3.887 \(\int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=491 \[ -\frac{\tan (c+d x) \left (-4 a^4 b^2 (42 A+23 C)-32 a^2 b^4 (49 A+39 C)-847 a^3 b^3 B+28 a^5 b B-8 a^6 C-896 a b^5 B-32 b^6 (7 A+6 C)\right )}{420 b^2 d}+\frac{\left (8 a^3 b (4 A+3 C)+36 a^2 b^2 B+8 a^4 B+4 a b^3 (6 A+5 C)+5 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\tan (c+d x) \left (2 a^2 C-7 a b B+42 A b^2+36 b^2 C\right ) (a+b \sec (c+d x))^4}{210 b^2 d}-\frac{\tan (c+d x) \left (28 a^2 b B-8 a^3 C-4 a b^2 (42 A+31 C)-175 b^3 B\right ) (a+b \sec (c+d x))^3}{840 b^2 d}-\frac{\tan (c+d x) \left (-12 a^2 b^2 (14 A+9 C)+28 a^3 b B-8 a^4 C-371 a b^3 B-32 b^4 (7 A+6 C)\right ) (a+b \sec (c+d x))^2}{840 b^2 d}-\frac{\tan (c+d x) \sec (c+d x) \left (-48 a^3 b^2 (7 A+4 C)-1246 a^2 b^3 B+56 a^4 b B-16 a^5 C-4 a b^4 (406 A+333 C)-525 b^5 B\right )}{1680 b d}+\frac{(7 b B-2 a C) \tan (c+d x) (a+b \sec (c+d x))^5}{42 b^2 d}+\frac{C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^5}{7 b d} \]

[Out]

((8*a^4*B + 36*a^2*b^2*B + 5*b^4*B + 8*a^3*b*(4*A + 3*C) + 4*a*b^3*(6*A + 5*C))*ArcTanh[Sin[c + d*x]])/(16*d)
- ((28*a^5*b*B - 847*a^3*b^3*B - 896*a*b^5*B - 8*a^6*C - 32*b^6*(7*A + 6*C) - 4*a^4*b^2*(42*A + 23*C) - 32*a^2
*b^4*(49*A + 39*C))*Tan[c + d*x])/(420*b^2*d) - ((56*a^4*b*B - 1246*a^2*b^3*B - 525*b^5*B - 16*a^5*C - 48*a^3*
b^2*(7*A + 4*C) - 4*a*b^4*(406*A + 333*C))*Sec[c + d*x]*Tan[c + d*x])/(1680*b*d) - ((28*a^3*b*B - 371*a*b^3*B
- 8*a^4*C - 32*b^4*(7*A + 6*C) - 12*a^2*b^2*(14*A + 9*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(840*b^2*d) - (
(28*a^2*b*B - 175*b^3*B - 8*a^3*C - 4*a*b^2*(42*A + 31*C))*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(840*b^2*d) +
((42*A*b^2 - 7*a*b*B + 2*a^2*C + 36*b^2*C)*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(210*b^2*d) + ((7*b*B - 2*a*C)
*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(42*b^2*d) + (C*Sec[c + d*x]*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(7*b*d
)

