\(\int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [887]

Optimal result

Integrand size = 41, antiderivative size = 491 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right ) \text {arctanh}(\sin (c+d x))}{16 d}-\frac {\left (28 a^5 b B-847 a^3 b^3 B-896 a b^5 B-8 a^6 C-32 b^6 (7 A+6 C)-4 a^4 b^2 (42 A+23 C)-32 a^2 b^4 (49 A+39 C)\right ) \tan (c+d x)}{420 b^2 d}-\frac {\left (56 a^4 b B-1246 a^2 b^3 B-525 b^5 B-16 a^5 C-48 a^3 b^2 (7 A+4 C)-4 a b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{1680 b d}-\frac {\left (28 a^3 b B-371 a b^3 B-8 a^4 C-32 b^4 (7 A+6 C)-12 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{840 b^2 d}-\frac {\left (28 a^2 b B-175 b^3 B-8 a^3 C-4 a b^2 (42 A+31 C)\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{840 b^2 d}+\frac {\left (42 A b^2-7 a b B+2 a^2 C+36 b^2 C\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{210 b^2 d}+\frac {(7 b B-2 a C) (a+b \sec (c+d x))^5 \tan (c+d x)}{42 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d} \]

[Out]

1/16*(8*B*a^4+36*B*a^2*b^2+5*B*b^4+8*a^3*b*(4*A+3*C)+4*a*b^3*(6*A+5*C))*arctanh(sin(d*x+c))/d-1/420*(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(d*x+c)/
b^2/d-1/1680*(56*B*a^4*b-1246*B*a^2*b^3-525*B*b^5-16*a^5*C-48*a^3*b^2*(7*A+4*C)-4*a*b^4*(406*A+333*C))*sec(d*x
+c)*tan(d*x+c)/b/d-1/840*(28*B*a^3*b-371*B*a*b^3-8*a^4*C-32*b^4*(7*A+6*C)-12*a^2*b^2*(14*A+9*C))*(a+b*sec(d*x+
c))^2*tan(d*x+c)/b^2/d-1/840*(28*B*a^2*b-175*B*b^3-8*a^3*C-4*a*b^2*(42*A+31*C))*(a+b*sec(d*x+c))^3*tan(d*x+c)/
b^2/d+1/210*(42*A*b^2-7*B*a*b+2*C*a^2+36*C*b^2)*(a+b*sec(d*x+c))^4*tan(d*x+c)/b^2/d+1/42*(7*B*b-2*C*a)*(a+b*se
c(d*x+c))^5*tan(d*x+c)/b^2/d+1/7*C*sec(d*x+c)*(a+b*sec(d*x+c))^5*tan(d*x+c)/b/d

Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {4177, 4167, 4087, 4082, 3872, 3855, 3852, 8} \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\tan (c+d x) \left (2 a^2 C-7 a b B+42 A b^2+36 b^2 C\right ) (a+b \sec (c+d x))^4}{210 b^2 d}-\frac {\tan (c+d x) \left (-8 a^3 C+28 a^2 b B-4 a b^2 (42 A+31 C)-175 b^3 B\right ) (a+b \sec (c+d x))^3}{840 b^2 d}+\frac {\left (8 a^4 B+8 a^3 b (4 A+3 C)+36 a^2 b^2 B+4 a b^3 (6 A+5 C)+5 b^4 B\right ) \text {arctanh}(\sin (c+d x))}{16 d}-\frac {\tan (c+d x) \left (-8 a^4 C+28 a^3 b B-12 a^2 b^2 (14 A+9 C)-371 a b^3 B-32 b^4 (7 A+6 C)\right ) (a+b \sec (c+d x))^2}{840 b^2 d}-\frac {\tan (c+d x) \sec (c+d x) \left (-16 a^5 C+56 a^4 b B-48 a^3 b^2 (7 A+4 C)-1246 a^2 b^3 B-4 a b^4 (406 A+333 C)-525 b^5 B\right )}{1680 b d}-\frac {\tan (c+d x) \left (-8 a^6 C+28 a^5 b B-4 a^4 b^2 (42 A+23 C)-847 a^3 b^3 B-32 a^2 b^4 (49 A+39 C)-896 a b^5 B-32 b^6 (7 A+6 C)\right )}{420 b^2 d}+\frac {(7 b B-2 a C) \tan (c+d x) (a+b \sec (c+d x))^5}{42 b^2 d}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^5}{7 b d} \]

