\(\int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx\) [92]

Optimal result

Integrand size = 28, antiderivative size = 515 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {63 a^5 x}{256}+\frac {35}{128} a^3 b^2 x+\frac {15}{256} a b^4 x-\frac {5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac {a^4 b \cos ^{10}(c+d x)}{2 d}+\frac {a^2 b^3 \cos ^{10}(c+d x)}{d}+\frac {63 a^5 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {35 a^3 b^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {15 a b^4 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {21 a^5 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {35 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {21 a^5 \cos ^5(c+d x) \sin (c+d x)}{160 d}+\frac {7 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a b^4 \cos ^5(c+d x) \sin (c+d x)}{32 d}+\frac {9 a^5 \cos ^7(c+d x) \sin (c+d x)}{80 d}+\frac {a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a b^4 \cos ^7(c+d x) \sin (c+d x)}{16 d}+\frac {a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac {a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac {a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac {b^5 \sin ^6(c+d x)}{6 d}-\frac {b^5 \sin ^8(c+d x)}{4 d}+\frac {b^5 \sin ^{10}(c+d x)}{10 d} \]

[Out]

63/256*a^5*x+35/128*a^3*b^2*x+15/256*a*b^4*x-5/4*a^2*b^3*cos(d*x+c)^8/d-1/2*a^4*b*cos(d*x+c)^10/d+a^2*b^3*cos(
d*x+c)^10/d+63/256*a^5*cos(d*x+c)*sin(d*x+c)/d+35/128*a^3*b^2*cos(d*x+c)*sin(d*x+c)/d+15/256*a*b^4*cos(d*x+c)*
sin(d*x+c)/d+21/128*a^5*cos(d*x+c)^3*sin(d*x+c)/d+35/192*a^3*b^2*cos(d*x+c)^3*sin(d*x+c)/d+5/128*a*b^4*cos(d*x
+c)^3*sin(d*x+c)/d+21/160*a^5*cos(d*x+c)^5*sin(d*x+c)/d+7/48*a^3*b^2*cos(d*x+c)^5*sin(d*x+c)/d+1/32*a*b^4*cos(
d*x+c)^5*sin(d*x+c)/d+9/80*a^5*cos(d*x+c)^7*sin(d*x+c)/d+1/8*a^3*b^2*cos(d*x+c)^7*sin(d*x+c)/d-3/16*a*b^4*cos(
d*x+c)^7*sin(d*x+c)/d+1/10*a^5*cos(d*x+c)^9*sin(d*x+c)/d-a^3*b^2*cos(d*x+c)^9*sin(d*x+c)/d-1/2*a*b^4*cos(d*x+c
)^7*sin(d*x+c)^3/d+1/6*b^5*sin(d*x+c)^6/d-1/4*b^5*sin(d*x+c)^8/d+1/10*b^5*sin(d*x+c)^10/d

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3169, 2715, 8, 2645, 30, 2648, 14, 2644, 272, 45} \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {a^5 \sin (c+d x) \cos ^9(c+d x)}{10 d}+\frac {9 a^5 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac {21 a^5 \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac {21 a^5 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {63 a^5 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {63 a^5 x}{256}-\frac {a^4 b \cos ^{10}(c+d x)}{2 d}-\frac {a^3 b^2 \sin (c+d x) \cos ^9(c+d x)}{d}+\frac {a^3 b^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {7 a^3 b^2 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {35 a^3 b^2 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {35 a^3 b^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {35}{128} a^3 b^2 x+\frac {a^2 b^3 \cos ^{10}(c+d x)}{d}-\frac {5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac {a b^4 \sin ^3(c+d x) \cos ^7(c+d x)}{2 d}-\frac {3 a b^4 \sin (c+d x) \cos ^7(c+d x)}{16 d}+\frac {a b^4 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac {5 a b^4 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {15 a b^4 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {15}{256} a b^4 x+\frac {b^5 \sin ^{10}(c+d x)}{10 d}-\frac {b^5 \sin ^8(c+d x)}{4 d}+\frac {b^5 \sin ^6(c+d x)}{6 d} \]

[In]

