\(\int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx\) [438]

Optimal result

Integrand size = 21, antiderivative size = 247 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {1}{4} a b \left (6 a^2+5 b^2\right ) x+\frac {\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin (c+d x)}{35 d}+\frac {a b \left (6 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a b \left (6 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{6 d}+\frac {b^2 \left (37 a^2+6 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{35 d}+\frac {8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}-\frac {\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin ^3(c+d x)}{105 d} \]

[Out]

1/4*a*b*(6*a^2+5*b^2)*x+1/35*(35*a^4+168*a^2*b^2+24*b^4)*sin(d*x+c)/d+1/4*a*b*(6*a^2+5*b^2)*cos(d*x+c)*sin(d*x
+c)/d+1/6*a*b*(6*a^2+5*b^2)*cos(d*x+c)^3*sin(d*x+c)/d+1/35*b^2*(37*a^2+6*b^2)*cos(d*x+c)^4*sin(d*x+c)/d+8/21*a
*b^3*cos(d*x+c)^5*sin(d*x+c)/d+1/7*b^2*cos(d*x+c)^4*(a+b*cos(d*x+c))^2*sin(d*x+c)/d-1/105*(35*a^4+168*a^2*b^2+
24*b^4)*sin(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2872, 3112, 3102, 2827, 2713, 2715, 8} \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {b^2 \left (37 a^2+6 b^2\right ) \sin (c+d x) \cos ^4(c+d x)}{35 d}+\frac {a b \left (6 a^2+5 b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{6 d}+\frac {a b \left (6 a^2+5 b^2\right ) \sin (c+d x) \cos (c+d x)}{4 d}+\frac {1}{4} a b x \left (6 a^2+5 b^2\right )-\frac {\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin ^3(c+d x)}{105 d}+\frac {\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin (c+d x)}{35 d}+\frac {8 a b^3 \sin (c+d x) \cos ^5(c+d x)}{21 d}+\frac {b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))^2}{7 d} \]

[In]

Int[Cos[c + d*x]^3*(a + b*Cos[c + d*x])^4,x]

[Out]

(a*b*(6*a^2 + 5*b^2)*x)/4 + ((35*a^4 + 168*a^2*b^2 + 24*b^4)*Sin[c + d*x])/(35*d) + (a*b*(6*a^2 + 5*b^2)*Cos[c
+ d*x]*Sin[c + d*x])/(4*d) + (a*b*(6*a^2 + 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(6*d) + (b^2*(37*a^2 + 6*b^2)*
Cos[c + d*x]^4*Sin[c + d*x])/(35*d) + (8*a*b^3*Cos[c + d*x]^5*Sin[c + d*x])/(21*d) + (b^2*Cos[c + d*x]^4*(a +
b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) - ((35*a^4 + 168*a^2*b^2 + 24*b^4)*Sin[c + d*x]^3)/(105*d)

Rule 8

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

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2715

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

Rule 2827

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 2872

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Dist[1/
(d*(m + n)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d*(m + n) + b^2*(b*c*(m - 2) + a
*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n
- 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
&& NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m]
|| (EqQ[a, 0] && NeQ[c, 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]

