3.961 \(\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^5(c+d x) \, dx\)

Optimal. Leaf size=223 \[ \frac {a \tan (c+d x) \sec (c+d x) \left (3 a^2 (3 A+4 C)+20 a b B+6 A b^2\right )}{24 d}+\frac {\tan (c+d x) \left (4 a^3 B+6 a^2 b (2 A+3 C)+16 a b^2 B+3 A b^3\right )}{6 d}+\frac {\left (a^3 (3 A+4 C)+12 a^2 b B+12 a b^2 (A+2 C)+8 b^3 B\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {(4 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}+b^3 C x \]

[Out]

b^3*C*x+1/8*(12*a^2*b*B+8*b^3*B+12*a*b^2*(A+2*C)+a^3*(3*A+4*C))*arctanh(sin(d*x+c))/d+1/6*(3*A*b^3+4*a^3*B+16*
a*b^2*B+6*a^2*b*(2*A+3*C))*tan(d*x+c)/d+1/24*a*(6*A*b^2+20*a*b*B+3*a^2*(3*A+4*C))*sec(d*x+c)*tan(d*x+c)/d+1/12
*(3*A*b+4*B*a)*(a+b*cos(d*x+c))^2*sec(d*x+c)^2*tan(d*x+c)/d+1/4*A*(a+b*cos(d*x+c))^3*sec(d*x+c)^3*tan(d*x+c)/d

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Rubi [A]  time = 0.67, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {3047, 3031, 3021, 2735, 3770} \[ \frac {\tan (c+d x) \left (6 a^2 b (2 A+3 C)+4 a^3 B+16 a b^2 B+3 A b^3\right )}{6 d}+\frac {\left (a^3 (3 A+4 C)+12 a^2 b B+12 a b^2 (A+2 C)+8 b^3 B\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a \tan (c+d x) \sec (c+d x) \left (3 a^2 (3 A+4 C)+20 a b B+6 A b^2\right )}{24 d}+\frac {(4 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}+b^3 C x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]

[Out]

b^3*C*x + ((12*a^2*b*B + 8*b^3*B + 12*a*b^2*(A + 2*C) + a^3*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) + ((3*A*
b^3 + 4*a^3*B + 16*a*b^2*B + 6*a^2*b*(2*A + 3*C))*Tan[c + d*x])/(6*d) + (a*(6*A*b^2 + 20*a*b*B + 3*a^2*(3*A +
4*C))*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((3*A*b + 4*a*B)*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])
/(12*d) + (A*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

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 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))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx &=\frac {A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \int (a+b \cos (c+d x))^2 \left (3 A b+4 a B+(3 a A+4 b B+4 a C) \cos (c+d x)+4 b C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac {(3 A b+4 a B) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{12} \int (a+b \cos (c+d x)) \left (6 A b^2+20 a b B+3 a^2 (3 A+4 C)+\left (15 a A b+8 a^2 B+12 b^2 B+24 a b C\right ) \cos (c+d x)+12 b^2 C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac {a \left (6 A b^2+20 a b B+3 a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(3 A b+4 a B) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{24} \int \left (-4 \left (3 A b^3+4 a^3 B+16 a b^2 B+6 a^2 b (2 A+3 C)\right )-3 \left (12 a^2 b B+8 b^3 B+12 a b^2 (A+2 C)+a^3 (3 A+4 C)\right ) \cos (c+d x)-24 b^3 C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac {\left (3 A b^3+4 a^3 B+16 a b^2 B+6 a^2 b (2 A+3 C)\right ) \tan (c+d x)}{6 d}+\frac {a \left (6 A b^2+20 a b B+3 a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(3 A b+4 a B) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{24} \int \left (-3 \left (12 a^2 b B+8 b^3 B+12 a b^2 (A+2 C)+a^3 (3 A+4 C)\right )-24 b^3 C \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=b^3 C x+\frac {\left (3 A b^3+4 a^3 B+16 a b^2 B+6 a^2 b (2 A+3 C)\right ) \tan (c+d x)}{6 d}+\frac {a \left (6 A b^2+20 a b B+3 a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(3 A b+4 a B) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{8} \left (-12 a^2 b B-8 b^3 B-12 a b^2 (A+2 C)-a^3 (3 A+4 C)\right ) \int \sec (c+d x) \, dx\\ &=b^3 C x+\frac {\left (12 a^2 b B+8 b^3 B+12 a b^2 (A+2 C)+a^3 (3 A+4 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\left (3 A b^3+4 a^3 B+16 a b^2 B+6 a^2 b (2 A+3 C)\right ) \tan (c+d x)}{6 d}+\frac {a \left (6 A b^2+20 a b B+3 a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(3 A b+4 a B) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}\\ \end {align*}

