\(\int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx\) [74]

Optimal result

Integrand size = 28, antiderivative size = 279 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {4 a b^3 \cos ^7(c+d x)}{7 d}-\frac {4 a^3 b \cos ^9(c+d x)}{9 d}+\frac {4 a b^3 \cos ^9(c+d x)}{9 d}+\frac {a^4 \sin (c+d x)}{d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {2 a^2 b^2 \sin ^3(c+d x)}{d}+\frac {6 a^4 \sin ^5(c+d x)}{5 d}-\frac {18 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac {b^4 \sin ^5(c+d x)}{5 d}-\frac {4 a^4 \sin ^7(c+d x)}{7 d}+\frac {18 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac {2 b^4 \sin ^7(c+d x)}{7 d}+\frac {a^4 \sin ^9(c+d x)}{9 d}-\frac {2 a^2 b^2 \sin ^9(c+d x)}{3 d}+\frac {b^4 \sin ^9(c+d x)}{9 d} \]

[Out]

-4/7*a*b^3*cos(d*x+c)^7/d-4/9*a^3*b*cos(d*x+c)^9/d+4/9*a*b^3*cos(d*x+c)^9/d+a^4*sin(d*x+c)/d-4/3*a^4*sin(d*x+c
)^3/d+2*a^2*b^2*sin(d*x+c)^3/d+6/5*a^4*sin(d*x+c)^5/d-18/5*a^2*b^2*sin(d*x+c)^5/d+1/5*b^4*sin(d*x+c)^5/d-4/7*a
^4*sin(d*x+c)^7/d+18/7*a^2*b^2*sin(d*x+c)^7/d-2/7*b^4*sin(d*x+c)^7/d+1/9*a^4*sin(d*x+c)^9/d-2/3*a^2*b^2*sin(d*
x+c)^9/d+1/9*b^4*sin(d*x+c)^9/d

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3169, 2713, 2645, 30, 2644, 276, 14} \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {a^4 \sin ^9(c+d x)}{9 d}-\frac {4 a^4 \sin ^7(c+d x)}{7 d}+\frac {6 a^4 \sin ^5(c+d x)}{5 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {a^4 \sin (c+d x)}{d}-\frac {4 a^3 b \cos ^9(c+d x)}{9 d}-\frac {2 a^2 b^2 \sin ^9(c+d x)}{3 d}+\frac {18 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac {18 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac {2 a^2 b^2 \sin ^3(c+d x)}{d}+\frac {4 a b^3 \cos ^9(c+d x)}{9 d}-\frac {4 a b^3 \cos ^7(c+d x)}{7 d}+\frac {b^4 \sin ^9(c+d x)}{9 d}-\frac {2 b^4 \sin ^7(c+d x)}{7 d}+\frac {b^4 \sin ^5(c+d x)}{5 d} \]

[In]

Int[Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]

[Out]

(-4*a*b^3*Cos[c + d*x]^7)/(7*d) - (4*a^3*b*Cos[c + d*x]^9)/(9*d) + (4*a*b^3*Cos[c + d*x]^9)/(9*d) + (a^4*Sin[c
+ d*x])/d - (4*a^4*Sin[c + d*x]^3)/(3*d) + (2*a^2*b^2*Sin[c + d*x]^3)/d + (6*a^4*Sin[c + d*x]^5)/(5*d) - (18*
a^2*b^2*Sin[c + d*x]^5)/(5*d) + (b^4*Sin[c + d*x]^5)/(5*d) - (4*a^4*Sin[c + d*x]^7)/(7*d) + (18*a^2*b^2*Sin[c
+ d*x]^7)/(7*d) - (2*b^4*Sin[c + d*x]^7)/(7*d) + (a^4*Sin[c + d*x]^9)/(9*d) - (2*a^2*b^2*Sin[c + d*x]^9)/(3*d)
+ (b^4*Sin[c + d*x]^9)/(9*d)

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 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