Int[Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

(63*a^5*x)/256 + (35*a^3*b^2*x)/128 + (15*a*b^4*x)/256 - (5*a^2*b^3*Cos[c + d*x]^8)/(4*d) - (a^4*b*Cos[c + d*x
]^10)/(2*d) + (a^2*b^3*Cos[c + d*x]^10)/d + (63*a^5*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (35*a^3*b^2*Cos[c + d
*x]*Sin[c + d*x])/(128*d) + (15*a*b^4*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (21*a^5*Cos[c + d*x]^3*Sin[c + d*x]
)/(128*d) + (35*a^3*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (5*a*b^4*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) +
(21*a^5*Cos[c + d*x]^5*Sin[c + d*x])/(160*d) + (7*a^3*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a*b^4*Cos[c
+ d*x]^5*Sin[c + d*x])/(32*d) + (9*a^5*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) + (a^3*b^2*Cos[c + d*x]^7*Sin[c + d
*x])/(8*d) - (3*a*b^4*Cos[c + d*x]^7*Sin[c + d*x])/(16*d) + (a^5*Cos[c + d*x]^9*Sin[c + d*x])/(10*d) - (a^3*b^
2*Cos[c + d*x]^9*Sin[c + d*x])/d - (a*b^4*Cos[c + d*x]^7*Sin[c + d*x]^3)/(2*d) + (b^5*Sin[c + d*x]^6)/(6*d) -
(b^5*Sin[c + d*x]^8)/(4*d) + (b^5*Sin[c + d*x]^10)/(10*d)

Rule 8

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 2645

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 2648

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-a)*(b*Cos[e
+ f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Dist[a^2*((m - 1)/(m + n)), Int[(b*Cos[e + f*x
])^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[
2*m, 2*n]

