3.954 \(\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^6(c+d x) \, dx\)

Optimal. Leaf size=232 \[ \frac {\tan (c+d x) \left (2 a^2 (4 A+5 C)+20 a b B+5 b^2 (2 A+3 C)\right )}{15 d}+\frac {\left (3 a^2 B+6 a A b+8 a b C+4 b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\tan (c+d x) \sec ^2(c+d x) \left (a^2 (4 A+5 C)+10 a b B+2 A b^2\right )}{15 d}+\frac {\tan (c+d x) \sec (c+d x) \left (3 a^2 B+6 a A b+8 a b C+4 b^2 B\right )}{8 d}+\frac {a (5 a B+2 A b) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d} \]

[Out]

1/8*(6*A*a*b+3*B*a^2+4*B*b^2+8*C*a*b)*arctanh(sin(d*x+c))/d+1/15*(20*a*b*B+5*b^2*(2*A+3*C)+2*a^2*(4*A+5*C))*ta
n(d*x+c)/d+1/8*(6*A*a*b+3*B*a^2+4*B*b^2+8*C*a*b)*sec(d*x+c)*tan(d*x+c)/d+1/15*(2*A*b^2+10*a*b*B+a^2*(4*A+5*C))
*sec(d*x+c)^2*tan(d*x+c)/d+1/20*a*(2*A*b+5*B*a)*sec(d*x+c)^3*tan(d*x+c)/d+1/5*A*(a+b*cos(d*x+c))^2*sec(d*x+c)^
4*tan(d*x+c)/d

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Rubi [A]  time = 0.51, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {3047, 3031, 3021, 2748, 3768, 3770, 3767, 8} \[ \frac {\tan (c+d x) \left (2 a^2 (4 A+5 C)+20 a b B+5 b^2 (2 A+3 C)\right )}{15 d}+\frac {\left (3 a^2 B+6 a A b+8 a b C+4 b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\tan (c+d x) \sec ^2(c+d x) \left (a^2 (4 A+5 C)+10 a b B+2 A b^2\right )}{15 d}+\frac {\tan (c+d x) \sec (c+d x) \left (3 a^2 B+6 a A b+8 a b C+4 b^2 B\right )}{8 d}+\frac {a (5 a B+2 A b) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]

[Out]

((6*a*A*b + 3*a^2*B + 4*b^2*B + 8*a*b*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((20*a*b*B + 5*b^2*(2*A + 3*C) + 2*a^2
*(4*A + 5*C))*Tan[c + d*x])/(15*d) + ((6*a*A*b + 3*a^2*B + 4*b^2*B + 8*a*b*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d)
+ ((2*A*b^2 + 10*a*b*B + a^2*(4*A + 5*C))*Sec[c + d*x]^2*Tan[c + d*x])/(15*d) + (a*(2*A*b + 5*a*B)*Sec[c + d*
x]^3*Tan[c + d*x])/(20*d) + (A*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d)

Rule 8

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

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 3021

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m +
1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B + a*
C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e,
f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

Rule 3031

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((b*c - a*d)*(A*b^2 - a*b*B + a^2*C)*
Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)),
Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b
^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] - b*C*d*(m +
1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && Ne
Q[a^2 - b^2, 0] && LtQ[m, -1]

Rule 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C - B*c*d + A*d^2)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(
c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c
*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*
d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)
))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2,
0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

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 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

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

Rubi steps

\begin {align*} \int (a+b \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx &=\frac {A (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{5} \int (a+b \cos (c+d x)) \left (2 A b+5 a B+(4 a A+5 b B+5 a C) \cos (c+d x)+b (2 A+5 C) \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx\\ &=\frac {a (2 A b+5 a B) \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {A (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac {1}{20} \int \left (-4 \left (2 A b^2+10 a b B+a^2 (4 A+5 C)\right )-5 \left (6 a A b+3 a^2 B+4 b^2 B+8 a b C\right ) \cos (c+d x)-4 b^2 (2 A+5 C) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac {\left (2 A b^2+10 a b B+a^2 (4 A+5 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{15 d}+\frac {a (2 A b+5 a B) \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {A (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac {1}{60} \int \left (-15 \left (6 a A b+3 a^2 B+4 b^2 B+8 a b C\right )-4 \left (20 a b B+5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right ) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac {\left (2 A b^2+10 a b B+a^2 (4 A+5 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{15 d}+\frac {a (2 A b+5 a B) \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {A (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac {1}{4} \left (-6 a A b-3 a^2 B-4 b^2 B-8 a b C\right ) \int \sec ^3(c+d x) \, dx-\frac {1}{15} \left (-20 a b B-5 b^2 (2 A+3 C)-2 a^2 (4 A+5 C)\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac {\left (6 a A b+3 a^2 B+4 b^2 B+8 a b C\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {\left (2 A b^2+10 a b B+a^2 (4 A+5 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{15 d}+\frac {a (2 A b+5 a B) \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {A (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac {1}{8} \left (-6 a A b-3 a^2 B-4 b^2 B-8 a b C\right ) \int \sec (c+d x) \, dx-\frac {\left (20 a b B+5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 d}\\ &=\frac {\left (6 a A b+3 a^2 B+4 b^2 B+8 a b C\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\left (20 a b B+5 b^2 (2 A+3 C)+2 a^2 (4 A+5 C)\right ) \tan (c+d x)}{15 d}+\frac {\left (6 a A b+3 a^2 B+4 b^2 B+8 a b C\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {\left (2 A b^2+10 a b B+a^2 (4 A+5 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{15 d}+\frac {a (2 A b+5 a B) \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {A (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{5 d}\\ \end {align*}