________________________________________________________________________________________

Rubi [A]  time = 1.23404, antiderivative size = 491, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.195, Rules used = {4092, 4082, 4002, 3997, 3787, 3770, 3767, 8} \[ -\frac{\tan (c+d x) \left (-4 a^4 b^2 (42 A+23 C)-32 a^2 b^4 (49 A+39 C)-847 a^3 b^3 B+28 a^5 b B-8 a^6 C-896 a b^5 B-32 b^6 (7 A+6 C)\right )}{420 b^2 d}+\frac{\left (8 a^3 b (4 A+3 C)+36 a^2 b^2 B+8 a^4 B+4 a b^3 (6 A+5 C)+5 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\tan (c+d x) \left (2 a^2 C-7 a b B+42 A b^2+36 b^2 C\right ) (a+b \sec (c+d x))^4}{210 b^2 d}-\frac{\tan (c+d x) \left (28 a^2 b B-8 a^3 C-4 a b^2 (42 A+31 C)-175 b^3 B\right ) (a+b \sec (c+d x))^3}{840 b^2 d}-\frac{\tan (c+d x) \left (-12 a^2 b^2 (14 A+9 C)+28 a^3 b B-8 a^4 C-371 a b^3 B-32 b^4 (7 A+6 C)\right ) (a+b \sec (c+d x))^2}{840 b^2 d}-\frac{\tan (c+d x) \sec (c+d x) \left (-48 a^3 b^2 (7 A+4 C)-1246 a^2 b^3 B+56 a^4 b B-16 a^5 C-4 a b^4 (406 A+333 C)-525 b^5 B\right )}{1680 b d}+\frac{(7 b B-2 a C) \tan (c+d x) (a+b \sec (c+d x))^5}{42 b^2 d}+\frac{C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^5}{7 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*a^4*B + 36*a^2*b^2*B + 5*b^4*B + 8*a^3*b*(4*A + 3*C) + 4*a*b^3*(6*A + 5*C))*ArcTanh[Sin[c + d*x]])/(16*d)
- ((28*a^5*b*B - 847*a^3*b^3*B - 896*a*b^5*B - 8*a^6*C - 32*b^6*(7*A + 6*C) - 4*a^4*b^2*(42*A + 23*C) - 32*a^2
*b^4*(49*A + 39*C))*Tan[c + d*x])/(420*b^2*d) - ((56*a^4*b*B - 1246*a^2*b^3*B - 525*b^5*B - 16*a^5*C - 48*a^3*
b^2*(7*A + 4*C) - 4*a*b^4*(406*A + 333*C))*Sec[c + d*x]*Tan[c + d*x])/(1680*b*d) - ((28*a^3*b*B - 371*a*b^3*B
- 8*a^4*C - 32*b^4*(7*A + 6*C) - 12*a^2*b^2*(14*A + 9*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(840*b^2*d) - (
(28*a^2*b*B - 175*b^3*B - 8*a^3*C - 4*a*b^2*(42*A + 31*C))*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(840*b^2*d) +
((42*A*b^2 - 7*a*b*B + 2*a^2*C + 36*b^2*C)*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(210*b^2*d) + ((7*b*B - 2*a*C)
*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(42*b^2*d) + (C*Sec[c + d*x]*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(7*b*d
)

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 4002

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

Rule 3997

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.
) + (A_)), x_Symbol] :> -Simp[(b*B*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*(n + 1)), x] + Dist[1/(n + 1), Int[(d*C
sc[e + f*x])^n*Simp[A*a*(n + 1) + B*b*n + (A*b + B*a)*(n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f
, A, B}, x] && NeQ[A*b - a*B, 0] &&  !LeQ[n, -1]