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 3169

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_.), x_Sym
bol] :> Int[ExpandTrig[cos[c + d*x]^m*(a*cos[c + d*x] + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] &&
IntegerQ[m] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^5 \cos ^{10}(c+d x)+5 a^4 b \cos ^9(c+d x) \sin (c+d x)+10 a^3 b^2 \cos ^8(c+d x) \sin ^2(c+d x)+10 a^2 b^3 \cos ^7(c+d x) \sin ^3(c+d x)+5 a b^4 \cos ^6(c+d x) \sin ^4(c+d x)+b^5 \cos ^5(c+d x) \sin ^5(c+d x)\right ) \, dx \\ & = a^5 \int \cos ^{10}(c+d x) \, dx+\left (5 a^4 b\right ) \int \cos ^9(c+d x) \sin (c+d x) \, dx+\left (10 a^3 b^2\right ) \int \cos ^8(c+d x) \sin ^2(c+d x) \, dx+\left (10 a^2 b^3\right ) \int \cos ^7(c+d x) \sin ^3(c+d x) \, dx+\left (5 a b^4\right ) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx+b^5 \int \cos ^5(c+d x) \sin ^5(c+d x) \, dx \\ & = \frac {a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac {a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac {a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac {1}{10} \left (9 a^5\right ) \int \cos ^8(c+d x) \, dx+\left (a^3 b^2\right ) \int \cos ^8(c+d x) \, dx+\frac {1}{2} \left (3 a b^4\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac {\left (5 a^4 b\right ) \text {Subst}\left (\int x^9 \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (10 a^2 b^3\right ) \text {Subst}\left (\int x^7 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {b^5 \text {Subst}\left (\int x^5 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {a^4 b \cos ^{10}(c+d x)}{2 d}+\frac {9 a^5 \cos ^7(c+d x) \sin (c+d x)}{80 d}+\frac {a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a b^4 \cos ^7(c+d x) \sin (c+d x)}{16 d}+\frac {a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac {a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac {a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac {1}{80} \left (63 a^5\right ) \int \cos ^6(c+d x) \, dx+\frac {1}{8} \left (7 a^3 b^2\right ) \int \cos ^6(c+d x) \, dx+\frac {1}{16} \left (3 a b^4\right ) \int \cos ^6(c+d x) \, dx-\frac {\left (10 a^2 b^3\right ) \text {Subst}\left (\int \left (x^7-x^9\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {b^5 \text {Subst}\left (\int (1-x)^2 x^2 \, dx,x,\sin ^2(c+d x)\right )}{2 d} \\ & = -\frac {5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac {a^4 b \cos ^{10}(c+d x)}{2 d}+\frac {a^2 b^3 \cos ^{10}(c+d x)}{d}+\frac {21 a^5 \cos ^5(c+d x) \sin (c+d x)}{160 d}+\frac {7 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a b^4 \cos ^5(c+d x) \sin (c+d x)}{32 d}+\frac {9 a^5 \cos ^7(c+d x) \sin (c+d x)}{80 d}+\frac {a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a b^4 \cos ^7(c+d x) \sin (c+d x)}{16 d}+\frac {a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac {a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac {a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac {1}{32} \left (21 a^5\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{48} \left (35 a^3 b^2\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{32} \left (5 a b^4\right ) \int \cos ^4(c+d x) \, dx+\frac {b^5 \text {Subst}\left (\int \left (x^2-2 x^3+x^4\right ) \, dx,x,\sin ^2(c+d x)\right )}{2 d} \\ & = -\frac {5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac {a^4 b \cos ^{10}(c+d x)}{2 d}+\frac {a^2 b^3 \cos ^{10}(c+d x)}{d}+\frac {21 a^5 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {35 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {21 a^5 \cos ^5(c+d x) \sin (c+d x)}{160 d}+\frac {7 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a b^4 \cos ^5(c+d x) \sin (c+d x)}{32 d}+\frac {9 a^5 \cos ^7(c+d x) \sin (c+d x)}{80 d}+\frac {a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a b^4 \cos ^7(c+d x) \sin (c+d x)}{16 d}+\frac {a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac {a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac {a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac {b^5 \sin ^6(c+d x)}{6 d}-\frac {b^5 \sin ^8(c+d x)}{4 d}+\frac {b^5 \sin ^{10}(c+d x)}{10 d}+\frac {1}{128} \left (63 a^5\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{64} \left (35 a^3 b^2\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{128} \left (15 a b^4\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac {a^4 b \cos ^{10}(c+d x)}{2 d}+\frac {a^2 b^3 \cos ^{10}(c+d x)}{d}+\frac {63 a^5 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {35 a^3 b^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {15 a b^4 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {21 a^5 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {35 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {21 a^5 \cos ^5(c+d x) \sin (c+d x)}{160 d}+\frac {7 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a b^4 \cos ^5(c+d x) \sin (c+d x)}{32 d}+\frac {9 a^5 \cos ^7(c+d x) \sin (c+d x)}{80 d}+\frac {a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a b^4 \cos ^7(c+d x) \sin (c+d x)}{16 d}+\frac {a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac {a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac {a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac {b^5 \sin ^6(c+d x)}{6 d}-\frac {b^5 \sin ^8(c+d x)}{4 d}+\frac {b^5 \sin ^{10}(c+d x)}{10 d}+\frac {1}{256} \left (63 a^5\right ) \int 1 \, dx+\frac {1}{128} \left (35 a^3 b^2\right ) \int 1 \, dx+\frac {1}{256} \left (15 a b^4\right ) \int 1 \, dx \\ & = \frac {63 a^5 x}{256}+\frac {35}{128} a^3 b^2 x+\frac {15}{256} a b^4 x-\frac {5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac {a^4 b \cos ^{10}(c+d x)}{2 d}+\frac {a^2 b^3 \cos ^{10}(c+d x)}{d}+\frac {63 a^5 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {35 a^3 b^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {15 a b^4 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {21 a^5 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {35 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {21 a^5 \cos ^5(c+d x) \sin (c+d x)}{160 d}+\frac {7 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a b^4 \cos ^5(c+d x) \sin (c+d x)}{32 d}+\frac {9 a^5 \cos ^7(c+d x) \sin (c+d x)}{80 d}+\frac {a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a b^4 \cos ^7(c+d x) \sin (c+d x)}{16 d}+\frac {a^5 \cos ^9(c+d x) \sin (c+d x)}{10 d}-\frac {a^3 b^2 \cos ^9(c+d x) \sin (c+d x)}{d}-\frac {a b^4 \cos ^7(c+d x) \sin ^3(c+d x)}{2 d}+\frac {b^5 \sin ^6(c+d x)}{6 d}-\frac {b^5 \sin ^8(c+d x)}{4 d}+\frac {b^5 \sin ^{10}(c+d x)}{10 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.34 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.60 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {120 a \left (63 a^4+70 a^2 b^2+15 b^4\right ) (c+d x)-300 b \left (21 a^4+14 a^2 b^2+b^4\right ) \cos (2 (c+d x))-1200 a^2 b \left (3 a^2+b^2\right ) \cos (4 (c+d x))+50 b \left (-27 a^4+6 a^2 b^2+b^4\right ) \cos (6 (c+d x))-300 a^2 b \left (a^2-b^2\right ) \cos (8 (c+d x))-6 b \left (5 a^4-10 a^2 b^2+b^4\right ) \cos (10 (c+d x))+300 a \left (21 a^4+14 a^2 b^2+b^4\right ) \sin (2 (c+d x))+600 a \left (3 a^4-2 a^2 b^2-b^4\right ) \sin (4 (c+d x))+50 a \left (9 a^4-26 a^2 b^2-3 b^4\right ) \sin (6 (c+d x))+75 a \left (a^4-6 a^2 b^2+b^4\right ) \sin (8 (c+d x))+6 a \left (a^4-10 a^2 b^2+5 b^4\right ) \sin (10 (c+d x))}{30720 d} \]

[In]