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Mathematica [A]  time = 2.51, size = 167, normalized size = 0.72 \[ \frac {15 \left (3 a^2 B+6 a A b+8 a b C+4 b^2 B\right ) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (8 \left (5 \tan ^2(c+d x) \left (a^2 (2 A+C)+2 a b B+A b^2\right )+15 \left (a^2 (A+C)+2 a b B+b^2 (A+C)\right )+3 a^2 A \tan ^4(c+d x)\right )+15 \sec (c+d x) \left (3 a^2 B+6 a A b+8 a b C+4 b^2 B\right )+30 a (a B+2 A b) \sec ^3(c+d x)\right )}{120 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]

[Out]

(15*(6*a*A*b + 3*a^2*B + 4*b^2*B + 8*a*b*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(6*a*A*b + 3*a^2*B + 4*b^
2*B + 8*a*b*C)*Sec[c + d*x] + 30*a*(2*A*b + a*B)*Sec[c + d*x]^3 + 8*(15*(2*a*b*B + a^2*(A + C) + b^2*(A + C))
+ 5*(A*b^2 + 2*a*b*B + a^2*(2*A + C))*Tan[c + d*x]^2 + 3*a^2*A*Tan[c + d*x]^4)))/(120*d)

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fricas [A]  time = 0.45, size = 243, normalized size = 1.05 \[ \frac {15 \, {\left (3 \, B a^{2} + 2 \, {\left (3 \, A + 4 \, C\right )} a b + 4 \, B b^{2}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (3 \, B a^{2} + 2 \, {\left (3 \, A + 4 \, C\right )} a b + 4 \, B b^{2}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (8 \, {\left (2 \, {\left (4 \, A + 5 \, C\right )} a^{2} + 20 \, B a b + 5 \, {\left (2 \, A + 3 \, C\right )} b^{2}\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (3 \, B a^{2} + 2 \, {\left (3 \, A + 4 \, C\right )} a b + 4 \, B b^{2}\right )} \cos \left (d x + c\right )^{3} + 24 \, A a^{2} + 8 \, {\left ({\left (4 \, A + 5 \, C\right )} a^{2} + 10 \, B a b + 5 \, A b^{2}\right )} \cos \left (d x + c\right )^{2} + 30 \, {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, algorithm="fricas")

[Out]

1/240*(15*(3*B*a^2 + 2*(3*A + 4*C)*a*b + 4*B*b^2)*cos(d*x + c)^5*log(sin(d*x + c) + 1) - 15*(3*B*a^2 + 2*(3*A
+ 4*C)*a*b + 4*B*b^2)*cos(d*x + c)^5*log(-sin(d*x + c) + 1) + 2*(8*(2*(4*A + 5*C)*a^2 + 20*B*a*b + 5*(2*A + 3*
C)*b^2)*cos(d*x + c)^4 + 15*(3*B*a^2 + 2*(3*A + 4*C)*a*b + 4*B*b^2)*cos(d*x + c)^3 + 24*A*a^2 + 8*((4*A + 5*C)
*a^2 + 10*B*a*b + 5*A*b^2)*cos(d*x + c)^2 + 30*(B*a^2 + 2*A*a*b)*cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^5
)

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giac [B]  time = 0.50, size = 766, normalized size = 3.30 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, algorithm="giac")

[Out]

