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

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

[Out]

1/16*(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(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*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(d*x
+c)*tan(d*x+c)/b/d-1/840*(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(d*x+
c))^2*tan(d*x+c)/b^2/d-1/840*(28*a^2*b*B-175*b^3*B-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

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Rubi [A]
time = 0.86, antiderivative size = 491, normalized size of antiderivative = 1.00, 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 = {4177, 4167, 4087, 4082, 3872, 3855, 3852, 8} \begin {gather*} \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 ) \tanh ^{-1}(\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} \end {gather*}

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 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*} \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 ) \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 ) \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*}

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Mathematica [A]
time = 4.62, size = 486, normalized size = 0.99 \begin {gather*} -\frac {\left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right ) \sec ^6(c+d x) \left (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 ) \cos ^6(c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-70 b \left (36 a^2 b B+5 b^3 B+24 a^3 C+4 a b^2 (6 A+5 C)\right ) \cos ^2(c+d x) \sin (c+d x)-16 \left (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)\right ) \cos ^3(c+d x) \sin (c+d x)-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 ) \cos ^4(c+d x) \sin (c+d x)-16 \left (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)\right ) \cos ^5(c+d x) \sin (c+d x)-8 b^2 \left (35 b (b B+4 a C) \sin (c+d x)+3 \left (7 A b^2+28 a b B+42 a^2 C+6 b^2 C\right ) \sin (2 (c+d x))+30 b^2 C \tan (c+d x)\right )\right )}{840 d (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x)))} \end {gather*}

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]

-1/840*((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])))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))

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Maple [A]
time = 0.20, size = 592, normalized size = 1.21 Too large to display

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,method=_RETURNVERBOSE)

[Out]

1/d*(-A*b^4*(-8/15-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c)+b^4*B*(-(-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)))-C*b^4*(-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)+4*a*A*b^3*(-(-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)))-4*a*b^3*B*(-8/15-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c)+4*C*b^3*a*(-(-1/6*sec(d*x+c)^5-5/24*s
ec(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)))-6*a^2*A*b^2*(-2/3-1/3*sec(d*x+c)^2)*ta
n(d*x+c)+6*a^2*b^2*B*(-(-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*C*a^2*b^
2*(-8/15-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c)+4*A*a^3*b*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)
+tan(d*x+c)))-4*a^3*b*B*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+4*a^3*b*C*(-(-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)))+A*a^4*tan(d*x+c)+a^4*B*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(
d*x+c)))-a^4*C*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c))

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Maxima [A]
time = 0.30, size = 746, normalized size = 1.52 \begin {gather*} \frac {1120 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} + 4480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} b + 6720 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} b^{2} + 1344 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C a^{2} b^{2} + 896 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B a b^{3} + 224 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A b^{4} + 96 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} C b^{4} - 140 \, C a b^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 35 \, B b^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 840 \, C a^{3} b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 1260 \, B a^{2} b^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 840 \, A a b^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 840 \, B a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 3360 \, A a^{3} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 3360 \, A a^{4} \tan \left (d x + c\right )}{3360 \, d} \end {gather*}

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 = 3.14, size = 450, normalized size = 0.92 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 1888 vs. \(2 (475) = 950\).
time = 0.60, size = 1888, normalized size = 3.85 \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)^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

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Mupad [B]
time = 7.75, size = 1044, normalized size = 2.13 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {B\,a^4}{2}+\frac {5\,B\,b^4}{16}+\frac {9\,B\,a^2\,b^2}{4}+\frac {3\,A\,a\,b^3}{2}+2\,A\,a^3\,b+\frac {5\,C\,a\,b^3}{4}+\frac {3\,C\,a^3\,b}{2}\right )}{2\,B\,a^4+\frac {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}\right )\,\left (B\,a^4+\frac {5\,B\,b^4}{8}+\frac {9\,B\,a^2\,b^2}{2}+3\,A\,a\,b^3+4\,A\,a^3\,b+\frac {5\,C\,a\,b^3}{2}+3\,C\,a^3\,b\right )}{d}-\frac {\left (2\,A\,a^4+2\,A\,b^4-B\,a^4-\frac {11\,B\,b^4}{8}+2\,C\,a^4+2\,C\,b^4+12\,A\,a^2\,b^2-\frac {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-\frac {11\,C\,a\,b^3}{2}-5\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (4\,B\,a^4-\frac {20\,A\,b^4}{3}-12\,A\,a^4+\frac {7\,B\,b^4}{6}-\frac {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-\frac {80\,B\,a\,b^3}{3}-\frac {112\,B\,a^3\,b}{3}+\frac {14\,C\,a\,b^3}{3}+12\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (30\,A\,a^4+\frac {226\,A\,b^4}{15}-5\,B\,a^4-\frac {85\,B\,b^4}{24}+\frac {58\,C\,a^4}{3}+\frac {86\,C\,b^4}{5}+116\,A\,a^2\,b^2-\frac {27\,B\,a^2\,b^2}{2}+\frac {452\,C\,a^2\,b^2}{5}-9\,A\,a\,b^3-20\,A\,a^3\,b+\frac {904\,B\,a\,b^3}{15}+\frac {232\,B\,a^3\,b}{3}-\frac {85\,C\,a\,b^3}{6}-9\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-40\,A\,a^4-\frac {104\,A\,b^4}{5}-24\,C\,a^4-\frac {424\,C\,b^4}{35}-144\,A\,a^2\,b^2-\frac {624\,C\,a^2\,b^2}{5}-\frac {416\,B\,a\,b^3}{5}-96\,B\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (30\,A\,a^4+\frac {226\,A\,b^4}{15}+5\,B\,a^4+\frac {85\,B\,b^4}{24}+\frac {58\,C\,a^4}{3}+\frac {86\,C\,b^4}{5}+116\,A\,a^2\,b^2+\frac {27\,B\,a^2\,b^2}{2}+\frac {452\,C\,a^2\,b^2}{5}+9\,A\,a\,b^3+20\,A\,a^3\,b+\frac {904\,B\,a\,b^3}{15}+\frac {232\,B\,a^3\,b}{3}+\frac {85\,C\,a\,b^3}{6}+9\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-12\,A\,a^4-\frac {20\,A\,b^4}{3}-4\,B\,a^4-\frac {7\,B\,b^4}{6}-\frac {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-\frac {80\,B\,a\,b^3}{3}-\frac {112\,B\,a^3\,b}{3}-\frac {14\,C\,a\,b^3}{3}-12\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a^4+2\,A\,b^4+B\,a^4+\frac {11\,B\,b^4}{8}+2\,C\,a^4+2\,C\,b^4+12\,A\,a^2\,b^2+\frac {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+\frac {11\,C\,a\,b^3}{2}+5\,C\,a^3\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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