Integrate[Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

(120*a*(63*a^4 + 70*a^2*b^2 + 15*b^4)*(c + d*x) - 300*b*(21*a^4 + 14*a^2*b^2 + b^4)*Cos[2*(c + d*x)] - 1200*a^
2*b*(3*a^2 + b^2)*Cos[4*(c + d*x)] + 50*b*(-27*a^4 + 6*a^2*b^2 + b^4)*Cos[6*(c + d*x)] - 300*a^2*b*(a^2 - b^2)
*Cos[8*(c + d*x)] - 6*b*(5*a^4 - 10*a^2*b^2 + b^4)*Cos[10*(c + d*x)] + 300*a*(21*a^4 + 14*a^2*b^2 + b^4)*Sin[2
*(c + d*x)] + 600*a*(3*a^4 - 2*a^2*b^2 - b^4)*Sin[4*(c + d*x)] + 50*a*(9*a^4 - 26*a^2*b^2 - 3*b^4)*Sin[6*(c +
d*x)] + 75*a*(a^4 - 6*a^2*b^2 + b^4)*Sin[8*(c + d*x)] + 6*a*(a^4 - 10*a^2*b^2 + 5*b^4)*Sin[10*(c + d*x)])/(307
20*d)

Maple [A] (verified)

Time = 2.05 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.63

method	result	size
parts	\(\frac {a^{5} \left (\frac {\left (\cos \left (d x +c \right )^{9}+\frac {9 \cos \left (d x +c \right )^{7}}{8}+\frac {21 \cos \left (d x +c \right )^{5}}{16}+\frac {105 \cos \left (d x +c \right )^{3}}{64}+\frac {315 \cos \left (d x +c \right )}{128}\right ) \sin \left (d x +c \right )}{10}+\frac {63 d x}{256}+\frac {63 c}{256}\right )}{d}+\frac {b^{5} \left (\frac {\sin \left (d x +c \right )^{10}}{10}-\frac {\sin \left (d x +c \right )^{8}}{4}+\frac {\sin \left (d x +c \right )^{6}}{6}\right )}{d}+\frac {5 a \,b^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{7}}{10}-\frac {3 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{7}}{80}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )}{d}-\frac {a^{4} b \cos \left (d x +c \right )^{10}}{2 d}+\frac {10 a^{3} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{9}}{10}+\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{80}+\frac {7 d x}{256}+\frac {7 c}{256}\right )}{d}+\frac {10 a^{2} b^{3} \left (\frac {\cos \left (d x +c \right )^{10}}{10}-\frac {\cos \left (d x +c \right )^{8}}{8}\right )}{d}\)	\(325\)
derivativedivides	\(\frac {a^{5} \left (\frac {\left (\cos \left (d x +c \right )^{9}+\frac {9 \cos \left (d x +c \right )^{7}}{8}+\frac {21 \cos \left (d x +c \right )^{5}}{16}+\frac {105 \cos \left (d x +c \right )^{3}}{64}+\frac {315 \cos \left (d x +c \right )}{128}\right ) \sin \left (d x +c \right )}{10}+\frac {63 d x}{256}+\frac {63 c}{256}\right )-\frac {a^{4} b \cos \left (d x +c \right )^{10}}{2}+10 a^{3} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{9}}{10}+\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{80}+\frac {7 d x}{256}+\frac {7 c}{256}\right )+10 a^{2} b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{8}}{10}-\frac {\cos \left (d x +c \right )^{8}}{40}\right )+5 a \,b^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{7}}{10}-\frac {3 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{7}}{80}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+b^{5} \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{6}}{10}-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{6}}{20}-\frac {\cos \left (d x +c \right )^{6}}{60}\right )}{d}\)	\(335\)
