B^2
*a^6*b^10 + 160*B^2*a^7*b^9 - 164*B^2*a^8*b^8 - 120*B^2*a^9*b^7 + 117*B^2*a^10*b^6 + 48*B^2*a^11*b^5 - 48*B^2*
a^12*b^4 - 8*B^2*a^13*b^3 + 8*B^2*a^14*b^2 + 64*C^2*a^2*b^14 - 128*C^2*a^3*b^13 + 80*C^2*a^4*b^12 + 768*C^2*a^
5*b^11 - 824*C^2*a^6*b^10 - 1920*C^2*a^7*b^9 + 2025*C^2*a^8*b^8 + 2560*C^2*a^9*b^7 - 2600*C^2*a^10*b^6 - 1920*
C^2*a^11*b^5 + 1920*C^2*a^12*b^4 + 768*C^2*a^13*b^3 - 768*C^2*a^14*b^2 - 32*A*B*a*b^15 - 32*B*C*a*b^15 - 64*B*
C*a^15*b - 16*A*B*a^3*b^13 + 20*A*B*a^5*b^11 - 34*A*B*a^7*b^9 + 12*A*B*a^9*b^7 + 80*A*C*a^2*b^14 - 20*A*C*a^4*
b^12 - 98*A*C*a^6*b^10 + 136*A*C*a^8*b^8 - 48*A*C*a^10*b^6 + 64*B*C*a^2*b^14 - 160*B*C*a^3*b^13 - 384*B*C*a^4*
b^12 + 592*B*C*a^5*b^11 + 960*B*C*a^6*b^10 - 1128*B*C*a^7*b^9 - 1280*B*C*a^8*b^8 + 1306*B*C*a^9*b^7 + 960*B*C*
a^10*b^6 - 948*B*C*a^11*b^5 - 384*B*C*a^12*b^4 + 384*B*C*a^13*b^3 + 64*B*C*a^14*b^2))/(a*b^18 + b^19 - 5*a^2*b
^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9
- a^11*b^8) + (((8*(4*A*b^24 + 4*B*b^24 - 6*A*a^2*b^22 + 6*A*a^3*b^21 - 6*A*a^4*b^20 + 6*A*a^5*b^19 + 14*A*a^6
*b^18 - 14*A*a^7*b^17 - 6*A*a^8*b^16 + 6*A*a^9*b^15 - 12*B*a^2*b^22 + 64*B*a^3*b^21 + 20*B*a^4*b^20 - 110*B*a^
5*b^19 - 30*B*a^6*b^18 + 110*B*a^7*b^17 + 30*B*a^8*b^16 - 70*B*a^9*b^15 - 14*B*a^10*b^14 + 26*B*a^11*b^13 + 2*
B*a^12*b^12 - 4*B*a^13*b^11 + 40*C*a^2*b^22 + 72*C*a^3*b^21 - 190*C*a^4*b^20 - 146*C*a^5*b^19 + 386*C*a^6*b^18
+ 174*C*a^7*b^17 - 434*C*a^8*b^16 - 126*C*a^9*b^15 + 286*C*a^10*b^14 + 50*C*a^11*b^13 - 104*C*a^12*b^12 - 8*C
*a^13*b^11 + 16*C*a^14*b^10 - 4*A*a*b^23 - 16*B*a*b^23 - 16*C*a*b^23))/(a*b^22 + b^23 - 5*a^2*b^21 - 5*a^3*b^2
0 + 10*a^4*b^19 + 10*a^5*b^18 - 10*a^6*b^17 - 10*a^7*b^16 + 5*a^8*b^15 + 5*a^9*b^14 - a^10*b^13 - a^11*b^12) +
(8*tan(c/2 + (d*x)/2)*(B*b - 4*C*a)*(8*a*b^23 - 8*a^2*b^22 - 48*a^3*b^21 + 48*a^4*b^20 + 120*a^5*b^19 - 120*a
^6*b^18 - 160*a^7*b^17 + 160*a^8*b^16 + 120*a^9*b^15 - 120*a^10*b^14 - 48*a^11*b^13 + 48*a^12*b^12 + 8*a^13*b^
11 - 8*a^14*b^10))/(b^5*(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 1
0*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8)))*(B*b - 4*C*a))/b^5)*(B*b - 4*C*a)*1i)/b^5 + (((8
*tan(c/2 + (d*x)/2)*(4*A^2*b^16 + 4*B^2*b^16 + 128*C^2*a^16 - 8*B^2*a*b^15 - 128*C^2*a^15*b + 12*A^2*a^2*b^14
+ 9*A^2*a^4*b^12 + 44*B^2*a^2*b^14 + 48*B^2*a^3*b^13 - 92*B^2*a^4*b^12 - 120*B^2*a^5*b^11 + 156*B^2*a^6*b^10 +
160*B^2*a^7*b^9 - 164*B^2*a^8*b^8 - 120*B^2*a^9*b^7 + 117*B^2*a^10*b^6 + 48*B^2*a^11*b^5 - 48*B^2*a^12*b^4 -