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 3770

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

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x))^4 \left (a C+b (7 A+6 C) \sec (c+d x)+(7 b B-2 a C) \sec ^2(c+d x)\right ) \, dx}{7 b}\\ &=\frac{(7 b B-2 a C) (a+b \sec (c+d x))^5 \tan (c+d x)}{42 b^2 d}+\frac{C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x))^4 \left (b (35 b B-4 a C)+\left (42 A b^2-7 a b B+2 a^2 C+36 b^2 C\right ) \sec (c+d x)\right ) \, dx}{42 b^2}\\ &=\frac{\left (42 A b^2-7 a b B+2 a^2 C+36 b^2 C\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{210 b^2 d}+\frac{(7 b B-2 a C) (a+b \sec (c+d x))^5 \tan (c+d x)}{42 b^2 d}+\frac{C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x))^3 \left (3 b \left (56 A b^2+49 a b B-4 a^2 C+48 b^2 C\right )-\left (28 a^2 b B-175 b^3 B-8 a^3 C-4 a b^2 (42 A+31 C)\right ) \sec (c+d x)\right ) \, dx}{210 b^2}\\ &=-\frac{\left (28 a^2 b B-175 b^3 B-8 a^3 C-4 a b^2 (42 A+31 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{840 b^2 d}+\frac{\left (42 A b^2-7 a b B+2 a^2 C+36 b^2 C\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{210 b^2 d}+\frac{(7 b B-2 a C) (a+b \sec (c+d x))^5 \tan (c+d x)}{42 b^2 d}+\frac{C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x))^2 \left (3 b \left (168 a^2 b B+175 b^3 B-8 a^3 C+4 a b^2 (98 A+79 C)\right )-3 \left (28 a^3 b B-371 a b^3 B-8 a^4 C-32 b^4 (7 A+6 C)-12 a^2 b^2 (14 A+9 C)\right ) \sec (c+d x)\right ) \, dx}{840 b^2}\\ &=-\frac{\left (28 a^3 b B-371 a b^3 B-8 a^4 C-32 b^4 (7 A+6 C)-12 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{840 b^2 d}-\frac{\left (28 a^2 b B-175 b^3 B-8 a^3 C-4 a b^2 (42 A+31 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{840 b^2 d}+\frac{\left (42 A b^2-7 a b B+2 a^2 C+36 b^2 C\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{210 b^2 d}+\frac{(7 b B-2 a C) (a+b \sec (c+d x))^5 \tan (c+d x)}{42 b^2 d}+\frac{C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x)) \left (3 b \left (448 a^3 b B+1267 a b^3 B-8 a^4 C+64 b^4 (7 A+6 C)+12 a^2 b^2 (126 A+97 C)\right )-3 \left (56 a^4 b B-1246 a^2 b^3 B-525 b^5 B-16 a^5 C-48 a^3 b^2 (7 A+4 C)-4 a b^4 (406 A+333 C)\right ) \sec (c+d x)\right ) \, dx}{2520 b^2}\\ &=-\frac{\left (56 a^4 b B-1246 a^2 b^3 B-525 b^5 B-16 a^5 C-48 a^3 b^2 (7 A+4 C)-4 a b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{1680 b d}-\frac{\left (28 a^3 b B-371 a b^3 B-8 a^4 C-32 b^4 (7 A+6 C)-12 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{840 b^2 d}-\frac{\left (28 a^2 b B-175 b^3 B-8 a^3 C-4 a b^2 (42 A+31 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{840 b^2 d}+\frac{\left (42 A b^2-7 a b B+2 a^2 C+36 b^2 C\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{210 b^2 d}+\frac{(7 b B-2 a C) (a+b \sec (c+d x))^5 \tan (c+d x)}{42 b^2 d}+\frac{C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac{\int \sec (c+d x) \left (315 