Rubi steps \begin{align*} \text {integral}& = \frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {1}{7} \int \cos ^3(c+d x) (a+b \cos (c+d x)) \left (a \left (7 a^2+4 b^2\right )+3 b \left (7 a^2+2 b^2\right ) \cos (c+d x)+16 a b^2 \cos ^2(c+d x)\right ) \, dx \\ & = \frac {8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {1}{42} \int \cos ^3(c+d x) \left (6 a^2 \left (7 a^2+4 b^2\right )+28 a b \left (6 a^2+5 b^2\right ) \cos (c+d x)+6 b^2 \left (37 a^2+6 b^2\right ) \cos ^2(c+d x)\right ) \, dx \\ & = \frac {b^2 \left (37 a^2+6 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{35 d}+\frac {8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {1}{210} \int \cos ^3(c+d x) \left (6 \left (35 a^4+168 a^2 b^2+24 b^4\right )+140 a b \left (6 a^2+5 b^2\right ) \cos (c+d x)\right ) \, dx \\ & = \frac {b^2 \left (37 a^2+6 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{35 d}+\frac {8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {1}{3} \left (2 a b \left (6 a^2+5 b^2\right )\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{35} \left (35 a^4+168 a^2 b^2+24 b^4\right ) \int \cos ^3(c+d x) \, dx \\ & = \frac {a b \left (6 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{6 d}+\frac {b^2 \left (37 a^2+6 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{35 d}+\frac {8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac {1}{2} \left (a b \left (6 a^2+5 b^2\right )\right ) \int \cos ^2(c+d x) \, dx-\frac {\left (35 a^4+168 a^2 b^2+24 b^4\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d} \\ & = \frac {\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin (c+d x)}{35 d}+\frac {a b \left (6 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a b \left (6 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{6 d}+\frac {b^2 \left (37 a^2+6 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{35 d}+\frac {8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}-\frac {\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin ^3(c+d x)}{105 d}+\frac {1}{4} \left (a b \left (6 a^2+5 b^2\right )\right ) \int 1 \, dx \\ & = \frac {1}{4} a b \left (6 a^2+5 b^2\right ) x+\frac {\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin (c+d x)}{35 d}+\frac {a b \left (6 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a b \left (6 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{6 d}+\frac {b^2 \left (37 a^2+6 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{35 d}+\frac {8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}-\frac {\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin ^3(c+d x)}{105 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.73 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {1680 a b \left (6 a^2+5 b^2\right ) (c+d x)+105 \left (48 a^4+240 a^2 b^2+35 b^4\right ) \sin (c+d x)+420 a b \left (16 a^2+15 b^2\right ) \sin (2 (c+d x))+35 \left (16 a^4+120 a^2 b^2+21 b^4\right ) \sin (3 (c+d x))+420 a b \left (2 a^2+3 b^2\right ) \sin (4 (c+d x))+21 b^2 \left (24 a^2+7 b^2\right ) \sin (5 (c+d x))+140 a b^3 \sin (6 (c+d x))+15 b^4 \sin (7 (c+d x))}{6720 d} \]

[In]

Integrate[Cos[c + d*x]^3*(a + b*Cos[c + d*x])^4,x]

[Out]

(1680*a*b*(6*a^2 + 5*b^2)*(c + d*x) + 105*(48*a^4 + 240*a^2*b^2 + 35*b^4)*Sin[c + d*x] + 420*a*b*(16*a^2 + 15*
b^2)*Sin[2*(c + d*x)] + 35*(16*a^4 + 120*a^2*b^2 + 21*b^4)*Sin[3*(c + d*x)] + 420*a*b*(2*a^2 + 3*b^2)*Sin[4*(c
+ d*x)] + 21*b^2*(24*a^2 + 7*b^2)*Sin[5*(c + d*x)] + 140*a*b^3*Sin[6*(c + d*x)] + 15*b^4*Sin[7*(c + d*x)])/(6
720*d)

Maple [A] (verified)

