\(\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec (c+d x) \, dx\) [957]

Optimal result

Integrand size = 39, antiderivative size = 207 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {1}{8} \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) x+\frac {a^3 A \text {arctanh}(\sin (c+d x))}{d}+\frac {\left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 d}+\frac {b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d} \]

[Out]

1/8*(8*B*a^3+12*B*a*b^2+12*a^2*b*(2*A+C)+b^3*(4*A+3*C))*x+a^3*A*arctanh(sin(d*x+c))/d+1/6*(16*B*a^2*b+4*B*b^3+
3*a^3*C+6*a*b^2*(3*A+2*C))*sin(d*x+c)/d+1/24*b*(12*A*b^2+20*B*a*b+6*C*a^2+9*C*b^2)*cos(d*x+c)*sin(d*x+c)/d+1/1
2*(4*B*b+3*C*a)*(a+b*cos(d*x+c))^2*sin(d*x+c)/d+1/4*C*(a+b*cos(d*x+c))^3*sin(d*x+c)/d

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {3128, 3112, 3102, 2814, 3855} \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {a^3 A \text {arctanh}(\sin (c+d x))}{d}+\frac {b \sin (c+d x) \cos (c+d x) \left (6 a^2 C+20 a b B+12 A b^2+9 b^2 C\right )}{24 d}+\frac {\sin (c+d x) \left (3 a^3 C+16 a^2 b B+6 a b^2 (3 A+2 C)+4 b^3 B\right )}{6 d}+\frac {1}{8} x \left (8 a^3 B+12 a^2 b (2 A+C)+12 a b^2 B+b^3 (4 A+3 C)\right )+\frac {(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d} \]

[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],x]

[Out]

((8*a^3*B + 12*a*b^2*B + 12*a^2*b*(2*A + C) + b^3*(4*A + 3*C))*x)/8 + (a^3*A*ArcTanh[Sin[c + d*x]])/d + ((16*a
^2*b*B + 4*b^3*B + 3*a^3*C + 6*a*b^2*(3*A + 2*C))*Sin[c + d*x])/(6*d) + (b*(12*A*b^2 + 20*a*b*B + 6*a^2*C + 9*
b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((4*b*B + 3*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(12*d) + (C*(
a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d)

Rule 2814

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 3102

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

Rule 3112

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[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a +
b*Sin[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*
c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e
+ f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!LtQ[m, -1]

Rule 3128

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

Rule 3855

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

Rubi steps \begin{align*} \text {integral}& = \frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{4} \int (a+b \cos (c+d x))^2 \left (4 a A+(4 A b+4 a B+3 b C) \cos (c+d x)+(4 b B+3 a C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {(4 b B+3 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{12} \int (a+b \cos (c+d x)) \left (12 a^2 A+\left (24 a A b+12 a^2 B+8 b^2 B+15 a b C\right ) \cos (c+d x)+\left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{24} \int \left (24 a^3 A+3 \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) \cos (c+d x)+4 \left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {\left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 d}+\frac {b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {1}{24} \int \left (24 a^3 A+3 \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {1}{8} \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) x+\frac {\left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 d}+\frac {b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\left (a^3 A\right ) \int \sec (c+d x) \, dx \\ & = \frac {1}{8} \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) x+\frac {a^3 A \text {arctanh}(\sin (c+d x))}{d}+\frac {\left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 d}+\frac {b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.00 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.05 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {12 \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) (c+d x)-96 a^3 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+96 a^3 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 \left (12 a^2 b B+3 b^3 B+4 a^3 C+3 a b^2 (4 A+3 C)\right ) \sin (c+d x)+24 b \left (A b^2+3 a b B+3 a^2 C+b^2 C\right ) \sin (2 (c+d x))+8 b^2 (b B+3 a C) \sin (3 (c+d x))+3 b^3 C \sin (4 (c+d x))}{96 d} \]

[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],x]

[Out]

(12*(8*a^3*B + 12*a*b^2*B + 12*a^2*b*(2*A + C) + b^3*(4*A + 3*C))*(c + d*x) - 96*a^3*A*Log[Cos[(c + d*x)/2] -
Sin[(c + d*x)/2]] + 96*a^3*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 24*(12*a^2*b*B + 3*b^3*B + 4*a^3*C + 3
*a*b^2*(4*A + 3*C))*Sin[c + d*x] + 24*b*(A*b^2 + 3*a*b*B + 3*a^2*C + b^2*C)*Sin[2*(c + d*x)] + 8*b^2*(b*B + 3*
a*C)*Sin[3*(c + d*x)] + 3*b^3*C*Sin[4*(c + d*x)])/(96*d)