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 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 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^4 \cos ^9(c+d x)+4 a^3 b \cos ^8(c+d x) \sin (c+d x)+6 a^2 b^2 \cos ^7(c+d x) \sin ^2(c+d x)+4 a b^3 \cos ^6(c+d x) \sin ^3(c+d x)+b^4 \cos ^5(c+d x) \sin ^4(c+d x)\right ) \, dx \\ & = a^4 \int \cos ^9(c+d x) \, dx+\left (4 a^3 b\right ) \int \cos ^8(c+d x) \sin (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \cos ^7(c+d x) \sin ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \cos ^6(c+d x) \sin ^3(c+d x) \, dx+b^4 \int \cos ^5(c+d x) \sin ^4(c+d x) \, dx \\ & = -\frac {a^4 \text {Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac {\left (4 a^3 b\right ) \text {Subst}\left (\int x^8 \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (6 a^2 b^2\right ) \text {Subst}\left (\int x^2 \left (1-x^2\right )^3 \, dx,x,\sin (c+d x)\right )}{d}-\frac {\left (4 a b^3\right ) \text {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {b^4 \text {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {4 a^3 b \cos ^9(c+d x)}{9 d}+\frac {a^4 \sin (c+d x)}{d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {6 a^4 \sin ^5(c+d x)}{5 d}-\frac {4 a^4 \sin ^7(c+d x)}{7 d}+\frac {a^4 \sin ^9(c+d x)}{9 d}+\frac {\left (6 a^2 b^2\right ) \text {Subst}\left (\int \left (x^2-3 x^4+3 x^6-x^8\right ) \, dx,x,\sin (c+d x)\right )}{d}-\frac {\left (4 a b^3\right ) \text {Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {b^4 \text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {4 a b^3 \cos ^7(c+d x)}{7 d}-\frac {4 a^3 b \cos ^9(c+d x)}{9 d}+\frac {4 a b^3 \cos ^9(c+d x)}{9 d}+\frac {a^4 \sin (c+d x)}{d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {2 a^2 b^2 \sin ^3(c+d x)}{d}+\frac {6 a^4 \sin ^5(c+d x)}{5 d}-\frac {18 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac {b^4 \sin ^5(c+d x)}{5 d}-\frac {4 a^4 \sin ^7(c+d x)}{7 d}+\frac {18 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac {2 b^4 \sin ^7(c+d x)}{7 d}+\frac {a^4 \sin ^9(c+d x)}{9 d}-\frac {2 a^2 b^2 \sin ^9(c+d x)}{3 d}+\frac {b^4 \sin ^9(c+d x)}{9 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.16 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.33 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {4 a^3 b \cos ^9(c+d x)}{9 d}+\frac {a^4 \sin (c+d x)}{d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {6 a^4 \sin ^5(c+d x)}{5 d}-\frac {4 a^4 \sin ^7(c+d x)}{7 d}+\frac {a^4 \sin ^9(c+d x)}{9 d}+\frac {2 a^2 b^2 \left (105 \sin ^3(c+d x)-189 \sin ^5(c+d x)+135 \sin ^7(c+d x)-35 \sin ^9(c+d x)\right )}{105 d}+\frac {b^4 \left (63 \sin ^5(c+d x)-90 \sin ^7(c+d x)+35 \sin ^9(c+d x)\right )}{315 d}+\frac {4 a b^3 \cos (c+d x) \sin ^8(c+d x) \left (2 \csc ^8(c+d x)+7 \sqrt {1-\sin ^2(c+d x)}-19 \csc ^2(c+d x) \sqrt {1-\sin ^2(c+d x)}+15 \csc ^4(c+d x) \sqrt {1-\sin ^2(c+d x)}-\csc ^6(c+d x) \sqrt {1-\sin ^2(c+d x)}-2 \csc ^8(c+d x) \sqrt {1-\sin ^2(c+d x)}\right )}{63 d \sqrt {\cos ^2(c+d x)}} \]

[In]

Integrate[Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]

[Out]

(-4*a^3*b*Cos[c + d*x]^9)/(9*d) + (a^4*Sin[c + d*x])/d - (4*a^4*Sin[c + d*x]^3)/(3*d) + (6*a^4*Sin[c + d*x]^5)
/(5*d) - (4*a^4*Sin[c + d*x]^7)/(7*d) + (a^4*Sin[c + d*x]^9)/(9*d) + (2*a^2*b^2*(105*Sin[c + d*x]^3 - 189*Sin[
c + d*x]^5 + 135*Sin[c + d*x]^7 - 35*Sin[c + d*x]^9))/(105*d) + (b^4*(63*Sin[c + d*x]^5 - 90*Sin[c + d*x]^7 +
35*Sin[c + d*x]^9))/(315*d) + (4*a*b^3*Cos[c + d*x]*Sin[c + d*x]^8*(2*Csc[c + d*x]^8 + 7*Sqrt[1 - Sin[c + d*x]
^2] - 19*Csc[c + d*x]^2*Sqrt[1 - Sin[c + d*x]^2] + 15*Csc[c + d*x]^4*Sqrt[1 - Sin[c + d*x]^2] - Csc[c + d*x]^6
*Sqrt[1 - Sin[c + d*x]^2] - 2*Csc[c + d*x]^8*Sqrt[1 - Sin[c + d*x]^2]))/(63*d*Sqrt[Cos[c + d*x]^2])