b^2 \left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right )-12 \left (28 a^5 b B-847 a^3 b^3 B-896 a b^5 B-8 a^6 C-32 b^6 (7 A+6 C)-4 a^4 b^2 (42 A+23 C)-32 a^2 b^4 (49 A+39 C)\right ) \sec (c+d x)\right ) \, dx}{5040 b^2}\\ &=-\frac{\left (56 a^4 b B-1246 a^2 b^3 B-525 b^5 B-16 a^5 C-48 a^3 b^2 (7 A+4 C)-4 a b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{1680 b d}-\frac{\left (28 a^3 b B-371 a b^3 B-8 a^4 C-32 b^4 (7 A+6 C)-12 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{840 b^2 d}-\frac{\left (28 a^2 b B-175 b^3 B-8 a^3 C-4 a b^2 (42 A+31 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{840 b^2 d}+\frac{\left (42 A b^2-7 a b B+2 a^2 C+36 b^2 C\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{210 b^2 d}+\frac{(7 b B-2 a C) (a+b \sec (c+d x))^5 \tan (c+d x)}{42 b^2 d}+\frac{C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac{1}{16} \left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right ) \int \sec (c+d x) \, dx-\frac{\left (28 a^5 b B-847 a^3 b^3 B-896 a b^5 B-8 a^6 C-32 b^6 (7 A+6 C)-4 a^4 b^2 (42 A+23 C)-32 a^2 b^4 (49 A+39 C)\right ) \int \sec ^2(c+d x) \, dx}{420 b^2}\\ &=\frac{\left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac{\left (56 a^4 b B-1246 a^2 b^3 B-525 b^5 B-16 a^5 C-48 a^3 b^2 (7 A+4 C)-4 a b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{1680 b d}-\frac{\left (28 a^3 b B-371 a b^3 B-8 a^4 C-32 b^4 (7 A+6 C)-12 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{840 b^2 d}-\frac{\left (28 a^2 b B-175 b^3 B-8 a^3 C-4 a b^2 (42 A+31 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{840 b^2 d}+\frac{\left (42 A b^2-7 a b B+2 a^2 C+36 b^2 C\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{210 b^2 d}+\frac{(7 b B-2 a C) (a+b \sec (c+d x))^5 \tan (c+d x)}{42 b^2 d}+\frac{C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac{\left (28 a^5 b B-847 a^3 b^3 B-896 a b^5 B-8 a^6 C-32 b^6 (7 A+6 C)-4 a^4 b^2 (42 A+23 C)-32 a^2 b^4 (49 A+39 C)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{420 b^2 d}\\ &=\frac{\left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac{\left (28 a^5 b B-847 a^3 b^3 B-896 a b^5 B-8 a^6 C-32 b^6 (7 A+6 C)-4 a^4 b^2 (42 A+23 C)-32 a^2 b^4 (49 A+39 C)\right ) \tan (c+d x)}{420 b^2 d}-\frac{\left (56 a^4 b B-1246 a^2 b^3 B-525 b^5 B-16 a^5 C-48 a^3 b^2 (7 A+4 C)-4 a b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{1680 b d}-\frac{\left (28 a^3 b B-371 a b^3 B-8 a^4 C-32 b^4 (7 A+6 C)-12 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{840 b^2 d}-\frac{\left (28 a^2 b B-175 b^3 B-8 a^3 C-4 a b^2 (42 A+31 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{840 b^2 d}+\frac{\left (42 A b^2-7 a b B+2 a^2 C+36 b^2 C\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{210 b^2 d}+\frac{(7 b B-2 a C) (a+b \sec (c+d x))^5 \tan (c+d x)}{42 b^2 d}+\frac{C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}\\ \end{align*}