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.92

method	result	size
parallelrisch	\(\frac {-96 A \,a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+96 A \,a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+24 \left (\left (A +C \right ) b^{2}+3 B a b +3 a^{2} C \right ) b \sin \left (2 d x +2 c \right )+\left (8 B \,b^{3}+24 C a \,b^{2}\right ) \sin \left (3 d x +3 c \right )+3 C \sin \left (4 d x +4 c \right ) b^{3}+\left (72 B \,b^{3}+288 a \left (A +\frac {3 C}{4}\right ) b^{2}+288 B \,a^{2} b +96 a^{3} C \right ) \sin \left (d x +c \right )+288 \left (\left (\frac {A}{6}+\frac {C}{8}\right ) b^{3}+\frac {B a \,b^{2}}{2}+a^{2} \left (A +\frac {C}{2}\right ) b +\frac {B \,a^{3}}{3}\right ) x d}{96 d}\)	\(190\)
parts	\(\frac {A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) \left (d x +c \right )}{d}+\frac {\left (B \,b^{3}+3 C a \,b^{2}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (A \,b^{3}+3 B a \,b^{2}+3 a^{2} b C \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (3 a A \,b^{2}+3 B \,a^{2} b +a^{3} C \right ) \sin \left (d x +c \right )}{d}+\frac {C \,b^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(197\)
derivativedivides	\(\frac {A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{3} \left (d x +c \right )+C \sin \left (d x +c \right ) a^{3}+3 A \,a^{2} b \left (d x +c \right )+3 B \sin \left (d x +c \right ) a^{2} b +3 a^{2} b C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 A \sin \left (d x +c \right ) a \,b^{2}+3 B a \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C a \,b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+A \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,b^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+C \,b^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(251\)
default	\(\frac {A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{3} \left (d x +c \right )+C \sin \left (d x +c \right ) a^{3}+3 A \,a^{2} b \left (d x +c \right )+3 B \sin \left (d x +c \right ) a^{2} b +3 a^{2} b C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 A \sin \left (d x +c \right ) a \,b^{2}+3 B a \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C a \,b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+A \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,b^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+C \,b^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(251\)
risch	\(\frac {\sin \left (3 d x +3 c \right ) B \,b^{3}}{12 d}+\frac {\sin \left (2 d x +2 c \right ) A \,b^{3}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) C \,b^{3}}{4 d}+3 x A \,a^{2} b +\frac {3 x B a \,b^{2}}{2}+\frac {3 x \,a^{2} b C}{2}+\frac {\sin \left (4 d x +4 c \right ) C \,b^{3}}{32 d}+\frac {3 b^{3} C x}{8}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} B \,b^{3}}{8 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} a^{3} C}{2 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B \,b^{3}}{8 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{3} C}{2 d}+\frac {\sin \left (3 d x +3 c \right ) C a \,b^{2}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) B a \,b^{2}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) a^{2} b C}{4 d}+\frac {x A \,b^{3}}{2}+x B \,a^{3}+\frac {9 i {\mathrm e}^{-i \left (d x +c \right )} C a \,b^{2}}{8 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a A \,b^{2}}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{2} b}{2 d}-\frac {9 i {\mathrm e}^{i \left (d x +c \right )} C a \,b^{2}}{8 d}+\frac {A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a A \,b^{2}}{2 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{2} b}{2 d}\)	\(414\)
norman	\(\frac {\left (3 A \,a^{2} b +B \,a^{3}+\frac {1}{2} A \,b^{3}+\frac {3}{2} B a \,b^{2}+\frac {3}{2} a^{2} b C +\frac {3}{8} C \,b^{3}\right ) x +\left (3 A \,a^{2} b +B \,a^{3}+\frac {1}{2} A \,b^{3}+\frac {3}{2} B a \,b^{2}+\frac {3}{2} a^{2} b C +\frac {3}{8} C \,b^{3}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 A \,a^{2} b +5 B \,a^{3}+\frac {5}{2} A \,b^{3}+\frac {15}{2} B a \,b^{2}+\frac {15}{2} a^{2} b C +\frac {15}{8} C \,b^{3}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (15 A \,a^{2} b +5 B \,a^{3}+\frac {5}{2} A \,b^{3}+\frac {15}{2} B a \,b^{2}+\frac {15}{2} a^{2} b C +\frac {15}{8} C \,b^{3}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 A \,a^{2} b +10 B \,a^{3}+5 A \,b^{3}+15 B a \,b^{2}+15 a^{2} b C +\frac {15}{4} C \,b^{3}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (30 A \,a^{2} b +10 B \,a^{3}+5 A \,b^{3}+15 B a \,b^{2}+15 a^{2} b C +\frac {15}{4} C \,b^{3}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 \left (27 a A \,b^{2}+27 B \,a^{2} b +5 B \,b^{3}+9 a^{3} C +15 C a \,b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {\left (24 a A \,b^{2}-4 A \,b^{3}+24 B \,a^{2} b -12 B a \,b^{2}+8 B \,b^{3}+8 a^{3} C -12 a^{2} b C +24 C a \,b^{2}-5 C \,b^{3}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {\left (24 a A \,b^{2}+4 A \,b^{3}+24 B \,a^{2} b +12 B a \,b^{2}+8 B \,b^{3}+8 a^{3} C +12 a^{2} b C +24 C a \,b^{2}+5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (144 a A \,b^{2}-12 A \,b^{3}+144 B \,a^{2} b -36 B a \,b^{2}+32 B \,b^{3}+48 a^{3} C -36 a^{2} b C +96 C a \,b^{2}-3 C \,b^{3}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {\left (144 a A \,b^{2}+12 A \,b^{3}+144 B \,a^{2} b +36 B a \,b^{2}+32 B \,b^{3}+48 a^{3} C +36 a^{2} b C +96 C a \,b^{2}+3 C \,b^{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {A \,a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {A \,a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\)	\(717\)