[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 8

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

Rule 3852

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 3855

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

Rule 3872

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 4082

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
*Csc[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 4087

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] /;
FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 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 4177

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {arctanh}(\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 ) \text {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 ) \text {arctanh}(\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] (verified)

Time = 11.30 (sec) , antiderivative size = 458, normalized size of antiderivative = 0.93 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {C \sec ^2(c+d x) (a+b \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {1}{7} \left (\frac {(7 b B+4 a C) \sec ^2(c+d x) (a+b \sec (c+d x))^3 \tan (c+d x)}{6 d}+\frac {1}{6} \left (\frac {3 \left (14 A b^2+21 a b B+4 a^2 C+12 b^2 C\right ) \sec ^2(c+d x) (a+b \sec (c+d x))^2 \tan (c+d x)}{5 d}+\frac {1}{5} \left (\frac {b \left (336 a^2 b B+175 b^3 B+24 a^3 C+4 a b^2 (126 A+103 C)\right ) \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \left (\frac {\left (48 \left (91 a^3 b B+112 a b^3 B+4 a^4 C+4 b^4 (7 A+6 C)+3 a^2 b^2 (63 A+50 C)\right )+24 a^2 \left (98 a b B+6 b^2 (7 A+6 C)+a^2 (105 A+62 C)\right )\right ) \tan (c+d x)}{3 d}+\frac {8 \left (91 a^3 b B+112 a b^3 B+4 a^4 C+4 b^4 (7 A+6 C)+3 a^2 b^2 (63 A+50 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{d}+105 \left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right ) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\sec (c+d x) \tan (c+d x)}{2 d}\right )\right )\right )\right )\right ) \]

[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*Sec[c + d*x]^2*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(7*d) + (((7*b*B + 4*a*C)*Sec[c + d*x]^2*(a + b*Sec[c +
d*x])^3*Tan[c + d*x])/(6*d) + ((3*(14*A*b^2 + 21*a*b*B + 4*a^2*C + 12*b^2*C)*Sec[c + d*x]^2*(a + b*Sec[c + d*
x])^2*Tan[c + d*x])/(5*d) + ((b*(336*a^2*b*B + 175*b^3*B + 24*a^3*C + 4*a*b^2*(126*A + 103*C))*Sec[c + d*x]^3*
Tan[c + d*x])/(4*d) + (((48*(91*a^3*b*B + 112*a*b^3*B + 4*a^4*C + 4*b^4*(7*A + 6*C) + 3*a^2*b^2*(63*A + 50*C))
+ 24*a^2*(98*a*b*B + 6*b^2*(7*A + 6*C) + a^2*(105*A + 62*C)))*Tan[c + d*x])/(3*d) + (8*(91*a^3*b*B + 112*a*b^
3*B + 4*a^4*C + 4*b^4*(7*A + 6*C) + 3*a^2*b^2*(63*A + 50*C))*Sec[c + d*x]^2*Tan[c + d*x])/d + 105*(8*a^4*B + 3
6*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]]/(2*d) + (Sec[c + d*x
]*Tan[c + d*x])/(2*d)))/4)/5)/6)/7

Maple [A] (verified)