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Mathematica [A]  time = 1.37, size = 165, normalized size = 0.74 \[ \frac {8 a^2 (a B+3 A b) \tan ^3(c+d x)+3 \left (a^3 (3 A+4 C)+12 a^2 b B+12 a b^2 (A+2 C)+8 b^3 B\right ) \tanh ^{-1}(\sin (c+d x))+3 \tan (c+d x) \left (2 a^3 A \sec ^3(c+d x)+a \sec (c+d x) \left (a^2 (3 A+4 C)+12 a b B+12 A b^2\right )+8 \left (a^3 B+3 a^2 b (A+C)+3 a b^2 B+A b^3\right )\right )+24 b^3 C d x}{24 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]

[Out]

(24*b^3*C*d*x + 3*(12*a^2*b*B + 8*b^3*B + 12*a*b^2*(A + 2*C) + a^3*(3*A + 4*C))*ArcTanh[Sin[c + d*x]] + 3*(8*(
A*b^3 + a^3*B + 3*a*b^2*B + 3*a^2*b*(A + C)) + a*(12*A*b^2 + 12*a*b*B + a^2*(3*A + 4*C))*Sec[c + d*x] + 2*a^3*
A*Sec[c + d*x]^3)*Tan[c + d*x] + 8*a^2*(3*A*b + a*B)*Tan[c + d*x]^3)/(24*d)

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fricas [A]  time = 0.46, size = 257, normalized size = 1.15 \[ \frac {48 \, C b^{3} d x \cos \left (d x + c\right )^{4} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \, B a^{2} b + 12 \, {\left (A + 2 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \, B a^{2} b + 12 \, {\left (A + 2 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, A a^{3} + 8 \, {\left (2 \, B a^{3} + 3 \, {\left (2 \, A + 3 \, C\right )} a^{2} b + 9 \, B a b^{2} + 3 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \, B a^{2} b + 12 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algorithm="fricas")

[Out]

1/48*(48*C*b^3*d*x*cos(d*x + c)^4 + 3*((3*A + 4*C)*a^3 + 12*B*a^2*b + 12*(A + 2*C)*a*b^2 + 8*B*b^3)*cos(d*x +
c)^4*log(sin(d*x + c) + 1) - 3*((3*A + 4*C)*a^3 + 12*B*a^2*b + 12*(A + 2*C)*a*b^2 + 8*B*b^3)*cos(d*x + c)^4*lo
g(-sin(d*x + c) + 1) + 2*(6*A*a^3 + 8*(2*B*a^3 + 3*(2*A + 3*C)*a^2*b + 9*B*a*b^2 + 3*A*b^3)*cos(d*x + c)^3 + 3
*((3*A + 4*C)*a^3 + 12*B*a^2*b + 12*A*a*b^2)*cos(d*x + c)^2 + 8*(B*a^3 + 3*A*a^2*b)*cos(d*x + c))*sin(d*x + c)
)/(d*cos(d*x + c)^4)

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giac [B]  time = 0.33, size = 759, normalized size = 3.40 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algorithm="giac")

[Out]