1/120*(15*(3*B*a^2 + 6*A*a*b + 8*C*a*b + 4*B*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(3*B*a^2 + 6*A*a*b +
8*C*a*b + 4*B*b^2)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(120*A*a^2*tan(1/2*d*x + 1/2*c)^9 - 75*B*a^2*tan(1/
2*d*x + 1/2*c)^9 + 120*C*a^2*tan(1/2*d*x + 1/2*c)^9 - 150*A*a*b*tan(1/2*d*x + 1/2*c)^9 + 240*B*a*b*tan(1/2*d*x
+ 1/2*c)^9 - 120*C*a*b*tan(1/2*d*x + 1/2*c)^9 + 120*A*b^2*tan(1/2*d*x + 1/2*c)^9 - 60*B*b^2*tan(1/2*d*x + 1/2
*c)^9 + 120*C*b^2*tan(1/2*d*x + 1/2*c)^9 - 160*A*a^2*tan(1/2*d*x + 1/2*c)^7 + 30*B*a^2*tan(1/2*d*x + 1/2*c)^7
- 320*C*a^2*tan(1/2*d*x + 1/2*c)^7 + 60*A*a*b*tan(1/2*d*x + 1/2*c)^7 - 640*B*a*b*tan(1/2*d*x + 1/2*c)^7 + 240*
C*a*b*tan(1/2*d*x + 1/2*c)^7 - 320*A*b^2*tan(1/2*d*x + 1/2*c)^7 + 120*B*b^2*tan(1/2*d*x + 1/2*c)^7 - 480*C*b^2
*tan(1/2*d*x + 1/2*c)^7 + 464*A*a^2*tan(1/2*d*x + 1/2*c)^5 + 400*C*a^2*tan(1/2*d*x + 1/2*c)^5 + 800*B*a*b*tan(
1/2*d*x + 1/2*c)^5 + 400*A*b^2*tan(1/2*d*x + 1/2*c)^5 + 720*C*b^2*tan(1/2*d*x + 1/2*c)^5 - 160*A*a^2*tan(1/2*d
*x + 1/2*c)^3 - 30*B*a^2*tan(1/2*d*x + 1/2*c)^3 - 320*C*a^2*tan(1/2*d*x + 1/2*c)^3 - 60*A*a*b*tan(1/2*d*x + 1/
2*c)^3 - 640*B*a*b*tan(1/2*d*x + 1/2*c)^3 - 240*C*a*b*tan(1/2*d*x + 1/2*c)^3 - 320*A*b^2*tan(1/2*d*x + 1/2*c)^
3 - 120*B*b^2*tan(1/2*d*x + 1/2*c)^3 - 480*C*b^2*tan(1/2*d*x + 1/2*c)^3 + 120*A*a^2*tan(1/2*d*x + 1/2*c) + 75*
B*a^2*tan(1/2*d*x + 1/2*c) + 120*C*a^2*tan(1/2*d*x + 1/2*c) + 150*A*a*b*tan(1/2*d*x + 1/2*c) + 240*B*a*b*tan(1
/2*d*x + 1/2*c) + 120*C*a*b*tan(1/2*d*x + 1/2*c) + 120*A*b^2*tan(1/2*d*x + 1/2*c) + 60*B*b^2*tan(1/2*d*x + 1/2
*c) + 120*C*b^2*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^5)/d

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maple [A]  time = 0.50, size = 404, normalized size = 1.74 \[ \frac {8 a^{2} A \tan \left (d x +c \right )}{15 d}+\frac {a^{2} A \tan \left (d x +c \right ) \left (\sec ^{4}\left (d x +c \right )\right )}{5 d}+\frac {4 a^{2} A \left (\sec ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{15 d}+\frac {a^{2} B \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{4 d}+\frac {3 a^{2} B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 B \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {2 a^{2} C \tan \left (d x +c \right )}{3 d}+\frac {a^{2} C \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {a A b \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{2 d}+\frac {3 a A b \sec \left (d x +c \right ) \tan \left (d x +c \right )}{4 d}+\frac {3 A a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{4 d}+\frac {4 B a b \tan \left (d x +c \right )}{3 d}+\frac {2 B a b \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {C a b \tan \left (d x +c \right ) \sec \left (d x +c \right )}{d}+\frac {C a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {2 A \,b^{2} \tan \left (d x +c \right )}{3 d}+\frac {A \,b^{2} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {b^{2} B \tan \left (d x +c \right ) \sec \left (d x +c \right )}{2 d}+\frac {b^{2} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {b^{2} C \tan \left (d x +c \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x)

[Out]

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

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maxima [A]  time = 0.34, size = 357, normalized size = 1.54 \[ \frac {16 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a^{2} + 80 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{2} + 160 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a b + 80 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A b^{2} - 15 \, B a^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 30 \, A a b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, C a b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, B b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 240 \, C b^{2} \tan \left (d x + c\right )}{240 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, algorithm="maxima")