Time = 2.19 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.72

method	result	size
parts	\(\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (B \,b^{4}+4 C a \,b^{3}\right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}-\frac {\left (A \,b^{4}+4 B a \,b^{3}+6 C \,a^{2} b^{2}\right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 a A \,b^{3}+6 B \,a^{2} b^{2}+4 a^{3} b C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b +a^{4} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}-\frac {C \,b^{4} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}+\frac {a^{4} A \tan \left (d x +c \right )}{d}\)	\(354\)
derivativedivides	\(\frac {a^{4} A \tan \left (d x +c \right )+B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 A \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 B \,a^{3} b \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{3} b C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 A \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 B \,a^{2} b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 C \,a^{2} b^{2} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+4 a A \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 B a \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+4 C a \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-A \,b^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+B \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-C \,b^{4} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}\)	\(592\)
default	\(\frac {a^{4} A \tan \left (d x +c \right )+B \,a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 A \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 B \,a^{3} b \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{3} b C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 A \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+6 B \,a^{2} b^{2} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 C \,a^{2} b^{2} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+4 a A \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 B a \,b^{3} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+4 C a \,b^{3} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-A \,b^{4} \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+B \,b^{4} \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-C \,b^{4} \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}\)	\(592\)
parallelrisch	\(\frac {-70560 \left (\frac {B \,a^{4}}{4}+b \left (A +\frac {3 C}{4}\right ) a^{3}+\frac {9 B \,a^{2} b^{2}}{8}+\frac {3 \left (A +\frac {5 C}{6}\right ) b^{3} a}{4}+\frac {5 B \,b^{4}}{32}\right ) \left (\frac {\cos \left (7 d x +7 c \right )}{21}+\frac {\cos \left (5 d x +5 c \right )}{3}+\cos \left (3 d x +3 c \right )+\frac {5 \cos \left (d x +c \right )}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+70560 \left (\frac {B \,a^{4}}{4}+b \left (A +\frac {3 C}{4}\right ) a^{3}+\frac {9 B \,a^{2} b^{2}}{8}+\frac {3 \left (A +\frac {5 C}{6}\right ) b^{3} a}{4}+\frac {5 B \,b^{4}}{32}\right ) \left (\frac {\cos \left (7 d x +7 c \right )}{21}+\frac {\cos \left (5 d x +5 c \right )}{3}+\cos \left (3 d x +3 c \right )+\frac {5 \cos \left (d x +c \right )}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (\left (15120 A +16800 C \right ) a^{4}+67200 B \,a^{3} b +100800 \left (A +\frac {28 C}{25}\right ) b^{2} a^{2}+75264 B a \,b^{3}+18816 b^{4} \left (A +\frac {6 C}{7}\right )\right ) \sin \left (3 d x +3 c \right )+\left (\left (8400 A +7840 C \right ) a^{4}+31360 B \,a^{3} b +47040 b^{2} \left (A +\frac {4 C}{5}\right ) a^{2}+25088 B a \,b^{3}+6272 b^{4} \left (A +\frac {6 C}{7}\right )\right ) \sin \left (5 d