1/24*(24*(d*x + c)*C*b^3 + 3*(3*A*a^3 + 4*C*a^3 + 12*B*a^2*b + 12*A*a*b^2 + 24*C*a*b^2 + 8*B*b^3)*log(abs(tan(
1/2*d*x + 1/2*c) + 1)) - 3*(3*A*a^3 + 4*C*a^3 + 12*B*a^2*b + 12*A*a*b^2 + 24*C*a*b^2 + 8*B*b^3)*log(abs(tan(1/
2*d*x + 1/2*c) - 1)) + 2*(15*A*a^3*tan(1/2*d*x + 1/2*c)^7 - 24*B*a^3*tan(1/2*d*x + 1/2*c)^7 + 12*C*a^3*tan(1/2
*d*x + 1/2*c)^7 - 72*A*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 36*B*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 72*C*a^2*b*tan(1/2*d
*x + 1/2*c)^7 + 36*A*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 72*B*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 24*A*b^3*tan(1/2*d*x +
1/2*c)^7 + 9*A*a^3*tan(1/2*d*x + 1/2*c)^5 + 40*B*a^3*tan(1/2*d*x + 1/2*c)^5 - 12*C*a^3*tan(1/2*d*x + 1/2*c)^5
+ 120*A*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 36*B*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 216*C*a^2*b*tan(1/2*d*x + 1/2*c)^5
- 36*A*a*b^2*tan(1/2*d*x + 1/2*c)^5 + 216*B*a*b^2*tan(1/2*d*x + 1/2*c)^5 + 72*A*b^3*tan(1/2*d*x + 1/2*c)^5 +
9*A*a^3*tan(1/2*d*x + 1/2*c)^3 - 40*B*a^3*tan(1/2*d*x + 1/2*c)^3 - 12*C*a^3*tan(1/2*d*x + 1/2*c)^3 - 120*A*a^2
*b*tan(1/2*d*x + 1/2*c)^3 - 36*B*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 216*C*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 36*A*a*b^
2*tan(1/2*d*x + 1/2*c)^3 - 216*B*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 72*A*b^3*tan(1/2*d*x + 1/2*c)^3 + 15*A*a^3*tan
(1/2*d*x + 1/2*c) + 24*B*a^3*tan(1/2*d*x + 1/2*c) + 12*C*a^3*tan(1/2*d*x + 1/2*c) + 72*A*a^2*b*tan(1/2*d*x + 1
/2*c) + 36*B*a^2*b*tan(1/2*d*x + 1/2*c) + 72*C*a^2*b*tan(1/2*d*x + 1/2*c) + 36*A*a*b^2*tan(1/2*d*x + 1/2*c) +
72*B*a*b^2*tan(1/2*d*x + 1/2*c) + 24*A*b^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^4)/d

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maple [A]  time = 0.45, size = 389, normalized size = 1.74 \[ \frac {A \,a^{3} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {3 A \,a^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {2 a^{3} B \tan \left (d x +c \right )}{3 d}+\frac {a^{3} B \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {C \,a^{3} \tan \left (d x +c \right ) \sec \left (d x +c \right )}{2 d}+\frac {C \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {2 A \,a^{2} b \tan \left (d x +c \right )}{d}+\frac {A \,a^{2} b \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{d}+\frac {3 a^{2} b B \tan \left (d x +c \right ) \sec \left (d x +c \right )}{2 d}+\frac {3 a^{2} b B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {3 C \,a^{2} b \tan \left (d x +c \right )}{d}+\frac {3 A a \,b^{2} \tan \left (d x +c \right ) \sec \left (d x +c \right )}{2 d}+\frac {3 A a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {3 B a \,b^{2} \tan \left (d x +c \right )}{d}+\frac {3 C a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {A \,b^{3} \tan \left (d x +c \right )}{d}+\frac {b^{3} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+b^{3} C x +\frac {b^{3} C c}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5,x)

[Out]

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

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maxima [A]  time = 0.35, size = 372, normalized size = 1.67 \[ \frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} + 48 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} b + 48 \, {\left (d x + c\right )} C b^{3} - 3 \, A a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, B a^{2} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, A a b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 72 \, C a b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, B b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 144 \, C a^{2} b \tan \left (d x + c\right ) + 144 \, B a b^{2} \tan \left (d x + c\right ) + 48 \, A b^{3} \tan \left (d x + c\right )}{48 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algorithm="maxima")

[Out]

1/48*(16*(tan(d*x + c)^3 + 3*tan(d*x + c))*B*a^3 + 48*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^2*b + 48*(d*x + c)
*C*b^3 - 3*A*a^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)) - 12*C*a^3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1)
+ log(sin(d*x + c) - 1)) - 36*B*a^2*b*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(
d*x + c) - 1)) - 36*A*a*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)) + 72*C*a*b^2*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 24*B*b^3*(log(sin(d*x + c) + 1) - log(sin(d
*x + c) - 1)) + 144*C*a^2*b*tan(d*x + c) + 144*B*a*b^2*tan(d*x + c) + 48*A*b^3*tan(d*x + c))/d

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mupad [B]  time = 4.34, size = 3210, normalized size = 14.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^5,x)