Maple [A] (verified)

Time = 1.73 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.69

method	result	size
parts	\(\frac {a^{4} \left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9 d}+\frac {b^{4} \left (\frac {\sin \left (d x +c \right )^{9}}{9}-\frac {2 \sin \left (d x +c \right )^{7}}{7}+\frac {\sin \left (d x +c \right )^{5}}{5}\right )}{d}+\frac {4 a \,b^{3} \left (\frac {\cos \left (d x +c \right )^{9}}{9}-\frac {\cos \left (d x +c \right )^{7}}{7}\right )}{d}-\frac {4 a^{3} b \cos \left (d x +c \right )^{9}}{9 d}-\frac {6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{9}}{9}-\frac {3 \sin \left (d x +c \right )^{7}}{7}+\frac {3 \sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}\right )}{d}\)	\(193\)
derivativedivides	\(\frac {\frac {a^{4} \left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9}-\frac {4 a^{3} b \cos \left (d x +c \right )^{9}}{9}+6 a^{2} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{8}}{9}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{63}\right )+4 a \,b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{7}}{9}-\frac {2 \cos \left (d x +c \right )^{7}}{63}\right )+b^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{6}}{9}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{6}}{21}+\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{105}\right )}{d}\)	\(236\)
default	\(\frac {\frac {a^{4} \left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9}-\frac {4 a^{3} b \cos \left (d x +c \right )^{9}}{9}+6 a^{2} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{8}}{9}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{63}\right )+4 a \,b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{7}}{9}-\frac {2 \cos \left (d x +c \right )^{7}}{63}\right )+b^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{6}}{9}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{6}}{21}+\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{105}\right )}{d}\)	\(236\)
risch	\(-\frac {7 a^{3} b \cos \left (d x +c \right )}{32 d}-\frac {3 a \,b^{3} \cos \left (d x +c \right )}{32 d}+\frac {63 a^{4} \sin \left (d x +c \right )}{128 d}+\frac {21 a^{2} b^{2} \sin \left (d x +c \right )}{64 d}+\frac {3 b^{4} \sin \left (d x +c \right )}{128 d}-\frac {a^{3} b \cos \left (9 d x +9 c \right )}{576 d}+\frac {a \,b^{3} \cos \left (9 d x +9 c \right )}{576 d}+\frac {\sin \left (9 d x +9 c \right ) a^{4}}{2304 d}-\frac {\sin \left (9 d x +9 c \right ) a^{2} b^{2}}{384 d}+\frac {\sin \left (9 d x +9 c \right ) b^{4}}{2304 d}-\frac {a^{3} b \cos \left (7 d x +7 c \right )}{64 d}+\frac {3 a \,b^{3} \cos \left (7 d x +7 c \right )}{448 d}+\frac {9 \sin \left (7 d x +7 c \right ) a^{4}}{1792 d}-\frac {15 \sin \left (7 d x +7 c \right ) a^{2} b^{2}}{896 d}+\frac {\sin \left (7 d x +7 c \right ) b^{4}}{1792 d}-\frac {a^{3} b \cos \left (5 d x +5 c \right )}{16 d}+\frac {9 \sin \left (5 d x +5 c \right ) a^{4}}{320 d}-\frac {3 \sin \left (5 d x +5 c \right ) a^{2} b^{2}}{80 d}-\frac {\sin \left (5 d x +5 c \right ) b^{4}}{320 d}-\frac {7 a^{3} b \cos \left (3 d x +3 c \right )}{48 d}-\frac {a \,b^{3} \cos \left (3 d x +3 c \right )}{24 d}+\frac {7 \sin \left (3 d x +3 c \right ) a^{4}}{64 d}-\frac {\sin \left (3 d x +3 c \right ) b^{4}}{192 d}\)	\(399\)
parallelrisch	\(\frac {630 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}-2520 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16} a^{3} b +\left (1680 a^{4}+5040 a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}-5040 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14} a \,b^{3}+\left (9576 a^{4}-6048 a^{2} b^{2}+2016 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}+\left (-23520 a^{3} b +8400 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (10224 a^{4}+34128 a^{2} b^{2}-3456 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}-25200 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} a \,b^{3}+\left (21316 a^{4}-17088 a^{2} b^{2}+6976 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (-35280 a^{3} b +15120 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (10224 a^{4}+34128 a^{2} b^{2}-3456 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-15120 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a \,b^{3}+\left (9576 a^{4}-6048 a^{2} b^{2}+2016 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-10080 a^{3} b +2160 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (1680 a^{4}+5040 a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-720 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a \,b^{3}+630 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-280 a^{3} b -80 a \,b^{3}}{315 d \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9}}\)	\(431\)
norman	\(\frac {-\frac {56 a^{3} b +16 a \,b^{3}}{63 d}+\frac {2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}}{d}-\frac {16 a \,b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{d}-\frac {48 a \,b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}-\frac {16 a \,b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{7 d}-\frac {80 a \,b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {8 a^{3} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{d}+\frac {8 \left (19 a^{4}-12 a^{2} b^{2}+4 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5 d}+\frac {8 \left (19 a^{4}-12 a^{2} b^{2}+4 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{5 d}+\frac {16 \left (71 a^{4}+237 a^{2} b^{2}-24 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d}+\frac {16 \left (71 a^{4}+237 a^{2} b^{2}-24 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{35 d}+\frac {4 \left (5329 a^{4}-4272 a^{2} b^{2}+1744 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{315 d}-\frac {2 \left (56 a^{3} b -24 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}-\frac {4 \left (56 a^{3} b -20 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{3 d}-\frac {4 \left (56 a^{3} b -12 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{7 d}+\frac {16 a^{2} \left (a^{2}+3 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {16 a^{2} \left (a^{2}+3 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9}}\)	\(490\)