8*B^2*a^13*b^3 + 8*B^2*a^14*b^2 + 64*C^2*a^2*b^14 - 128*C^2*a^3*b^13 + 80*C^2*a^4*b^12 + 768*C^2*a^5*b^11 - 82
4*C^2*a^6*b^10 - 1920*C^2*a^7*b^9 + 2025*C^2*a^8*b^8 + 2560*C^2*a^9*b^7 - 2600*C^2*a^10*b^6 - 1920*C^2*a^11*b^
5 + 1920*C^2*a^12*b^4 + 768*C^2*a^13*b^3 - 768*C^2*a^14*b^2 - 32*A*B*a*b^15 - 32*B*C*a*b^15 - 64*B*C*a^15*b -
16*A*B*a^3*b^13 + 20*A*B*a^5*b^11 - 34*A*B*a^7*b^9 + 12*A*B*a^9*b^7 + 80*A*C*a^2*b^14 - 20*A*C*a^4*b^12 - 98*A
*C*a^6*b^10 + 136*A*C*a^8*b^8 - 48*A*C*a^10*b^6 + 64*B*C*a^2*b^14 - 160*B*C*a^3*b^13 - 384*B*C*a^4*b^12 + 592*
B*C*a^5*b^11 + 960*B*C*a^6*b^10 - 1128*B*C*a^7*b^9 - 1280*B*C*a^8*b^8 + 1306*B*C*a^9*b^7 + 960*B*C*a^10*b^6 -
948*B*C*a^11*b^5 - 384*B*C*a^12*b^4 + 384*B*C*a^13*b^3 + 64*B*C*a^14*b^2))/(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3
*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12 + 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8)
- (((8*(4*A*b^24 + 4*B*b^24 - 6*A*a^2*b^22 + 6*A*a^3*b^21 - 6*A*a^4*b^20 + 6*A*a^5*b^19 + 14*A*a^6*b^18 - 14*
A*a^7*b^17 - 6*A*a^8*b^16 + 6*A*a^9*b^15 - 12*B*a^2*b^22 + 64*B*a^3*b^21 + 20*B*a^4*b^20 - 110*B*a^5*b^19 - 30
*B*a^6*b^18 + 110*B*a^7*b^17 + 30*B*a^8*b^16 - 70*B*a^9*b^15 - 14*B*a^10*b^14 + 26*B*a^11*b^13 + 2*B*a^12*b^12
- 4*B*a^13*b^11 + 40*C*a^2*b^22 + 72*C*a^3*b^21 - 190*C*a^4*b^20 - 146*C*a^5*b^19 + 386*C*a^6*b^18 + 174*C*a^
7*b^17 - 434*C*a^8*b^16 - 126*C*a^9*b^15 + 286*C*a^10*b^14 + 50*C*a^11*b^13 - 104*C*a^12*b^12 - 8*C*a^13*b^11
+ 16*C*a^14*b^10 - 4*A*a*b^23 - 16*B*a*b^23 - 16*C*a*b^23))/(a*b^22 + b^23 - 5*a^2*b^21 - 5*a^3*b^20 + 10*a^4*
b^19 + 10*a^5*b^18 - 10*a^6*b^17 - 10*a^7*b^16 + 5*a^8*b^15 + 5*a^9*b^14 - a^10*b^13 - a^11*b^12) - (8*tan(c/2
+ (d*x)/2)*(B*b - 4*C*a)*(8*a*b^23 - 8*a^2*b^22 - 48*a^3*b^21 + 48*a^4*b^20 + 120*a^5*b^19 - 120*a^6*b^18 - 1
60*a^7*b^17 + 160*a^8*b^16 + 120*a^9*b^15 - 120*a^10*b^14 - 48*a^11*b^13 + 48*a^12*b^12 + 8*a^13*b^11 - 8*a^14
*b^10))/(b^5*(a*b^18 + b^19 - 5*a^2*b^17 - 5*a^3*b^16 + 10*a^4*b^15 + 10*a^5*b^14 - 10*a^6*b^13 - 10*a^7*b^12
+ 5*a^8*b^11 + 5*a^9*b^10 - a^10*b^9 - a^11*b^8)))*(B*b - 4*C*a))/b^5)*(B*b - 4*C*a)*1i)/b^5)/((16*(256*C^3*a^
16 + 4*A*B^2*b^16 - 4*A^2*B*b^16 - 16*B^3*a*b^15 - 128*C^3*a^15*b - 48*B^3*a^2*b^14 + 64*B^3*a^3*b^13 + 64*B^3
*a^4*b^12 - 110*B^3*a^5*b^11 - 66*B^3*a^6*b^10 ...

________________________________________________________________________________________