x +5 c \right )+\left (\left (1680 A +1120 C \right ) a^{4}+4480 B \,a^{3} b +6720 b^{2} \left (A +\frac {4 C}{5}\right ) a^{2}+3584 B a \,b^{3}+896 b^{4} \left (A +\frac {6 C}{7}\right )\right ) \sin \left (7 d x +7 c \right )+\left (8400 B \,a^{4}+33600 b \left (A +\frac {31 C}{20}\right ) a^{3}+78120 B \,a^{2} b^{2}+52080 b^{3} \left (A +\frac {283 C}{186}\right ) a +19810 B \,b^{4}\right ) \sin \left (2 d x +2 c \right )+\left (6720 B \,a^{4}+26880 b \left (A +\frac {5 C}{4}\right ) a^{3}+50400 B \,a^{2} b^{2}+33600 \left (A +\frac {5 C}{6}\right ) b^{3} a +7000 B \,b^{4}\right ) \sin \left (4 d x +4 c \right )+\left (1680 B \,a^{4}+6720 b \left (A +\frac {3 C}{4}\right ) a^{3}+7560 B \,a^{2} b^{2}+5040 \left (A +\frac {5 C}{6}\right ) b^{3} a +1050 B \,b^{4}\right ) \sin \left (6 d x +6 c \right )+8400 \left (\left (A +\frac {6 C}{5}\right ) a^{4}+\frac {24 B \,a^{3} b}{5}+\frac {36 b^{2} \left (A +\frac {4 C}{3}\right ) a^{2}}{5}+\frac {32 B a \,b^{3}}{5}+\frac {8 b^{4} \left (A +2 C \right )}{5}\right ) \sin \left (d x +c \right )}{1680 d \left (\cos \left (7 d x +7 c \right )+7 \cos \left (5 d x +5 c \right )+21 \cos \left (3 d x +3 c \right )+35 \cos \left (d x +c \right )\right )}\)	\(637\)
norman	\(\frac {\frac {8 \left (175 a^{4} A +630 A \,a^{2} b^{2}+91 A \,b^{4}+420 B \,a^{3} b +364 B a \,b^{3}+105 a^{4} C +546 C \,a^{2} b^{2}+53 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d}-\frac {\left (16 a^{4} A -32 A \,a^{3} b +96 A \,a^{2} b^{2}-40 a A \,b^{3}+16 A \,b^{4}-8 B \,a^{4}+64 B \,a^{3} b -60 B \,a^{2} b^{2}+64 B a \,b^{3}-11 B \,b^{4}+16 a^{4} C -40 a^{3} b C +96 C \,a^{2} b^{2}-44 C a \,b^{3}+16 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{8 d}-\frac {\left (16 a^{4} A +32 A \,a^{3} b +96 A \,a^{2} b^{2}+40 a A \,b^{3}+16 A \,b^{4}+8 B \,a^{4}+64 B \,a^{3} b +60 B \,a^{2} b^{2}+64 B a \,b^{3}+11 B \,b^{4}+16 a^{4} C +40 a^{3} b C +96 C \,a^{2} b^{2}+44 C a \,b^{3}+16 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (72 a^{4} A -96 A \,a^{3} b +336 A \,a^{2} b^{2}-72 a A \,b^{3}+40 A \,b^{4}-24 B \,a^{4}+224 B \,a^{3} b -108 B \,a^{2} b^{2}+160 B a \,b^{3}-7 B \,b^{4}+56 a^{4} C -72 a^{3} b C +240 C \,a^{2} b^{2}-28 C a \,b^{3}+24 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{6 d}+\frac {\left (72 a^{4} A +96 A \,a^{3} b +336 A \,a^{2} b^{2}+72 a A \,b^{3}+40 A \,b^{4}+24 B \,a^{4}+224 B \,a^{3} b +108 B \,a^{2} b^{2}+160 B a \,b^{3}+7 B \,b^{4}+56 a^{4} C +72 a^{3} b C +240 C \,a^{2} b^{2}+28 C a \,b^{3}+24 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}-\frac {\left (3600 a^{4} A -2400 A \,a^{3} b +13920 A \,a^{2} b^{2}-1080 a A \,b^{3}+1808 A \,b^{4}-600 B \,a^{4}+9280 B \,a^{3} b -1620 B \,a^{2} b^{2}+7232 B a \,b^{3}-425 B \,b^{4}+2320 a^{4} C -1080 a^{3} b C +10848 C \,a^{2} b^{2}-1700 C a \,b^{3}+2064 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{120 d}-\frac {\left (3600 a^{4} A +2400 A \,a^{3} b +13920 A \,a^{2} b^{2}+1080 a A \,b^{3}+1808 A \,b^{4}+600 B \,a^{4}+9280 B \,a^{3} b +1620 B \,a^{2} b^{2}+7232 B a \,b^{3}+425 B \,b^{4}+2320 a^{4} C +1080 a^{3} b C +10848 C \,a^{2} b^{2}+1700 C a \,b^{3}+2064 C \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{120 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{7}}-\frac {\left (32 A \,a^{3} b +24 a A \,b^{3}+8 B \,a^{4}+36 B \,a^{2} b^{2}+5 B \,b^{4}+24 a^{3} b C +20 C a \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {\left (32 A \,a^{3} b +24 a A \,b^{3}+8 B \,a^{4}+36 B \,a^{2} b^{2}+5 B \,b^{4}+24 a^{3} b C +20 C a \,b^{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\)	\(956\)
risch	\(\text {Expression too large to display}\)	\(1648\)