[In]

int(cos(d*x+c)^5*(cos(d*x+c)*a+b*sin(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/9*a^4/d*(128/35+cos(d*x+c)^8+8/7*cos(d*x+c)^6+48/35*cos(d*x+c)^4+64/35*cos(d*x+c)^2)*sin(d*x+c)+b^4/d*(1/9*s
in(d*x+c)^9-2/7*sin(d*x+c)^7+1/5*sin(d*x+c)^5)+4*a*b^3/d*(1/9*cos(d*x+c)^9-1/7*cos(d*x+c)^7)-4/9*a^3*b*cos(d*x
+c)^9/d-6*a^2*b^2/d*(1/9*sin(d*x+c)^9-3/7*sin(d*x+c)^7+3/5*sin(d*x+c)^5-1/3*sin(d*x+c)^3)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.63 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {180 \, a b^{3} \cos \left (d x + c\right )^{7} + 140 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{9} - {\left (35 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{8} + 10 \, {\left (4 \, a^{4} + 3 \, a^{2} b^{2} - 5 \, b^{4}\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (16 \, a^{4} + 12 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + 128 \, a^{4} + 96 \, a^{2} b^{2} + 8 \, b^{4} + 4 \, {\left (16 \, a^{4} + 12 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{315 \, d} \]

[In]

integrate(cos(d*x+c)^5*(a*cos(d*x+c)+b*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/315*(180*a*b^3*cos(d*x + c)^7 + 140*(a^3*b - a*b^3)*cos(d*x + c)^9 - (35*(a^4 - 6*a^2*b^2 + b^4)*cos(d*x +
c)^8 + 10*(4*a^4 + 3*a^2*b^2 - 5*b^4)*cos(d*x + c)^6 + 3*(16*a^4 + 12*a^2*b^2 + b^4)*cos(d*x + c)^4 + 128*a^4
+ 96*a^2*b^2 + 8*b^4 + 4*(16*a^4 + 12*a^2*b^2 + b^4)*cos(d*x + c)^2)*sin(d*x + c))/d

Sympy [A] (verification not implemented)

Time = 1.07 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.32 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\begin {cases} \frac {128 a^{4} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {64 a^{4} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {16 a^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {8 a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac {a^{4} \sin {\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{d} - \frac {4 a^{3} b \cos ^{9}{\left (c + d x \right )}}{9 d} + \frac {32 a^{2} b^{2} \sin ^{9}{\left (c + d x \right )}}{105 d} + \frac {48 a^{2} b^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {12 a^{2} b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {2 a^{2} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {4 a b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {8 a b^{3} \cos ^{9}{\left (c + d x \right )}}{63 d} + \frac {8 b^{4} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {4 b^{4} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + b \sin {\left (c \right )}\right )^{4} \cos ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**5*(a*cos(d*x+c)+b*sin(d*x+c))**4,x)