Time = 5.95 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.71

method	result	size
parallelrisch	\(\frac {\left (560 a^{4}+4200 a^{2} b^{2}+735 b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (6720 a^{3} b +6300 a \,b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (840 a^{3} b +1260 a \,b^{3}\right ) \sin \left (4 d x +4 c \right )+\left (504 a^{2} b^{2}+147 b^{4}\right ) \sin \left (5 d x +5 c \right )+140 a \,b^{3} \sin \left (6 d x +6 c \right )+15 b^{4} \sin \left (7 d x +7 c \right )+\left (5040 a^{4}+25200 a^{2} b^{2}+3675 b^{4}\right ) \sin \left (d x +c \right )+10080 b \left (a^{2}+\frac {5 b^{2}}{6}\right ) d x a}{6720 d}\)	\(176\)
derivativedivides	\(\frac {\frac {a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{3} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {6 a^{2} b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a \,b^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {b^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\)	\(190\)
default	\(\frac {\frac {a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{3} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {6 a^{2} b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a \,b^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {b^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\)	\(190\)
parts	\(\frac {a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {b^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7 d}+\frac {4 a \,b^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {6 a^{2} b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {4 a^{3} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(201\)
risch	\(\frac {3 a^{3} b x}{2}+\frac {5 a \,b^{3} x}{4}+\frac {3 a^{4} \sin \left (d x +c \right )}{4 d}+\frac {15 \sin \left (d x +c \right ) a^{2} b^{2}}{4 d}+\frac {35 \sin \left (d x +c \right ) b^{4}}{64 d}+\frac {b^{4} \sin \left (7 d x +7 c \right )}{448 d}+\frac {a \,b^{3} \sin \left (6 d x +6 c \right )}{48 d}+\frac {3 \sin \left (5 d x +5 c \right ) a^{2} b^{2}}{40 d}+\frac {7 \sin \left (5 d x +5 c \right ) b^{4}}{320 d}+\frac {\sin \left (4 d x +4 c \right ) a^{3} b}{8 d}+\frac {3 \sin \left (4 d x +4 c \right ) a \,b^{3}}{16 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4}}{12 d}+\frac {5 \sin \left (3 d x +3 c \right ) a^{2} b^{2}}{8 d}+\frac {7 \sin \left (3 d x +3 c \right ) b^{4}}{64 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} b}{d}+\frac {15 \sin \left (2 d x +2 c \right ) a \,b^{3}}{16 d}\)	\(258\)
norman	\(\frac {\left (\frac {3}{2} a^{3} b +\frac {5}{4} a \,b^{3}\right ) x +\left (\frac {3}{2} a^{3} b +\frac {5}{4} a \,b^{3}\right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {21}{2} a^{3} b +\frac {35}{4} a \,b^{3}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {21}{2} a^{3} b +\frac {35}{4} a \,b^{3}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {63}{2} a^{3} b +\frac {105}{4} a \,b^{3}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {63}{2} a^{3} b +\frac {105}{4} a \,b^{3}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {105}{2} a^{3} b +\frac {175}{4} a \,b^{3}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {105}{2} a^{3} b +\frac {175}{4} a \,b^{3}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8 \left (105 a^{4}+546 a^{2} b^{2}+53 b^{4}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {\left (4 a^{4}-10 a^{3} b +24 a^{2} b^{2}-11 a \,b^{3}+4 b^{4}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {\left (4 a^{4}+10 a^{3} b +24 a^{2} b^{2}+11 a \,b^{3}+4 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}+\frac {2 \left (14 a^{4}-18 a^{3} b +60 a^{2} b^{2}-7 a \,b^{3}+6 b^{4}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 \left (14 a^{4}+18 a^{3} b +60 a^{2} b^{2}+7 a \,b^{3}+6 b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {\left (580 a^{4}-270 a^{3} b +2712 a^{2} b^{2}-425 a \,b^{3}+516 b^{4}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 d}+\frac {\left (580 a^{4}+270 a^{3} b +2712 a^{2} b^{2}+425 a \,b^{3}+516 b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{30 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\)	\(530\)

[In]

int(cos(d*x+c)^3*(a+cos(d*x+c)*b)^4,x,method=_RETURNVERBOSE)

[Out]

1/6720*((560*a^4+4200*a^2*b^2+735*b^4)*sin(3*d*x+3*c)+(6720*a^3*b+6300*a*b^3)*sin(2*d*x+2*c)+(840*a^3*b+1260*a
*b^3)*sin(4*d*x+4*c)+(504*a^2*b^2+147*b^4)*sin(5*d*x+5*c)+140*a*b^3*sin(6*d*x+6*c)+15*b^4*sin(7*d*x+7*c)+(5040
*a^4+25200*a^2*b^2+3675*b^4)*sin(d*x+c)+10080*b*(a^2+5/6*b^2)*d*x*a)/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.69 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {105 \, {\left (6 \, a^{3} b + 5 \, a b^{3}\right )} d x + {\left (60 \, b^{4} \cos \left (d x + c\right )^{6} + 280 \, a b^{3} \cos \left (d x + c\right )^{5} + 72 \, {\left (7 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + 280 \, a^{4} + 1344 \, a^{2} b^{2} + 192 \, b^{4} + 70 \, {\left (6 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (35 \, a^{4} + 168 \, a^{2} b^{2} + 24 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + 105 \, {\left (6 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{420 \, d} \]

[In]

integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

1/420*(105*(6*a^3*b + 5*a*b^3)*d*x + (60*b^4*cos(d*x + c)^6 + 280*a*b^3*cos(d*x + c)^5 + 72*(7*a^2*b^2 + b^4)*
cos(d*x + c)^4 + 280*a^4 + 1344*a^2*b^2 + 192*b^4 + 70*(6*a^3*b + 5*a*b^3)*cos(d*x + c)^3 + 4*(35*a^4 + 168*a^
2*b^2 + 24*b^4)*cos(d*x + c)^2 + 105*(6*a^3*b + 5*a*b^3)*cos(d*x + c))*sin(d*x + c))/d

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (231) = 462\).