Mathematica [A]  time = 4.61019, size = 486, normalized size = 0.99 \[ -\frac{\sec ^6(c+d x) \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right ) \left (-8 b^2 \left (3 \sin (2 (c+d x)) \left (42 a^2 C+28 a b B+7 A b^2+6 b^2 C\right )+35 b (4 a C+b B) \sin (c+d x)+30 b^2 C \tan (c+d x)\right )-16 \sin (c+d x) \cos ^5(c+d x) \left (84 a^2 b^2 (5 A+4 C)+35 a^4 (3 A+2 C)+280 a^3 b B+224 a b^3 B+8 b^4 (7 A+6 C)\right )-105 \sin (c+d x) \cos ^4(c+d x) \left (8 a^3 b (4 A+3 C)+36 a^2 b^2 B+8 a^4 B+4 a b^3 (6 A+5 C)+5 b^4 B\right )-16 \sin (c+d x) \cos ^3(c+d x) \left (42 a^2 b^2 (5 A+4 C)+140 a^3 b B+35 a^4 C+112 a b^3 B+4 b^4 (7 A+6 C)\right )-70 b \sin (c+d x) \cos ^2(c+d x) \left (36 a^2 b B+24 a^3 C+4 a b^2 (6 A+5 C)+5 b^3 B\right )+105 \cos ^6(c+d x) \left (8 a^3 b (4 A+3 C)+36 a^2 b^2 B+8 a^4 B+4 a b^3 (6 A+5 C)+5 b^4 B\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{840 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[c + d*x]^6*(105*(8*a^4*B + 36*a^2*b^2*B + 5*b^4*B + 8*a^3*b*(4*A
+ 3*C) + 4*a*b^3*(6*A + 5*C))*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2]
+ Sin[(c + d*x)/2]]) - 70*b*(36*a^2*b*B + 5*b^3*B + 24*a^3*C + 4*a*b^2*(6*A + 5*C))*Cos[c + d*x]^2*Sin[c + d*
x] - 16*(140*a^3*b*B + 112*a*b^3*B + 35*a^4*C + 42*a^2*b^2*(5*A + 4*C) + 4*b^4*(7*A + 6*C))*Cos[c + d*x]^3*Sin
[c + d*x] - 105*(8*a^4*B + 36*a^2*b^2*B + 5*b^4*B + 8*a^3*b*(4*A + 3*C) + 4*a*b^3*(6*A + 5*C))*Cos[c + d*x]^4*
Sin[c + d*x] - 16*(280*a^3*b*B + 224*a*b^3*B + 35*a^4*(3*A + 2*C) + 84*a^2*b^2*(5*A + 4*C) + 8*b^4*(7*A + 6*C)
)*Cos[c + d*x]^5*Sin[c + d*x] - 8*b^2*(35*b*(b*B + 4*a*C)*Sin[c + d*x] + 3*(7*A*b^2 + 28*a*b*B + 42*a^2*C + 6*
b^2*C)*Sin[2*(c + d*x)] + 30*b^2*C*Tan[c + d*x])))/(840*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))