[Out]

Piecewise((128*a**4*sin(c + d*x)**9/(315*d) + 64*a**4*sin(c + d*x)**7*cos(c + d*x)**2/(35*d) + 16*a**4*sin(c +
d*x)**5*cos(c + d*x)**4/(5*d) + 8*a**4*sin(c + d*x)**3*cos(c + d*x)**6/(3*d) + a**4*sin(c + d*x)*cos(c + d*x)
**8/d - 4*a**3*b*cos(c + d*x)**9/(9*d) + 32*a**2*b**2*sin(c + d*x)**9/(105*d) + 48*a**2*b**2*sin(c + d*x)**7*c
os(c + d*x)**2/(35*d) + 12*a**2*b**2*sin(c + d*x)**5*cos(c + d*x)**4/(5*d) + 2*a**2*b**2*sin(c + d*x)**3*cos(c
+ d*x)**6/d - 4*a*b**3*sin(c + d*x)**2*cos(c + d*x)**7/(7*d) - 8*a*b**3*cos(c + d*x)**9/(63*d) + 8*b**4*sin(c
+ d*x)**9/(315*d) + 4*b**4*sin(c + d*x)**7*cos(c + d*x)**2/(35*d) + b**4*sin(c + d*x)**5*cos(c + d*x)**4/(5*d
), Ne(d, 0)), (x*(a*cos(c) + b*sin(c))**4*cos(c)**5, True))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.67 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {140 \, a^{3} b \cos \left (d x + c\right )^{9} - {\left (35 \, \sin \left (d x + c\right )^{9} - 180 \, \sin \left (d x + c\right )^{7} + 378 \, \sin \left (d x + c\right )^{5} - 420 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )\right )} a^{4} + 6 \, {\left (35 \, \sin \left (d x + c\right )^{9} - 135 \, \sin \left (d x + c\right )^{7} + 189 \, \sin \left (d x + c\right )^{5} - 105 \, \sin \left (d x + c\right )^{3}\right )} a^{2} b^{2} - 20 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a b^{3} - {\left (35 \, \sin \left (d x + c\right )^{9} - 90 \, \sin \left (d x + c\right )^{7} + 63 \, \sin \left (d x + c\right )^{5}\right )} b^{4}}{315 \, d} \]

[In]

integrate(cos(d*x+c)^5*(a*cos(d*x+c)+b*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/315*(140*a^3*b*cos(d*x + c)^9 - (35*sin(d*x + c)^9 - 180*sin(d*x + c)^7 + 378*sin(d*x + c)^5 - 420*sin(d*x
+ c)^3 + 315*sin(d*x + c))*a^4 + 6*(35*sin(d*x + c)^9 - 135*sin(d*x + c)^7 + 189*sin(d*x + c)^5 - 105*sin(d*x
+ c)^3)*a^2*b^2 - 20*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a*b^3 - (35*sin(d*x + c)^9 - 90*sin(d*x + c)^7 + 63
*sin(d*x + c)^5)*b^4)/d

Giac [A] (verification not implemented)

none

Time = 0.53 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.96 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {a^{3} b \cos \left (5 \, d x + 5 \, c\right )}{16 \, d} - \frac {{\left (a^{3} b - a b^{3}\right )} \cos \left (9 \, d x + 9 \, c\right )}{576 \, d} - \frac {{\left (7 \, a^{3} b - 3 \, a b^{3}\right )} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {{\left (7 \, a^{3} b + 2 \, a b^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {{\left (7 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )}{32 \, d} + \frac {{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {{\left (9 \, a^{4} - 30 \, a^{2} b^{2} + b^{4}\right )} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {{\left (9 \, a^{4} - 12 \, a^{2} b^{2} - b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (21 \, a^{4} - b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {3 \, {\left (21 \, a^{4} + 14 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )}{128 \, d} \]

[In]

integrate(cos(d*x+c)^5*(a*cos(d*x+c)+b*sin(d*x+c))^4,x, algorithm="giac")

[Out]