[Out]

(atan(((((3*A*a^3)/8 + B*b^3 + (C*a^3)/2 + (3*A*a*b^2)/2 + (3*B*a^2*b)/2 + 3*C*a*b^2)*(12*A*a^3 + 32*B*b^3 + 1
6*C*a^3 + 32*C*b^3 + 48*A*a*b^2 + 48*B*a^2*b + 96*C*a*b^2) + tan(c/2 + (d*x)/2)*((9*A^2*a^6)/2 + 32*B^2*b^6 +
8*C^2*a^6 + 32*C^2*b^6 + 72*A^2*a^2*b^4 + 36*A^2*a^4*b^2 + 96*B^2*a^2*b^4 + 72*B^2*a^4*b^2 + 288*C^2*a^2*b^4 +
96*C^2*a^4*b^2 + 12*A*C*a^6 + 96*A*B*a*b^5 + 36*A*B*a^5*b + 192*B*C*a*b^5 + 48*B*C*a^5*b + 168*A*B*a^3*b^3 +
288*A*C*a^2*b^4 + 120*A*C*a^4*b^2 + 320*B*C*a^3*b^3))*((3*A*a^3)/8 + B*b^3 + (C*a^3)/2 + (3*A*a*b^2)/2 + (3*B*
a^2*b)/2 + 3*C*a*b^2)*1i - (((3*A*a^3)/8 + B*b^3 + (C*a^3)/2 + (3*A*a*b^2)/2 + (3*B*a^2*b)/2 + 3*C*a*b^2)*(12*
A*a^3 + 32*B*b^3 + 16*C*a^3 + 32*C*b^3 + 48*A*a*b^2 + 48*B*a^2*b + 96*C*a*b^2) - tan(c/2 + (d*x)/2)*((9*A^2*a^
6)/2 + 32*B^2*b^6 + 8*C^2*a^6 + 32*C^2*b^6 + 72*A^2*a^2*b^4 + 36*A^2*a^4*b^2 + 96*B^2*a^2*b^4 + 72*B^2*a^4*b^2
+ 288*C^2*a^2*b^4 + 96*C^2*a^4*b^2 + 12*A*C*a^6 + 96*A*B*a*b^5 + 36*A*B*a^5*b + 192*B*C*a*b^5 + 48*B*C*a^5*b
+ 168*A*B*a^3*b^3 + 288*A*C*a^2*b^4 + 120*A*C*a^4*b^2 + 320*B*C*a^3*b^3))*((3*A*a^3)/8 + B*b^3 + (C*a^3)/2 + (
3*A*a*b^2)/2 + (3*B*a^2*b)/2 + 3*C*a*b^2)*1i)/(64*B^2*C*b^9 - (((3*A*a^3)/8 + B*b^3 + (C*a^3)/2 + (3*A*a*b^2)/
2 + (3*B*a^2*b)/2 + 3*C*a*b^2)*(12*A*a^3 + 32*B*b^3 + 16*C*a^3 + 32*C*b^3 + 48*A*a*b^2 + 48*B*a^2*b + 96*C*a*b
^2) - tan(c/2 + (d*x)/2)*((9*A^2*a^6)/2 + 32*B^2*b^6 + 8*C^2*a^6 + 32*C^2*b^6 + 72*A^2*a^2*b^4 + 36*A^2*a^4*b^
2 + 96*B^2*a^2*b^4 + 72*B^2*a^4*b^2 + 288*C^2*a^2*b^4 + 96*C^2*a^4*b^2 + 12*A*C*a^6 + 96*A*B*a*b^5 + 36*A*B*a^
5*b + 192*B*C*a*b^5 + 48*B*C*a^5*b + 168*A*B*a^3*b^3 + 288*A*C*a^2*b^4 + 120*A*C*a^4*b^2 + 320*B*C*a^3*b^3))*(
(3*A*a^3)/8 + B*b^3 + (C*a^3)/2 + (3*A*a*b^2)/2 + (3*B*a^2*b)/2 + 3*C*a*b^2) - 64*B*C^2*b^9 - (((3*A*a^3)/8 +
B*b^3 + (C*a^3)/2 + (3*A*a*b^2)/2 + (3*B*a^2*b)/2 + 3*C*a*b^2)*(12*A*a^3 + 32*B*b^3 + 16*C*a^3 + 32*C*b^3 + 48
*A*a*b^2 + 48*B*a^2*b + 96*C*a*b^2) + tan(c/2 + (d*x)/2)*((9*A^2*a^6)/2 + 32*B^2*b^6 + 8*C^2*a^6 + 32*C^2*b^6