default	\(\frac {a^{5} \left (\frac {\left (\cos \left (d x +c \right )^{9}+\frac {9 \cos \left (d x +c \right )^{7}}{8}+\frac {21 \cos \left (d x +c \right )^{5}}{16}+\frac {105 \cos \left (d x +c \right )^{3}}{64}+\frac {315 \cos \left (d x +c \right )}{128}\right ) \sin \left (d x +c \right )}{10}+\frac {63 d x}{256}+\frac {63 c}{256}\right )-\frac {a^{4} b \cos \left (d x +c \right )^{10}}{2}+10 a^{3} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{9}}{10}+\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{80}+\frac {7 d x}{256}+\frac {7 c}{256}\right )+10 a^{2} b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{8}}{10}-\frac {\cos \left (d x +c \right )^{8}}{40}\right )+5 a \,b^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{7}}{10}-\frac {3 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{7}}{80}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+b^{5} \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{6}}{10}-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{6}}{20}-\frac {\cos \left (d x +c \right )^{6}}{60}\right )}{d}\)	\(335\)
parallelrisch	\(\frac {6 \left (-5 a^{4} b +10 a^{2} b^{3}-b^{5}\right ) \cos \left (10 d x +10 c \right )+6 \left (a^{5}-10 a^{3} b^{2}+5 a \,b^{4}\right ) \sin \left (10 d x +10 c \right )+300 \left (-21 a^{4} b -14 a^{2} b^{3}-b^{5}\right ) \cos \left (2 d x +2 c \right )+50 \left (-27 a^{4} b +6 a^{2} b^{3}+b^{5}\right ) \cos \left (6 d x +6 c \right )+300 \left (21 a^{5}+14 a^{3} b^{2}+a \,b^{4}\right ) \sin \left (2 d x +2 c \right )+600 \left (3 a^{5}-2 a^{3} b^{2}-a \,b^{4}\right ) \sin \left (4 d x +4 c \right )+50 \left (9 a^{5}-26 a^{3} b^{2}-3 a \,b^{4}\right ) \sin \left (6 d x +6 c \right )+75 \left (a^{5}-6 a^{3} b^{2}+a \,b^{4}\right ) \sin \left (8 d x +8 c \right )+1200 \left (-3 a^{4} b -a^{2} b^{3}\right ) \cos \left (4 d x +4 c \right )+300 \left (-a^{4} b +a^{2} b^{3}\right ) \cos \left (8 d x +8 c \right )+7560 a^{5} d x +8400 a^{3} b^{2} d x +1800 a \,b^{4} d x +11580 a^{4} b +4740 a^{2} b^{3}+256 b^{5}}{30720 d}\)	\(342\)
risch	\(\frac {35 a^{3} b^{2} x}{128}+\frac {15 a \,b^{4} x}{256}+\frac {5 b^{5} \cos \left (6 d x +6 c \right )}{3072 d}-\frac {5 b^{5} \cos \left (2 d x +2 c \right )}{512 d}+\frac {105 a^{5} \sin \left (2 d x +2 c \right )}{512 d}-\frac {5 a^{3} \sin \left (4 d x +4 c \right ) b^{2}}{128 d}-\frac {5 a \sin \left (4 d x +4 c \right ) b^{4}}{256 d}-\frac {105 b \cos \left (2 d x +2 c \right ) a^{4}}{512 d}-\frac {35 b^{3} \cos \left (2 d x +2 c \right ) a^{2}}{256 d}+\frac {35 a^{3} \sin \left (2 d x +2 c \right ) b^{2}}{256 d}+\frac {5 a \sin \left (2 d x +2 c \right ) b^{4}}{512 d}-\frac {b \cos \left (10 d x +10 c \right ) a^{4}}{1024 d}+\frac {b^{3} \cos \left (10 d x +10 c \right ) a^{2}}{512 d}-\frac {a^{3} \sin \left (10 d x +10 c \right ) b^{2}}{512 d}+\frac {a \sin \left (10 d x +10 c \right ) b^{4}}{1024 d}-\frac {5 a^{4} b \cos \left (8 d x +8 c \right )}{512 d}+\frac {5 a^{2} b^{3} \cos \left (8 d x +8 c \right )}{512 d}-\frac {15 a^{3} \sin \left (8 d x +8 c \right ) b^{2}}{1024 d}+\frac {5 a \sin \left (8 d x +8 c \right ) b^{4}}{2048 d}-\frac {45 b \cos \left (6 d x +6 c \right ) a^{4}}{1024 d}+\frac {5 b^{3} \cos \left (6 d x +6 c \right ) a^{2}}{512 d}-\frac {65 a^{3} \sin \left (6 d x +6 c \right ) b^{2}}{1536 d}-\frac {5 a \sin \left (6 d x +6 c \right ) b^{4}}{1024 d}-\frac {15 a^{4} b \cos \left (4 d x +4 c \right )}{128 d}-\frac {5 a^{2} b^{3} \cos \left (4 d x +4 c \right )}{128 d}-\frac {b^{5} \cos \left (10 d x +10 c \right )}{5120 d}+\frac {a^{5} \sin \left (10 d x +10 c \right )}{5120 d}+\frac {5 a^{5} \sin \left (8 d x +8 c \right )}{2048 d}+\frac {63 a^{5} x}{256}+\frac {15 a^{5} \sin \left (6 d x +6 c \right )}{1024 d}+\frac {15 a^{5} \sin \left (4 d x +4 c \right )}{256 d}\)	\(540\)

[In]

int(cos(d*x+c)^5*(cos(d*x+c)*a+b*sin(d*x+c))^5,x,method=_RETURNVERBOSE)