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Maple [A]  time = 0.077, size = 905, normalized size = 1.8 \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)^2*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

1/3/d*a^4*C*tan(d*x+c)*sec(d*x+c)^2+2/d*A*a^3*b*ln(sec(d*x+c)+tan(d*x+c))+3/2/d*a^3*b*C*ln(sec(d*x+c)+tan(d*x+
c))+16/5/d*C*a^2*b^2*tan(d*x+c)+1/5/d*A*b^4*tan(d*x+c)*sec(d*x+c)^4+4/15/d*A*b^4*tan(d*x+c)*sec(d*x+c)^2+1/7/d
*C*b^4*tan(d*x+c)*sec(d*x+c)^6+6/35/d*C*b^4*tan(d*x+c)*sec(d*x+c)^4+8/35/d*C*b^4*tan(d*x+c)*sec(d*x+c)^2+32/15
/d*a*b^3*B*tan(d*x+c)+8/3/d*B*a^3*b*tan(d*x+c)+4/d*A*a^2*b^2*tan(d*x+c)+5/4/d*C*a*b^3*ln(sec(d*x+c)+tan(d*x+c)
)+1/6/d*B*b^4*tan(d*x+c)*sec(d*x+c)^5+5/24/d*B*b^4*tan(d*x+c)*sec(d*x+c)^3+5/16/d*B*b^4*sec(d*x+c)*tan(d*x+c)+
3/2/d*A*a*b^3*ln(sec(d*x+c)+tan(d*x+c))+2/3/d*a^4*C*tan(d*x+c)+1/d*A*a^4*tan(d*x+c)+1/2/d*B*a^4*ln(sec(d*x+c)+
tan(d*x+c))+16/35/d*C*b^4*tan(d*x+c)+8/15/d*A*b^4*tan(d*x+c)+5/16/d*B*b^4*ln(sec(d*x+c)+tan(d*x+c))+1/2/d*B*a^
4*sec(d*x+c)*tan(d*x+c)+4/5/d*a*b^3*B*tan(d*x+c)*sec(d*x+c)^4+16/15/d*a*b^3*B*tan(d*x+c)*sec(d*x+c)^2+4/3/d*B*
a^3*b*tan(d*x+c)*sec(d*x+c)^2+2/d*A*a^2*b^2*tan(d*x+c)*sec(d*x+c)^2+2/3/d*C*a*b^3*tan(d*x+c)*sec(d*x+c)^5+5/6/
d*C*a*b^3*tan(d*x+c)*sec(d*x+c)^3+5/4/d*C*a*b^3*sec(d*x+c)*tan(d*x+c)+9/4/d*a^2*b^2*B*ln(sec(d*x+c)+tan(d*x+c)
)+2/d*A*a^3*b*sec(d*x+c)*tan(d*x+c)+1/d*a^3*b*C*tan(d*x+c)*sec(d*x+c)^3+3/2/d*a^3*b*C*sec(d*x+c)*tan(d*x+c)+3/
2/d*a^2*b^2*B*tan(d*x+c)*sec(d*x+c)^3+9/4/d*a^2*b^2*B*sec(d*x+c)*tan(d*x+c)+1/d*A*a*b^3*tan(d*x+c)*sec(d*x+c)^
3+3/2/d*A*a*b^3*sec(d*x+c)*tan(d*x+c)+6/5/d*C*a^2*b^2*tan(d*x+c)*sec(d*x+c)^4+8/5/d*C*a^2*b^2*tan(d*x+c)*sec(d
*x+c)^2