[Out]

1/240*(16*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*A*a^2 + 80*(tan(d*x + c)^3 + 3*tan(d*x + c)
)*C*a^2 + 160*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a*b + 80*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*b^2 - 15*B*a^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*l
og(sin(d*x + c) - 1)) - 30*A*a*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)) - 120*C*a*b*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(
sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 60*B*b^2*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c)
+ 1) + log(sin(d*x + c) - 1)) + 240*C*b^2*tan(d*x + c))/d

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mupad [B]  time = 5.17, size = 455, normalized size = 1.96 \[ \frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,B\,a^2}{8}+\frac {B\,b^2}{2}+\frac {3\,A\,a\,b}{4}+C\,a\,b\right )}{\frac {3\,B\,a^2}{2}+2\,B\,b^2+3\,A\,a\,b+4\,C\,a\,b}\right )\,\left (\frac {3\,B\,a^2}{4}+B\,b^2+\frac {3\,A\,a\,b}{2}+2\,C\,a\,b\right )}{d}-\frac {\left (2\,A\,a^2+2\,A\,b^2-\frac {5\,B\,a^2}{4}-B\,b^2+2\,C\,a^2+2\,C\,b^2-\frac {5\,A\,a\,b}{2}+4\,B\,a\,b-2\,C\,a\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {B\,a^2}{2}-\frac {16\,A\,b^2}{3}-\frac {8\,A\,a^2}{3}+2\,B\,b^2-\frac {16\,C\,a^2}{3}-8\,C\,b^2+A\,a\,b-\frac {32\,B\,a\,b}{3}+4\,C\,a\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {116\,A\,a^2}{15}+\frac {20\,A\,b^2}{3}+\frac {20\,C\,a^2}{3}+12\,C\,b^2+\frac {40\,B\,a\,b}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {8\,A\,a^2}{3}-\frac {16\,A\,b^2}{3}-\frac {B\,a^2}{2}-2\,B\,b^2-\frac {16\,C\,a^2}{3}-8\,C\,b^2-A\,a\,b-\frac {32\,B\,a\,b}{3}-4\,C\,a\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a^2+2\,A\,b^2+\frac {5\,B\,a^2}{4}+B\,b^2+2\,C\,a^2+2\,C\,b^2+\frac {5\,A\,a\,b}{2}+4\,B\,a\,b+2\,C\,a\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*cos(c + d*x))^2*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^6,x)

[Out]

(atanh((4*tan(c/2 + (d*x)/2)*((3*B*a^2)/8 + (B*b^2)/2 + (3*A*a*b)/4 + C*a*b))/((3*B*a^2)/2 + 2*B*b^2 + 3*A*a*b
+ 4*C*a*b))*((3*B*a^2)/4 + B*b^2 + (3*A*a*b)/2 + 2*C*a*b))/d - (tan(c/2 + (d*x)/2)*(2*A*a^2 + 2*A*b^2 + (5*B*
a^2)/4 + B*b^2 + 2*C*a^2 + 2*C*b^2 + (5*A*a*b)/2 + 4*B*a*b + 2*C*a*b) + tan(c/2 + (d*x)/2)^9*(2*A*a^2 + 2*A*b^
2 - (5*B*a^2)/4 - B*b^2 + 2*C*a^2 + 2*C*b^2 - (5*A*a*b)/2 + 4*B*a*b - 2*C*a*b) - tan(c/2 + (d*x)/2)^3*((8*A*a^
2)/3 + (16*A*b^2)/3 + (B*a^2)/2 + 2*B*b^2 + (16*C*a^2)/3 + 8*C*b^2 + A*a*b + (32*B*a*b)/3 + 4*C*a*b) - tan(c/2
+ (d*x)/2)^7*((8*A*a^2)/3 + (16*A*b^2)/3 - (B*a^2)/2 - 2*B*b^2 + (16*C*a^2)/3 + 8*C*b^2 - A*a*b + (32*B*a*b)/
3 - 4*C*a*b) + tan(c/2 + (d*x)/2)^5*((116*A*a^2)/15 + (20*A*b^2)/3 + (20*C*a^2)/3 + 12*C*b^2 + (40*B*a*b)/3))/
(d*(5*tan(c/2 + (d*x)/2)^2 - 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2)^6 - 5*tan(c/2 + (d*x)/2)^8 + tan(
c/2 + (d*x)/2)^10 - 1))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))**2*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**6,x)

[Out]

Timed out

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