[Out]

a^5/d*(1/10*(cos(d*x+c)^9+9/8*cos(d*x+c)^7+21/16*cos(d*x+c)^5+105/64*cos(d*x+c)^3+315/128*cos(d*x+c))*sin(d*x+
c)+63/256*d*x+63/256*c)+b^5/d*(1/10*sin(d*x+c)^10-1/4*sin(d*x+c)^8+1/6*sin(d*x+c)^6)+5*a*b^4/d*(-1/10*sin(d*x+
c)^3*cos(d*x+c)^7-3/80*sin(d*x+c)*cos(d*x+c)^7+1/160*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c
)+3/256*d*x+3/256*c)-1/2*a^4*b*cos(d*x+c)^10/d+10*a^3*b^2/d*(-1/10*sin(d*x+c)*cos(d*x+c)^9+1/80*(cos(d*x+c)^7+
7/6*cos(d*x+c)^5+35/24*cos(d*x+c)^3+35/16*cos(d*x+c))*sin(d*x+c)+7/256*d*x+7/256*c)+10*a^2*b^3/d*(1/10*cos(d*x
+c)^10-1/8*cos(d*x+c)^8)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.49 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {640 \, b^{5} \cos \left (d x + c\right )^{6} + 384 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{10} + 960 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{8} - 15 \, {\left (63 \, a^{5} + 70 \, a^{3} b^{2} + 15 \, a b^{4}\right )} d x - {\left (384 \, {\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{9} + 48 \, {\left (9 \, a^{5} + 10 \, a^{3} b^{2} - 55 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} + 8 \, {\left (63 \, a^{5} + 70 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 10 \, {\left (63 \, a^{5} + 70 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (63 \, a^{5} + 70 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3840 \, d} \]

[In]

integrate(cos(d*x+c)^5*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="fricas")

[Out]

-1/3840*(640*b^5*cos(d*x + c)^6 + 384*(5*a^4*b - 10*a^2*b^3 + b^5)*cos(d*x + c)^10 + 960*(5*a^2*b^3 - b^5)*cos
(d*x + c)^8 - 15*(63*a^5 + 70*a^3*b^2 + 15*a*b^4)*d*x - (384*(a^5 - 10*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^9 + 48*
(9*a^5 + 10*a^3*b^2 - 55*a*b^4)*cos(d*x + c)^7 + 8*(63*a^5 + 70*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^5 + 10*(63*a^
5 + 70*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^3 + 15*(63*a^5 + 70*a^3*b^2 + 15*a*b^4)*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. 1037 vs. \(2 (498) = 996\).

Time = 1.51 (sec) , antiderivative size = 1037, normalized size of antiderivative = 2.01 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)**5*(a*cos(d*x+c)+b*sin(d*x+c))**5,x)

[Out]

Piecewise((63*a**5*x*sin(c + d*x)**10/256 + 315*a**5*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 315*a**5*x*sin(c
+ d*x)**6*cos(c + d*x)**4/128 + 315*a**5*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 315*a**5*x*sin(c + d*x)**2*co
s(c + d*x)**8/256 + 63*a**5*x*cos(c + d*x)**10/256 + 63*a**5*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 147*a**5*s
in(c + d*x)**7*cos(c + d*x)**3/(128*d) + 21*a**5*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) + 237*a**5*sin(c + d*x
)**3*cos(c + d*x)**7/(128*d) + 193*a**5*sin(c + d*x)*cos(c + d*x)**9/(256*d) - a**4*b*cos(c + d*x)**10/(2*d) +
35*a**3*b**2*x*sin(c + d*x)**10/128 + 175*a**3*b**2*x*sin(c + d*x)**8*cos(c + d*x)**2/128 + 175*a**3*b**2*x*s
in(c + d*x)**6*cos(c + d*x)**4/64 + 175*a**3*b**2*x*sin(c + d*x)**4*cos(c + d*x)**6/64 + 175*a**3*b**2*x*sin(c
+ d*x)**2*cos(c + d*x)**8/128 + 35*a**3*b**2*x*cos(c + d*x)**10/128 + 35*a**3*b**2*sin(c + d*x)**9*cos(c + d*
x)/(128*d) + 245*a**3*b**2*sin(c + d*x)**7*cos(c + d*x)**3/(192*d) + 7*a**3*b**2*sin(c + d*x)**5*cos(c + d*x)*
*5/(3*d) + 395*a**3*b**2*sin(c + d*x)**3*cos(c + d*x)**7/(192*d) - 35*a**3*b**2*sin(c + d*x)*cos(c + d*x)**9/(
128*d) + a**2*b**3*sin(c + d*x)**10/(4*d) + 5*a**2*b**3*sin(c + d*x)**8*cos(c + d*x)**2/(4*d) + 5*a**2*b**3*si
n(c + d*x)**6*cos(c + d*x)**4/(2*d) + 5*a**2*b**3*sin(c + d*x)**4*cos(c + d*x)**6/(2*d) + 15*a*b**4*x*sin(c +
d*x)**10/256 + 75*a*b**4*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 75*a*b**4*x*sin(c + d*x)**6*cos(c + d*x)**4/1
28 + 75*a*b**4*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 75*a*b**4*x*sin(c + d*x)**2*cos(c + d*x)**8/256 + 15*a*
b**4*x*cos(c + d*x)**10/256 + 15*a*b**4*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 35*a*b**4*sin(c + d*x)**7*cos(c
+ d*x)**3/(128*d) + a*b**4*sin(c + d*x)**5*cos(c + d*x)**5/(2*d) - 35*a*b**4*sin(c + d*x)**3*cos(c + d*x)**7/
(128*d) - 15*a*b**4*sin(c + d*x)*cos(c + d*x)**9/(256*d) + b**5*sin(c + d*x)**10/(60*d) + b**5*sin(c + d*x)**8
*cos(c + d*x)**2/(12*d) + b**5*sin(c + d*x)**6*cos(c + d*x)**4/(6*d), Ne(d, 0)), (x*(a*cos(c) + b*sin(c))**5*c
os(c)**5, True))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.56 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {15360 \, a^{4} b \cos \left (d x + c\right )^{10} - 3 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 640 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 2520 \, d x + 2520 \, c + 25 \, \sin \left (8 \, d x + 8 \, c\right ) + 600 \, \sin \left (4 \, d x + 4 \, c\right ) + 2560 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{5} + 10 \, {\left (96 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 640 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c + 45 \, \sin \left (8 \, d x + 8 \, c\right ) + 120 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} b^{2} + 7680 \, {\left (4 \, \sin \left (d x + c\right )^{10} - 15 \, \sin \left (d x + c\right )^{8} + 20 \, \sin \left (d x + c\right )^{6} - 10 \, \sin \left (d x + c\right )^{4}\right )} a^{2} b^{3} - 15 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{4} - 512 \, {\left (6 \, \sin \left (d x + c\right )^{10} - 15 \, \sin \left (d x + c\right )^{8} + 10 \, \sin \left (d x + c\right )^{6}\right )} b^{5}}{30720 \, d} \]