+ 72*A^2*a^2*b^4 + 36*A^2*a^4*b^2 + 96*B^2*a^2*b^4 + 72*B^2*a^4*b^2 + 288*C^2*a^2*b^4 + 96*C^2*a^4*b^2 + 12*A*
C*a^6 + 96*A*B*a*b^5 + 36*A*B*a^5*b + 192*B*C*a*b^5 + 48*B*C*a^5*b + 168*A*B*a^3*b^3 + 288*A*C*a^2*b^4 + 120*A
*C*a^4*b^2 + 320*B*C*a^3*b^3))*((3*A*a^3)/8 + B*b^3 + (C*a^3)/2 + (3*A*a*b^2)/2 + (3*B*a^2*b)/2 + 3*C*a*b^2) -
192*C^3*a*b^8 + 576*C^3*a^2*b^7 - 32*C^3*a^3*b^6 + 192*C^3*a^4*b^5 + 16*C^3*a^6*b^3 - 96*A*C^2*a*b^8 + 384*B*
C^2*a*b^8 + 576*A*C^2*a^2*b^7 - 24*A*C^2*a^3*b^6 + 240*A*C^2*a^4*b^5 + 24*A*C^2*a^6*b^3 + 144*A^2*C*a^2*b^7 +
72*A^2*C*a^4*b^5 + 9*A^2*C*a^6*b^3 - 96*B*C^2*a^2*b^7 + 640*B*C^2*a^3*b^6 + 96*B*C^2*a^5*b^4 + 192*B^2*C*a^2*b
^7 + 144*B^2*C*a^4*b^5 + 192*A*B*C*a*b^8 + 336*A*B*C*a^3*b^6 + 72*A*B*C*a^5*b^4))*((A*a^3*3i)/4 + B*b^3*2i + C
*a^3*1i + A*a*b^2*3i + B*a^2*b*3i + C*a*b^2*6i))/d - (tan(c/2 + (d*x)/2)^7*(2*A*b^3 - (5*A*a^3)/4 + 2*B*a^3 -
C*a^3 - 3*A*a*b^2 + 6*A*a^2*b + 6*B*a*b^2 - 3*B*a^2*b + 6*C*a^2*b) + tan(c/2 + (d*x)/2)^3*(6*A*b^3 - (3*A*a^3)
/4 + (10*B*a^3)/3 + C*a^3 + 3*A*a*b^2 + 10*A*a^2*b + 18*B*a*b^2 + 3*B*a^2*b + 18*C*a^2*b) - tan(c/2 + (d*x)/2)
^5*((3*A*a^3)/4 + 6*A*b^3 + (10*B*a^3)/3 - C*a^3 - 3*A*a*b^2 + 10*A*a^2*b + 18*B*a*b^2 - 3*B*a^2*b + 18*C*a^2*
b) - tan(c/2 + (d*x)/2)*((5*A*a^3)/4 + 2*A*b^3 + 2*B*a^3 + C*a^3 + 3*A*a*b^2 + 6*A*a^2*b + 6*B*a*b^2 + 3*B*a^2
*b + 6*C*a^2*b))/(d*(6*tan(c/2 + (d*x)/2)^4 - 4*tan(c/2 + (d*x)/2)^2 - 4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x
)/2)^8 + 1)) + (2*C*b^3*atan((C*b^3*(tan(c/2 + (d*x)/2)*((9*A^2*a^6)/2 + 32*B^2*b^6 + 8*C^2*a^6 + 32*C^2*b^6 +
72*A^2*a^2*b^4 + 36*A^2*a^4*b^2 + 96*B^2*a^2*b^4 + 72*B^2*a^4*b^2 + 288*C^2*a^2*b^4 + 96*C^2*a^4*b^2 + 12*A*C
*a^6 + 96*A*B*a*b^5 + 36*A*B*a^5*b + 192*B*C*a*b^5 + 48*B*C*a^5*b + 168*A*B*a^3*b^3 + 288*A*C*a^2*b^4 + 120*A*
C*a^4*b^2 + 320*B*C*a^3*b^3) - C*b^3*(12*A*a^3 + 32*B*b^3 + 16*C*a^3 + 32*C*b^3 + 48*A*a*b^2 + 48*B*a^2*b + 96