Time = 0.52 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.00 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx=\begin {cases} \frac {2 a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 a^{3} b x \sin ^{4}{\left (c + d x \right )}}{2} + 3 a^{3} b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + \frac {3 a^{3} b x \cos ^{4}{\left (c + d x \right )}}{2} + \frac {3 a^{3} b \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {5 a^{3} b \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {16 a^{2} b^{2} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {8 a^{2} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {6 a^{2} b^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 a b^{3} x \sin ^{6}{\left (c + d x \right )}}{4} + \frac {15 a b^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {15 a b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4} + \frac {5 a b^{3} x \cos ^{6}{\left (c + d x \right )}}{4} + \frac {5 a b^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} + \frac {10 a b^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac {11 a b^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{4 d} + \frac {16 b^{4} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {b^{4} \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\left (c \right )}\right )^{4} \cos ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**3*(a+b*cos(d*x+c))**4,x)

[Out]

Piecewise((2*a**4*sin(c + d*x)**3/(3*d) + a**4*sin(c + d*x)*cos(c + d*x)**2/d + 3*a**3*b*x*sin(c + d*x)**4/2 +
3*a**3*b*x*sin(c + d*x)**2*cos(c + d*x)**2 + 3*a**3*b*x*cos(c + d*x)**4/2 + 3*a**3*b*sin(c + d*x)**3*cos(c +
d*x)/(2*d) + 5*a**3*b*sin(c + d*x)*cos(c + d*x)**3/(2*d) + 16*a**2*b**2*sin(c + d*x)**5/(5*d) + 8*a**2*b**2*si
n(c + d*x)**3*cos(c + d*x)**2/d + 6*a**2*b**2*sin(c + d*x)*cos(c + d*x)**4/d + 5*a*b**3*x*sin(c + d*x)**6/4 +
15*a*b**3*x*sin(c + d*x)**4*cos(c + d*x)**2/4 + 15*a*b**3*x*sin(c + d*x)**2*cos(c + d*x)**4/4 + 5*a*b**3*x*cos
(c + d*x)**6/4 + 5*a*b**3*sin(c + d*x)**5*cos(c + d*x)/(4*d) + 10*a*b**3*sin(c + d*x)**3*cos(c + d*x)**3/(3*d)
+ 11*a*b**3*sin(c + d*x)*cos(c + d*x)**5/(4*d) + 16*b**4*sin(c + d*x)**7/(35*d) + 8*b**4*sin(c + d*x)**5*cos(
c + d*x)**2/(5*d) + 2*b**4*sin(c + d*x)**3*cos(c + d*x)**4/d + b**4*sin(c + d*x)*cos(c + d*x)**6/d, Ne(d, 0)),
(x*(a + b*cos(c))**4*cos(c)**3, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.78 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx=-\frac {560 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{4} - 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} b - 672 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{2} b^{2} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a b^{3} + 48 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} b^{4}}{1680 \, d} \]

[In]

integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/1680*(560*(sin(d*x + c)^3 - 3*sin(d*x + c))*a^4 - 210*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c
))*a^3*b - 672*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*a^2*b^2 + 35*(4*sin(2*d*x + 2*c)^3 - 6
0*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*a*b^3 + 48*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35
*sin(d*x + c)^3 - 35*sin(d*x + c))*b^4)/d

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.80 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {b^{4} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {a b^{3} \sin \left (6 \, d x + 6 \, c\right )}{48 \, d} + \frac {1}{4} \, {\left (6 \, a^{3} b + 5 \, a b^{3}\right )} x + \frac {{\left (24 \, a^{2} b^{2} + 7 \, b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{16 \, d} + \frac {{\left (16 \, a^{4} + 120 \, a^{2} b^{2} + 21 \, b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (16 \, a^{3} b + 15 \, a b^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{16 \, d} + \frac {{\left (48 \, a^{4} + 240 \, a^{2} b^{2} + 35 \, b^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \]