[In]

integrate(cos(d*x+c)^5*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="maxima")

[Out]

-1/30720*(15360*a^4*b*cos(d*x + c)^10 - 3*(32*sin(2*d*x + 2*c)^5 - 640*sin(2*d*x + 2*c)^3 + 2520*d*x + 2520*c
+ 25*sin(8*d*x + 8*c) + 600*sin(4*d*x + 4*c) + 2560*sin(2*d*x + 2*c))*a^5 + 10*(96*sin(2*d*x + 2*c)^5 - 640*si
n(2*d*x + 2*c)^3 - 840*d*x - 840*c + 45*sin(8*d*x + 8*c) + 120*sin(4*d*x + 4*c))*a^3*b^2 + 7680*(4*sin(d*x + c
)^10 - 15*sin(d*x + c)^8 + 20*sin(d*x + c)^6 - 10*sin(d*x + c)^4)*a^2*b^3 - 15*(32*sin(2*d*x + 2*c)^5 + 120*d*
x + 120*c + 5*sin(8*d*x + 8*c) - 40*sin(4*d*x + 4*c))*a*b^4 - 512*(6*sin(d*x + c)^10 - 15*sin(d*x + c)^8 + 10*
sin(d*x + c)^6)*b^5)/d

Giac [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.66 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {1}{256} \, {\left (63 \, a^{5} + 70 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x - \frac {{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (10 \, d x + 10 \, c\right )}{5120 \, d} - \frac {5 \, {\left (a^{4} b - a^{2} b^{3}\right )} \cos \left (8 \, d x + 8 \, c\right )}{512 \, d} - \frac {5 \, {\left (27 \, a^{4} b - 6 \, a^{2} b^{3} - b^{5}\right )} \cos \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac {5 \, {\left (3 \, a^{4} b + a^{2} b^{3}\right )} \cos \left (4 \, d x + 4 \, c\right )}{128 \, d} - \frac {5 \, {\left (21 \, a^{4} b + 14 \, a^{2} b^{3} + b^{5}\right )} \cos \left (2 \, d x + 2 \, c\right )}{512 \, d} + \frac {{\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {5 \, {\left (a^{5} - 6 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac {5 \, {\left (9 \, a^{5} - 26 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} + \frac {5 \, {\left (3 \, a^{5} - 2 \, a^{3} b^{2} - a b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {5 \, {\left (21 \, a^{5} + 14 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]

[In]

integrate(cos(d*x+c)^5*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="giac")

[Out]