[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),x,method=_RETURNVERBOSE)

[Out]

1/96*(-96*A*a^3*ln(tan(1/2*d*x+1/2*c)-1)+96*A*a^3*ln(tan(1/2*d*x+1/2*c)+1)+24*((A+C)*b^2+3*B*a*b+3*a^2*C)*b*si
n(2*d*x+2*c)+(8*B*b^3+24*C*a*b^2)*sin(3*d*x+3*c)+3*C*sin(4*d*x+4*c)*b^3+(72*B*b^3+288*a*(A+3/4*C)*b^2+288*B*a^
2*b+96*a^3*C)*sin(d*x+c)+288*((1/6*A+1/8*C)*b^3+1/2*B*a*b^2+a^2*(A+1/2*C)*b+1/3*B*a^3)*x*d)/d

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.91 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {12 \, A a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, A a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (8 \, B a^{3} + 12 \, {\left (2 \, A + C\right )} a^{2} b + 12 \, B a b^{2} + {\left (4 \, A + 3 \, C\right )} b^{3}\right )} d x + {\left (6 \, C b^{3} \cos \left (d x + c\right )^{3} + 24 \, C a^{3} + 72 \, B a^{2} b + 24 \, {\left (3 \, A + 2 \, C\right )} a b^{2} + 16 \, B b^{3} + 8 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (12 \, C a^{2} b + 12 \, B a b^{2} + {\left (4 \, A + 3 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]

[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),x, algorithm="fricas")

[Out]

1/24*(12*A*a^3*log(sin(d*x + c) + 1) - 12*A*a^3*log(-sin(d*x + c) + 1) + 3*(8*B*a^3 + 12*(2*A + C)*a^2*b + 12*
B*a*b^2 + (4*A + 3*C)*b^3)*d*x + (6*C*b^3*cos(d*x + c)^3 + 24*C*a^3 + 72*B*a^2*b + 24*(3*A + 2*C)*a*b^2 + 16*B
*b^3 + 8*(3*C*a*b^2 + B*b^3)*cos(d*x + c)^2 + 3*(12*C*a^2*b + 12*B*a*b^2 + (4*A + 3*C)*b^3)*cos(d*x + c))*sin(
d*x + c))/d

Sympy [F]

\[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\int \left (a + b \cos {\left (c + d x \right )}\right )^{3} \left (A + B \cos {\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]

[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),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.15 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {96 \, {\left (d x + c\right )} B a^{3} + 288 \, {\left (d x + c\right )} A a^{2} b + 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} b + 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b^{2} - 96 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a b^{2} + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{3} - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B b^{3} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{3} + 96 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 96 \, C a^{3} \sin \left (d x + c\right ) + 288 \, B a^{2} b \sin \left (d x + c\right ) + 288 \, A a b^{2} \sin \left (d x + c\right )}{96 \, d} \]

[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),x, algorithm="maxima")

[Out]

1/96*(96*(d*x + c)*B*a^3 + 288*(d*x + c)*A*a^2*b + 72*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^2*b + 72*(2*d*x + 2
*c + sin(2*d*x + 2*c))*B*a*b^2 - 96*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a*b^2 + 24*(2*d*x + 2*c + sin(2*d*x +
2*c))*A*b^3 - 32*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*b^3 + 3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x +
2*c))*C*b^3 + 96*A*a^3*log(sec(d*x + c) + tan(d*x + c)) + 96*C*a^3*sin(d*x + c) + 288*B*a^2*b*sin(d*x + c) +
288*A*a*b^2*sin(d*x + c))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 723 vs. \(2 (197) = 394\).

Time = 0.34 (sec) , antiderivative size = 723, normalized size of antiderivative = 3.49 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\text {Too large to display} \]

[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),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.70 (sec) , antiderivative size = 3250, normalized size of antiderivative = 15.70 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\text {Too large to display} \]

[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),x)

[Out]

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