[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,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 450, normalized size of antiderivative = 0.92 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (8 \, B a^{4} + 8 \, {\left (4 \, A + 3 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 4 \, {\left (6 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (8 \, B a^{4} + 8 \, {\left (4 \, A + 3 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 4 \, {\left (6 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (35 \, {\left (3 \, A + 2 \, C\right )} a^{4} + 280 \, B a^{3} b + 84 \, {\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 224 \, B a b^{3} + 8 \, {\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} + 105 \, {\left (8 \, B a^{4} + 8 \, {\left (4 \, A + 3 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 4 \, {\left (6 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} + 240 \, C b^{4} + 16 \, {\left (35 \, C a^{4} + 140 \, B a^{3} b + 42 \, {\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 112 \, B a b^{3} + 4 \, {\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 70 \, {\left (24 \, C a^{3} b + 36 \, B a^{2} b^{2} + 4 \, {\left (6 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 48 \, {\left (42 \, C a^{2} b^{2} + 28 \, B a b^{3} + {\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 280 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3360 \, d \cos \left (d x + c\right )^{7}} \]

[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)

Sympy [F]

\[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{4} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 746, normalized size of antiderivative = 1.52 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1888 vs. \(2 (475) = 950\).

Time = 0.41 (sec) , antiderivative size = 1888, normalized size of antiderivative = 3.85 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]

[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

Mupad [B] (verification not implemented)

Time = 20.34 (sec) , antiderivative size = 1044, normalized size of antiderivative = 2.13 \[ \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

(atanh((4*tan(c/2 + (d*x)/2)*((B*a^4)/2 + (5*B*b^4)/16 + (9*B*a^2*b^2)/4 + (3*A*a*b^3)/2 + 2*A*a^3*b + (5*C*a*
b^3)/4 + (3*C*a^3*b)/2))/(2*B*a^4 + (5*B*b^4)/4 + 9*B*a^2*b^2 + 6*A*a*b^3 + 8*A*a^3*b + 5*C*a*b^3 + 6*C*a^3*b)
)*(B*a^4 + (5*B*b^4)/8 + (9*B*a^2*b^2)/2 + 3*A*a*b^3 + 4*A*a^3*b + (5*C*a*b^3)/2 + 3*C*a^3*b))/d - (tan(c/2 +
(d*x)/2)^13*(2*A*a^4 + 2*A*b^4 - B*a^4 - (11*B*b^4)/8 + 2*C*a^4 + 2*C*b^4 + 12*A*a^2*b^2 - (15*B*a^2*b^2)/2 +
12*C*a^2*b^2 - 5*A*a*b^3 - 4*A*a^3*b + 8*B*a*b^3 + 8*B*a^3*b - (11*C*a*b^3)/2 - 5*C*a^3*b) - tan(c/2 + (d*x)/2
)^3*(12*A*a^4 + (20*A*b^4)/3 + 4*B*a^4 + (7*B*b^4)/6 + (28*C*a^4)/3 + 4*C*b^4 + 56*A*a^2*b^2 + 18*B*a^2*b^2 +
40*C*a^2*b^2 + 12*A*a*b^3 + 16*A*a^3*b + (80*B*a*b^3)/3 + (112*B*a^3*b)/3 + (14*C*a*b^3)/3 + 12*C*a^3*b) - tan
(c/2 + (d*x)/2)^11*(12*A*a^4 + (20*A*b^4)/3 - 4*B*a^4 - (7*B*b^4)/6 + (28*C*a^4)/3 + 4*C*b^4 + 56*A*a^2*b^2 -
18*B*a^2*b^2 + 40*C*a^2*b^2 - 12*A*a*b^3 - 16*A*a^3*b + (80*B*a*b^3)/3 + (112*B*a^3*b)/3 - (14*C*a*b^3)/3 - 12
*C*a^3*b) + tan(c/2 + (d*x)/2)^5*(30*A*a^4 + (226*A*b^4)/15 + 5*B*a^4 + (85*B*b^4)/24 + (58*C*a^4)/3 + (86*C*b
^4)/5 + 116*A*a^2*b^2 + (27*B*a^2*b^2)/2 + (452*C*a^2*b^2)/5 + 9*A*a*b^3 + 20*A*a^3*b + (904*B*a*b^3)/15 + (23
2*B*a^3*b)/3 + (85*C*a*b^3)/6 + 9*C*a^3*b) + tan(c/2 + (d*x)/2)^9*(30*A*a^4 + (226*A*b^4)/15 - 5*B*a^4 - (85*B
*b^4)/24 + (58*C*a^4)/3 + (86*C*b^4)/5 + 116*A*a^2*b^2 - (27*B*a^2*b^2)/2 + (452*C*a^2*b^2)/5 - 9*A*a*b^3 - 20
*A*a^3*b + (904*B*a*b^3)/15 + (232*B*a^3*b)/3 - (85*C*a*b^3)/6 - 9*C*a^3*b) - tan(c/2 + (d*x)/2)^7*(40*A*a^4 +
(104*A*b^4)/5 + 24*C*a^4 + (424*C*b^4)/35 + 144*A*a^2*b^2 + (624*C*a^2*b^2)/5 + (416*B*a*b^3)/5 + 96*B*a^3*b)
+ tan(c/2 + (d*x)/2)*(2*A*a^4 + 2*A*b^4 + B*a^4 + (11*B*b^4)/8 + 2*C*a^4 + 2*C*b^4 + 12*A*a^2*b^2 + (15*B*a^2
*b^2)/2 + 12*C*a^2*b^2 + 5*A*a*b^3 + 4*A*a^3*b + 8*B*a*b^3 + 8*B*a^3*b + (11*C*a*b^3)/2 + 5*C*a^3*b))/(d*(7*ta
n(c/2 + (d*x)/2)^2 - 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 - 35*tan(c/2 + (d*x)/2)^8 + 21*tan(c/2
+ (d*x)/2)^10 - 7*tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^14 - 1))