1/256*(63*a^5 + 70*a^3*b^2 + 15*a*b^4)*x - 1/5120*(5*a^4*b - 10*a^2*b^3 + b^5)*cos(10*d*x + 10*c)/d - 5/512*(a
^4*b - a^2*b^3)*cos(8*d*x + 8*c)/d - 5/3072*(27*a^4*b - 6*a^2*b^3 - b^5)*cos(6*d*x + 6*c)/d - 5/128*(3*a^4*b +
a^2*b^3)*cos(4*d*x + 4*c)/d - 5/512*(21*a^4*b + 14*a^2*b^3 + b^5)*cos(2*d*x + 2*c)/d + 1/5120*(a^5 - 10*a^3*b
^2 + 5*a*b^4)*sin(10*d*x + 10*c)/d + 5/2048*(a^5 - 6*a^3*b^2 + a*b^4)*sin(8*d*x + 8*c)/d + 5/3072*(9*a^5 - 26*
a^3*b^2 - 3*a*b^4)*sin(6*d*x + 6*c)/d + 5/256*(3*a^5 - 2*a^3*b^2 - a*b^4)*sin(4*d*x + 4*c)/d + 5/512*(21*a^5 +
14*a^3*b^2 + a*b^4)*sin(2*d*x + 2*c)/d

Mupad [B] (verification not implemented)

Time = 24.16 (sec) , antiderivative size = 801, normalized size of antiderivative = 1.56 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\text {Too large to display} \]

[In]

int(cos(c + d*x)^5*(a*cos(c + d*x) + b*sin(c + d*x))^5,x)

[Out]

(tan(c/2 + (d*x)/2)^19*((15*a*b^4)/128 - (193*a^5)/128 + (35*a^3*b^2)/64) - tan(c/2 + (d*x)/2)*((15*a*b^4)/128
- (193*a^5)/128 + (35*a^3*b^2)/64) + tan(c/2 + (d*x)/2)^3*((159*a^5)/128 - (145*a*b^4)/128 + (4105*a^3*b^2)/1
92) - tan(c/2 + (d*x)/2)^17*((159*a^5)/128 - (145*a*b^4)/128 + (4105*a^3*b^2)/192) - tan(c/2 + (d*x)/2)^7*((25
95*a*b^4)/32 + (147*a^5)/32 - (2905*a^3*b^2)/16) + tan(c/2 + (d*x)/2)^13*((2595*a*b^4)/32 + (147*a^5)/32 - (29
05*a^3*b^2)/16) + tan(c/2 + (d*x)/2)^5*((867*a*b^4)/32 + (2847*a^5)/160 - (2891*a^3*b^2)/48) - tan(c/2 + (d*x)
/2)^15*((867*a*b^4)/32 + (2847*a^5)/160 - (2891*a^3*b^2)/48) + tan(c/2 + (d*x)/2)^9*((9395*a*b^4)/64 + (1827*a
^5)/64 - (7945*a^3*b^2)/32) - tan(c/2 + (d*x)/2)^11*((9395*a*b^4)/64 + (1827*a^5)/64 - (7945*a^3*b^2)/32) + ta
n(c/2 + (d*x)/2)^6*(120*a^4*b + (32*b^5)/3 - 80*a^2*b^3) + tan(c/2 + (d*x)/2)^14*(120*a^4*b + (32*b^5)/3 - 80*
a^2*b^3) + tan(c/2 + (d*x)/2)^10*(252*a^4*b + (192*b^5)/5 - 224*a^2*b^3) - tan(c/2 + (d*x)/2)^8*((64*b^5)/3 -
280*a^2*b^3) - tan(c/2 + (d*x)/2)^12*((64*b^5)/3 - 280*a^2*b^3) + 40*a^2*b^3*tan(c/2 + (d*x)/2)^4 + 40*a^2*b^3
*tan(c/2 + (d*x)/2)^16 + 10*a^4*b*tan(c/2 + (d*x)/2)^2 + 10*a^4*b*tan(c/2 + (d*x)/2)^18)/(d*(10*tan(c/2 + (d*x
)/2)^2 + 45*tan(c/2 + (d*x)/2)^4 + 120*tan(c/2 + (d*x)/2)^6 + 210*tan(c/2 + (d*x)/2)^8 + 252*tan(c/2 + (d*x)/2
)^10 + 210*tan(c/2 + (d*x)/2)^12 + 120*tan(c/2 + (d*x)/2)^14 + 45*tan(c/2 + (d*x)/2)^16 + 10*tan(c/2 + (d*x)/2
)^18 + tan(c/2 + (d*x)/2)^20 + 1)) + (a*atan((a*tan(c/2 + (d*x)/2)*(63*a^4 + 15*b^4 + 70*a^2*b^2))/(128*((15*a
*b^4)/128 + (63*a^5)/128 + (35*a^3*b^2)/64)))*(63*a^4 + 15*b^4 + 70*a^2*b^2))/(128*d) - (a*(atan(tan(c/2 + (d*
x)/2)) - (d*x)/2)*(63*a^4 + 15*b^4 + 70*a^2*b^2))/(128*d)