[In]

integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^4,x, algorithm="giac")

[Out]

1/448*b^4*sin(7*d*x + 7*c)/d + 1/48*a*b^3*sin(6*d*x + 6*c)/d + 1/4*(6*a^3*b + 5*a*b^3)*x + 1/320*(24*a^2*b^2 +
7*b^4)*sin(5*d*x + 5*c)/d + 1/16*(2*a^3*b + 3*a*b^3)*sin(4*d*x + 4*c)/d + 1/192*(16*a^4 + 120*a^2*b^2 + 21*b^
4)*sin(3*d*x + 3*c)/d + 1/16*(16*a^3*b + 15*a*b^3)*sin(2*d*x + 2*c)/d + 1/64*(48*a^4 + 240*a^2*b^2 + 35*b^4)*s
in(d*x + c)/d

Mupad [B] (verification not implemented)

Time = 15.71 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.93 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx=\frac {\left (2\,a^4-5\,a^3\,b+12\,a^2\,b^2-\frac {11\,a\,b^3}{2}+2\,b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {28\,a^4}{3}-12\,a^3\,b+40\,a^2\,b^2-\frac {14\,a\,b^3}{3}+4\,b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {58\,a^4}{3}-9\,a^3\,b+\frac {452\,a^2\,b^2}{5}-\frac {85\,a\,b^3}{6}+\frac {86\,b^4}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (24\,a^4+\frac {624\,a^2\,b^2}{5}+\frac {424\,b^4}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {58\,a^4}{3}+9\,a^3\,b+\frac {452\,a^2\,b^2}{5}+\frac {85\,a\,b^3}{6}+\frac {86\,b^4}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {28\,a^4}{3}+12\,a^3\,b+40\,a^2\,b^2+\frac {14\,a\,b^3}{3}+4\,b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a^4+5\,a^3\,b+12\,a^2\,b^2+\frac {11\,a\,b^3}{2}+2\,b^4\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a\,b\,\mathrm {atan}\left (\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^2+5\,b^2\right )}{2\,\left (3\,a^3\,b+\frac {5\,a\,b^3}{2}\right )}\right )\,\left (6\,a^2+5\,b^2\right )}{2\,d}-\frac {a\,b\,\left (6\,a^2+5\,b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{2\,d} \]

[In]

int(cos(c + d*x)^3*(a + b*cos(c + d*x))^4,x)

[Out]

(tan(c/2 + (d*x)/2)^7*(24*a^4 + (424*b^4)/35 + (624*a^2*b^2)/5) + tan(c/2 + (d*x)/2)^13*(2*a^4 - 5*a^3*b - (11
*a*b^3)/2 + 2*b^4 + 12*a^2*b^2) + tan(c/2 + (d*x)/2)^3*((14*a*b^3)/3 + 12*a^3*b + (28*a^4)/3 + 4*b^4 + 40*a^2*
b^2) + tan(c/2 + (d*x)/2)^11*((28*a^4)/3 - 12*a^3*b - (14*a*b^3)/3 + 4*b^4 + 40*a^2*b^2) + tan(c/2 + (d*x)/2)^
5*((85*a*b^3)/6 + 9*a^3*b + (58*a^4)/3 + (86*b^4)/5 + (452*a^2*b^2)/5) + tan(c/2 + (d*x)/2)^9*((58*a^4)/3 - 9*
a^3*b - (85*a*b^3)/6 + (86*b^4)/5 + (452*a^2*b^2)/5) + tan(c/2 + (d*x)/2)*((11*a*b^3)/2 + 5*a^3*b + 2*a^4 + 2*
b^4 + 12*a^2*b^2))/(d*(7*tan(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)) + (a*b*atan((
a*b*tan(c/2 + (d*x)/2)*(6*a^2 + 5*b^2))/(2*((5*a*b^3)/2 + 3*a^3*b)))*(6*a^2 + 5*b^2))/(2*d) - (a*b*(6*a^2 + 5*
b^2)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(2*d)