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Maxima [A]  time = 1.07927, size = 1007, normalized size = 2.05 \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)^2*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

1/3360*(1120*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^4 + 4480*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a^3*b + 6720*(
tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^2*b^2 + 1344*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*C*a
^2*b^2 + 896*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*B*a*b^3 + 224*(3*tan(d*x + c)^5 + 10*tan
(d*x + c)^3 + 15*tan(d*x + c))*A*b^4 + 96*(5*tan(d*x + c)^7 + 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 35*tan(d
*x + c))*C*b^4 - 140*C*a*b^3*(2*(15*sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 33*sin(d*x + c))/(sin(d*x + c)^6 - 3*
sin(d*x + c)^4 + 3*sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 35*B*b^4*(2*(1
5*sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 33*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^4 + 3*sin(d*x + c)^2
- 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 840*C*a^3*b*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c
))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 1260*B*a^2*b
^2*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) +
3*log(sin(d*x + c) - 1)) - 840*A*a*b^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)
^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 840*B*a^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1)
- log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 3360*A*a^3*b*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin
(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 3360*A*a^4*tan(d*x + c))/d

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Fricas [A]  time = 0.676581, size = 1085, normalized size = 2.21 \begin{align*} \frac{105 \,{\left (8 \, B a^{4} + 8 \,{\left (4 \, A + 3 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 4 \,{\left (6 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \,{\left (8 \, B a^{4} + 8 \,{\left (4 \, A + 3 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 4 \,{\left (6 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (16 \,{\left (35 \,{\left (3 \, A + 2 \, C\right )} a^{4} + 280 \, B a^{3} b + 84 \,{\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 224 \, B a b^{3} + 8 \,{\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} + 105 \,{\left (8 \, B a^{4} + 8 \,{\left (4 \, A + 3 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 4 \,{\left (6 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} + 240 \, C b^{4} + 16 \,{\left (35 \, C a^{4} + 140 \, B a^{3} b + 42 \,{\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 112 \, B a b^{3} + 4 \,{\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 70 \,{\left (24 \, C a^{3} b + 36 \, B a^{2} b^{2} + 4 \,{\left (6 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 48 \,{\left (42 \, C a^{2} b^{2} + 28 \, B a b^{3} +{\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 280 \,{\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3360 \, d \cos \left (d x + c\right )^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3360*(105*(8*B*a^4 + 8*(4*A + 3*C)*a^3*b + 36*B*a^2*b^2 + 4*(6*A + 5*C)*a*b^3 + 5*B*b^4)*cos(d*x + c)^7*log(
sin(d*x + c) + 1) - 105*(8*B*a^4 + 8*(4*A + 3*C)*a^3*b + 36*B*a^2*b^2 + 4*(6*A + 5*C)*a*b^3 + 5*B*b^4)*cos(d*x
+ c)^7*log(-sin(d*x + c) + 1) + 2*(16*(35*(3*A + 2*C)*a^4 + 280*B*a^3*b + 84*(5*A + 4*C)*a^2*b^2 + 224*B*a*b^
3 + 8*(7*A + 6*C)*b^4)*cos(d*x + c)^6 + 105*(8*B*a^4 + 8*(4*A + 3*C)*a^3*b + 36*B*a^2*b^2 + 4*(6*A + 5*C)*a*b^
3 + 5*B*b^4)*cos(d*x + c)^5 + 240*C*b^4 + 16*(35*C*a^4 + 140*B*a^3*b + 42*(5*A + 4*C)*a^2*b^2 + 112*B*a*b^3 +
4*(7*A + 6*C)*b^4)*cos(d*x + c)^4 + 70*(24*C*a^3*b + 36*B*a^2*b^2 + 4*(6*A + 5*C)*a*b^3 + 5*B*b^4)*cos(d*x + c
)^3 + 48*(42*C*a^2*b^2 + 28*B*a*b^3 + (7*A + 6*C)*b^4)*cos(d*x + c)^2 + 280*(4*C*a*b^3 + B*b^4)*cos(d*x + c))*
sin(d*x + c))/(d*cos(d*x + c)^7)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.46661, size = 2549, normalized size = 5.19 \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)^2*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/1680*(105*(8*B*a^4 + 32*A*a^3*b + 24*C*a^3*b + 36*B*a^2*b^2 + 24*A*a*b^3 + 20*C*a*b^3 + 5*B*b^4)*log(abs(tan
(1/2*d*x + 1/2*c) + 1)) - 105*(8*B*a^4 + 32*A*a^3*b + 24*C*a^3*b + 36*B*a^2*b^2 + 24*A*a*b^3 + 20*C*a*b^3 + 5*