-1/16*a^3*b*cos(5*d*x + 5*c)/d - 1/576*(a^3*b - a*b^3)*cos(9*d*x + 9*c)/d - 1/448*(7*a^3*b - 3*a*b^3)*cos(7*d*
x + 7*c)/d - 1/48*(7*a^3*b + 2*a*b^3)*cos(3*d*x + 3*c)/d - 1/32*(7*a^3*b + 3*a*b^3)*cos(d*x + c)/d + 1/2304*(a
^4 - 6*a^2*b^2 + b^4)*sin(9*d*x + 9*c)/d + 1/1792*(9*a^4 - 30*a^2*b^2 + b^4)*sin(7*d*x + 7*c)/d + 1/320*(9*a^4
- 12*a^2*b^2 - b^4)*sin(5*d*x + 5*c)/d + 1/192*(21*a^4 - b^4)*sin(3*d*x + 3*c)/d + 3/128*(21*a^4 + 14*a^2*b^2
+ b^4)*sin(d*x + c)/d

Mupad [B] (verification not implemented)

Time = 24.67 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.20 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {\frac {b^4\,\sin \left (3\,c+3\,d\,x\right )}{192}-\frac {3\,b^4\,\sin \left (c+d\,x\right )}{128}-\frac {7\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{64}-\frac {9\,a^4\,\sin \left (5\,c+5\,d\,x\right )}{320}-\frac {9\,a^4\,\sin \left (7\,c+7\,d\,x\right )}{1792}-\frac {a^4\,\sin \left (9\,c+9\,d\,x\right )}{2304}-\frac {63\,a^4\,\sin \left (c+d\,x\right )}{128}+\frac {b^4\,\sin \left (5\,c+5\,d\,x\right )}{320}-\frac {b^4\,\sin \left (7\,c+7\,d\,x\right )}{1792}-\frac {b^4\,\sin \left (9\,c+9\,d\,x\right )}{2304}+\frac {a\,b^3\,\cos \left (3\,c+3\,d\,x\right )}{24}+\frac {7\,a^3\,b\,\cos \left (3\,c+3\,d\,x\right )}{48}+\frac {a^3\,b\,\cos \left (5\,c+5\,d\,x\right )}{16}-\frac {3\,a\,b^3\,\cos \left (7\,c+7\,d\,x\right )}{448}+\frac {a^3\,b\,\cos \left (7\,c+7\,d\,x\right )}{64}-\frac {a\,b^3\,\cos \left (9\,c+9\,d\,x\right )}{576}+\frac {a^3\,b\,\cos \left (9\,c+9\,d\,x\right )}{576}-\frac {21\,a^2\,b^2\,\sin \left (c+d\,x\right )}{64}+\frac {3\,a^2\,b^2\,\sin \left (5\,c+5\,d\,x\right )}{80}+\frac {15\,a^2\,b^2\,\sin \left (7\,c+7\,d\,x\right )}{896}+\frac {a^2\,b^2\,\sin \left (9\,c+9\,d\,x\right )}{384}+\frac {3\,a\,b^3\,\cos \left (c+d\,x\right )}{32}+\frac {7\,a^3\,b\,\cos \left (c+d\,x\right )}{32}}{d} \]

[In]

int(cos(c + d*x)^5*(a*cos(c + d*x) + b*sin(c + d*x))^4,x)

[Out]

-((b^4*sin(3*c + 3*d*x))/192 - (3*b^4*sin(c + d*x))/128 - (7*a^4*sin(3*c + 3*d*x))/64 - (9*a^4*sin(5*c + 5*d*x
))/320 - (9*a^4*sin(7*c + 7*d*x))/1792 - (a^4*sin(9*c + 9*d*x))/2304 - (63*a^4*sin(c + d*x))/128 + (b^4*sin(5*
c + 5*d*x))/320 - (b^4*sin(7*c + 7*d*x))/1792 - (b^4*sin(9*c + 9*d*x))/2304 + (a*b^3*cos(3*c + 3*d*x))/24 + (7
*a^3*b*cos(3*c + 3*d*x))/48 + (a^3*b*cos(5*c + 5*d*x))/16 - (3*a*b^3*cos(7*c + 7*d*x))/448 + (a^3*b*cos(7*c +
7*d*x))/64 - (a*b^3*cos(9*c + 9*d*x))/576 + (a^3*b*cos(9*c + 9*d*x))/576 - (21*a^2*b^2*sin(c + d*x))/64 + (3*a
^2*b^2*sin(5*c + 5*d*x))/80 + (15*a^2*b^2*sin(7*c + 7*d*x))/896 + (a^2*b^2*sin(9*c + 9*d*x))/384 + (3*a*b^3*co
s(c + d*x))/32 + (7*a^3*b*cos(c + d*x))/32)/d