*C*a*b^2)*1i) + C*b^3*(tan(c/2 + (d*x)/2)*((9*A^2*a^6)/2 + 32*B^2*b^6 + 8*C^2*a^6 + 32*C^2*b^6 + 72*A^2*a^2*b^
4 + 36*A^2*a^4*b^2 + 96*B^2*a^2*b^4 + 72*B^2*a^4*b^2 + 288*C^2*a^2*b^4 + 96*C^2*a^4*b^2 + 12*A*C*a^6 + 96*A*B*
a*b^5 + 36*A*B*a^5*b + 192*B*C*a*b^5 + 48*B*C*a^5*b + 168*A*B*a^3*b^3 + 288*A*C*a^2*b^4 + 120*A*C*a^4*b^2 + 32
0*B*C*a^3*b^3) + C*b^3*(12*A*a^3 + 32*B*b^3 + 16*C*a^3 + 32*C*b^3 + 48*A*a*b^2 + 48*B*a^2*b + 96*C*a*b^2)*1i))
/(64*B^2*C*b^9 - 64*B*C^2*b^9 - 192*C^3*a*b^8 + C*b^3*(tan(c/2 + (d*x)/2)*((9*A^2*a^6)/2 + 32*B^2*b^6 + 8*C^2*
a^6 + 32*C^2*b^6 + 72*A^2*a^2*b^4 + 36*A^2*a^4*b^2 + 96*B^2*a^2*b^4 + 72*B^2*a^4*b^2 + 288*C^2*a^2*b^4 + 96*C^
2*a^4*b^2 + 12*A*C*a^6 + 96*A*B*a*b^5 + 36*A*B*a^5*b + 192*B*C*a*b^5 + 48*B*C*a^5*b + 168*A*B*a^3*b^3 + 288*A*
C*a^2*b^4 + 120*A*C*a^4*b^2 + 320*B*C*a^3*b^3) - C*b^3*(12*A*a^3 + 32*B*b^3 + 16*C*a^3 + 32*C*b^3 + 48*A*a*b^2
+ 48*B*a^2*b + 96*C*a*b^2)*1i)*1i - C*b^3*(tan(c/2 + (d*x)/2)*((9*A^2*a^6)/2 + 32*B^2*b^6 + 8*C^2*a^6 + 32*C^
2*b^6 + 72*A^2*a^2*b^4 + 36*A^2*a^4*b^2 + 96*B^2*a^2*b^4 + 72*B^2*a^4*b^2 + 288*C^2*a^2*b^4 + 96*C^2*a^4*b^2 +
12*A*C*a^6 + 96*A*B*a*b^5 + 36*A*B*a^5*b + 192*B*C*a*b^5 + 48*B*C*a^5*b + 168*A*B*a^3*b^3 + 288*A*C*a^2*b^4 +
120*A*C*a^4*b^2 + 320*B*C*a^3*b^3) + C*b^3*(12*A*a^3 + 32*B*b^3 + 16*C*a^3 + 32*C*b^3 + 48*A*a*b^2 + 48*B*a^2
*b + 96*C*a*b^2)*1i)*1i + 576*C^3*a^2*b^7 - 32*C^3*a^3*b^6 + 192*C^3*a^4*b^5 + 16*C^3*a^6*b^3 - 96*A*C^2*a*b^8
+ 384*B*C^2*a*b^8 + 576*A*C^2*a^2*b^7 - 24*A*C^2*a^3*b^6 + 240*A*C^2*a^4*b^5 + 24*A*C^2*a^6*b^3 + 144*A^2*C*a
^2*b^7 + 72*A^2*C*a^4*b^5 + 9*A^2*C*a^6*b^3 - 96*B*C^2*a^2*b^7 + 640*B*C^2*a^3*b^6 + 96*B*C^2*a^5*b^4 + 192*B^
2*C*a^2*b^7 + 144*B^2*C*a^4*b^5 + 192*A*B*C*a*b^8 + 336*A*B*C*a^3*b^6 + 72*A*B*C*a^5*b^4)))/d

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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))**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**5,x)

[Out]

Timed out

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