B*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(1680*A*a^4*tan(1/2*d*x + 1/2*c)^13 - 840*B*a^4*tan(1/2*d*x + 1/
2*c)^13 + 1680*C*a^4*tan(1/2*d*x + 1/2*c)^13 - 3360*A*a^3*b*tan(1/2*d*x + 1/2*c)^13 + 6720*B*a^3*b*tan(1/2*d*x
+ 1/2*c)^13 - 4200*C*a^3*b*tan(1/2*d*x + 1/2*c)^13 + 10080*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^13 - 6300*B*a^2*b^2
*tan(1/2*d*x + 1/2*c)^13 + 10080*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^13 - 4200*A*a*b^3*tan(1/2*d*x + 1/2*c)^13 + 67
20*B*a*b^3*tan(1/2*d*x + 1/2*c)^13 - 4620*C*a*b^3*tan(1/2*d*x + 1/2*c)^13 + 1680*A*b^4*tan(1/2*d*x + 1/2*c)^13
- 1155*B*b^4*tan(1/2*d*x + 1/2*c)^13 + 1680*C*b^4*tan(1/2*d*x + 1/2*c)^13 - 10080*A*a^4*tan(1/2*d*x + 1/2*c)^
11 + 3360*B*a^4*tan(1/2*d*x + 1/2*c)^11 - 7840*C*a^4*tan(1/2*d*x + 1/2*c)^11 + 13440*A*a^3*b*tan(1/2*d*x + 1/2
*c)^11 - 31360*B*a^3*b*tan(1/2*d*x + 1/2*c)^11 + 10080*C*a^3*b*tan(1/2*d*x + 1/2*c)^11 - 47040*A*a^2*b^2*tan(1
/2*d*x + 1/2*c)^11 + 15120*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 - 33600*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 + 10080
*A*a*b^3*tan(1/2*d*x + 1/2*c)^11 - 22400*B*a*b^3*tan(1/2*d*x + 1/2*c)^11 + 3920*C*a*b^3*tan(1/2*d*x + 1/2*c)^1
1 - 5600*A*b^4*tan(1/2*d*x + 1/2*c)^11 + 980*B*b^4*tan(1/2*d*x + 1/2*c)^11 - 3360*C*b^4*tan(1/2*d*x + 1/2*c)^1
1 + 25200*A*a^4*tan(1/2*d*x + 1/2*c)^9 - 4200*B*a^4*tan(1/2*d*x + 1/2*c)^9 + 16240*C*a^4*tan(1/2*d*x + 1/2*c)^
9 - 16800*A*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 64960*B*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 7560*C*a^3*b*tan(1/2*d*x + 1
/2*c)^9 + 97440*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 11340*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 75936*C*a^2*b^2*ta
n(1/2*d*x + 1/2*c)^9 - 7560*A*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 50624*B*a*b^3*tan(1/2*d*x + 1/2*c)^9 - 11900*C*a*
b^3*tan(1/2*d*x + 1/2*c)^9 + 12656*A*b^4*tan(1/2*d*x + 1/2*c)^9 - 2975*B*b^4*tan(1/2*d*x + 1/2*c)^9 + 14448*C*
b^4*tan(1/2*d*x + 1/2*c)^9 - 33600*A*a^4*tan(1/2*d*x + 1/2*c)^7 - 20160*C*a^4*tan(1/2*d*x + 1/2*c)^7 - 80640*B
*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 120960*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 104832*C*a^2*b^2*tan(1/2*d*x + 1/2*c
)^7 - 69888*B*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 17472*A*b^4*tan(1/2*d*x + 1/2*c)^7 - 10176*C*b^4*tan(1/2*d*x + 1/
2*c)^7 + 25200*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 4200*B*a^4*tan(1/2*d*x + 1/2*c)^5 + 16240*C*a^4*tan(1/2*d*x + 1/
2*c)^5 + 16800*A*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 64960*B*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 7560*C*a^3*b*tan(1/2*d*
x + 1/2*c)^5 + 97440*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 11340*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 75936*C*a^2*b
^2*tan(1/2*d*x + 1/2*c)^5 + 7560*A*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 50624*B*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 11900
*C*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 12656*A*b^4*tan(1/2*d*x + 1/2*c)^5 + 2975*B*b^4*tan(1/2*d*x + 1/2*c)^5 + 144
48*C*b^4*tan(1/2*d*x + 1/2*c)^5 - 10080*A*a^4*tan(1/2*d*x + 1/2*c)^3 - 3360*B*a^4*tan(1/2*d*x + 1/2*c)^3 - 784
0*C*a^4*tan(1/2*d*x + 1/2*c)^3 - 13440*A*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 31360*B*a^3*b*tan(1/2*d*x + 1/2*c)^3 -
10080*C*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 47040*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 15120*B*a^2*b^2*tan(1/2*d*x +
1/2*c)^3 - 33600*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 10080*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 22400*B*a*b^3*tan(
1/2*d*x + 1/2*c)^3 - 3920*C*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 5600*A*b^4*tan(1/2*d*x + 1/2*c)^3 - 980*B*b^4*tan(1
/2*d*x + 1/2*c)^3 - 3360*C*b^4*tan(1/2*d*x + 1/2*c)^3 + 1680*A*a^4*tan(1/2*d*x + 1/2*c) + 840*B*a^4*tan(1/2*d*
x + 1/2*c) + 1680*C*a^4*tan(1/2*d*x + 1/2*c) + 3360*A*a^3*b*tan(1/2*d*x + 1/2*c) + 6720*B*a^3*b*tan(1/2*d*x +
1/2*c) + 4200*C*a^3*b*tan(1/2*d*x + 1/2*c) + 10080*A*a^2*b^2*tan(1/2*d*x + 1/2*c) + 6300*B*a^2*b^2*tan(1/2*d*x
+ 1/2*c) + 10080*C*a^2*b^2*tan(1/2*d*x + 1/2*c) + 4200*A*a*b^3*tan(1/2*d*x + 1/2*c) + 6720*B*a*b^3*tan(1/2*d*
x + 1/2*c) + 4620*C*a*b^3*tan(1/2*d*x + 1/2*c) + 1680*A*b^4*tan(1/2*d*x + 1/2*c) + 1155*B*b^4*tan(1/2*d*x + 1/
2*c) + 1680*C*b^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^7)/d