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

Optimal. Leaf size=279 \[ \frac {a^4 \sin ^9(c+d x)}{9 d}-\frac {4 a^4 \sin ^7(c+d x)}{7 d}+\frac {6 a^4 \sin ^5(c+d x)}{5 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {a^4 \sin (c+d x)}{d}-\frac {4 a^3 b \cos ^9(c+d x)}{9 d}-\frac {2 a^2 b^2 \sin ^9(c+d x)}{3 d}+\frac {18 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac {18 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac {2 a^2 b^2 \sin ^3(c+d x)}{d}+\frac {4 a b^3 \cos ^9(c+d x)}{9 d}-\frac {4 a b^3 \cos ^7(c+d x)}{7 d}+\frac {b^4 \sin ^9(c+d x)}{9 d}-\frac {2 b^4 \sin ^7(c+d x)}{7 d}+\frac {b^4 \sin ^5(c+d x)}{5 d} \]

[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

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Rubi [A]  time = 0.26, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3090, 2633, 2565, 30, 2564, 270, 14} \[ -\frac {2 a^2 b^2 \sin ^9(c+d x)}{3 d}+\frac {18 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac {18 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac {2 a^2 b^2 \sin ^3(c+d x)}{d}-\frac {4 a^3 b \cos ^9(c+d x)}{9 d}+\frac {a^4 \sin ^9(c+d x)}{9 d}-\frac {4 a^4 \sin ^7(c+d x)}{7 d}+\frac {6 a^4 \sin ^5(c+d x)}{5 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {a^4 \sin (c+d x)}{d}+\frac {4 a b^3 \cos ^9(c+d x)}{9 d}-\frac {4 a b^3 \cos ^7(c+d x)}{7 d}+\frac {b^4 \sin ^9(c+d x)}{9 d}-\frac {2 b^4 \sin ^7(c+d x)}{7 d}+\frac {b^4 \sin ^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[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 270

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 2564

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 2565

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 2633

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 3090

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*} \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx &=\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 \operatorname {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 ) \operatorname {Subst}\left (\int x^8 \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (6 a^2 b^2\right ) \operatorname {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 ) \operatorname {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {b^4 \operatorname {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 ) \operatorname {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 ) \operatorname {Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {b^4 \operatorname {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*}

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Mathematica [A]  time = 0.71, size = 237, normalized size = 0.85 \[ \frac {420 \left (21 a^4-b^4\right ) \sin (3 (c+d x))-5040 a^3 b \cos (5 (c+d x))-2520 a b \left (7 a^2+3 b^2\right ) \cos (c+d x)-1680 a b \left (7 a^2+2 b^2\right ) \cos (3 (c+d x))-180 a b \left (7 a^2-3 b^2\right ) \cos (7 (c+d x))-140 a b \left (a^2-b^2\right ) \cos (9 (c+d x))+1890 \left (21 a^4+14 a^2 b^2+b^4\right ) \sin (c+d x)+252 \left (9 a^4-12 a^2 b^2-b^4\right ) \sin (5 (c+d x))+45 \left (9 a^4-30 a^2 b^2+b^4\right ) \sin (7 (c+d x))+35 \left (a^4-6 a^2 b^2+b^4\right ) \sin (9 (c+d x))}{80640 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2520*a*b*(7*a^2 + 3*b^2)*Cos[c + d*x] - 1680*a*b*(7*a^2 + 2*b^2)*Cos[3*(c + d*x)] - 5040*a^3*b*Cos[5*(c + d*
x)] - 180*a*b*(7*a^2 - 3*b^2)*Cos[7*(c + d*x)] - 140*a*b*(a^2 - b^2)*Cos[9*(c + d*x)] + 1890*(21*a^4 + 14*a^2*
b^2 + b^4)*Sin[c + d*x] + 420*(21*a^4 - b^4)*Sin[3*(c + d*x)] + 252*(9*a^4 - 12*a^2*b^2 - b^4)*Sin[5*(c + d*x)
] + 45*(9*a^4 - 30*a^2*b^2 + b^4)*Sin[7*(c + d*x)] + 35*(a^4 - 6*a^2*b^2 + b^4)*Sin[9*(c + d*x)])/(80640*d)

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fricas [A]  time = 0.66, size = 177, normalized size = 0.63 \[ -\frac {180 \, a b^{3} \cos \left (d x + c\right )^{7} + 140 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{9} - {\left (35 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{8} + 10 \, {\left (4 \, a^{4} + 3 \, a^{2} b^{2} - 5 \, b^{4}\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (16 \, a^{4} + 12 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + 128 \, a^{4} + 96 \, a^{2} b^{2} + 8 \, b^{4} + 4 \, {\left (16 \, a^{4} + 12 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{315 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [A]  time = 0.70, size = 269, normalized size = 0.96 \[ -\frac {a^{3} b \cos \left (5 \, d x + 5 \, c\right )}{16 \, d} - \frac {{\left (a^{3} b - a b^{3}\right )} \cos \left (9 \, d x + 9 \, c\right )}{576 \, d} - \frac {{\left (7 \, a^{3} b - 3 \, a b^{3}\right )} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {{\left (7 \, a^{3} b + 2 \, a b^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {{\left (7 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )}{32 \, d} + \frac {{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {{\left (9 \, a^{4} - 30 \, a^{2} b^{2} + b^{4}\right )} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {{\left (9 \, a^{4} - 12 \, a^{2} b^{2} - b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (21 \, a^{4} - b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {3 \, {\left (21 \, a^{4} + 14 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )}{128 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 11.18, size = 236, normalized size = 0.85 \[ \frac {b^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{9}-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{21}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{105}\right )+4 a \,b^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+6 a^{2} b^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )}{9}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{63}\right )-\frac {4 a^{3} b \left (\cos ^{9}\left (d x +c \right )\right )}{9}+\frac {a^{4} \left (\frac {128}{35}+\cos ^{8}\left (d x +c \right )+\frac {8 \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\cos ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\cos ^{2}\left (d x +c \right )\right )}{35}\right ) \sin \left (d x +c \right )}{9}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b^4*(-1/9*sin(d*x+c)^3*cos(d*x+c)^6-1/21*sin(d*x+c)*cos(d*x+c)^6+1/105*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2
)*sin(d*x+c))+4*a*b^3*(-1/9*sin(d*x+c)^2*cos(d*x+c)^7-2/63*cos(d*x+c)^7)+6*a^2*b^2*(-1/9*sin(d*x+c)*cos(d*x+c)
^8+1/63*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c))-4/9*a^3*b*cos(d*x+c)^9+1/9*a^4*(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))

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maxima [A]  time = 0.33, size = 186, normalized size = 0.67 \[ -\frac {140 \, a^{3} b \cos \left (d x + c\right )^{9} - {\left (35 \, \sin \left (d x + c\right )^{9} - 180 \, \sin \left (d x + c\right )^{7} + 378 \, \sin \left (d x + c\right )^{5} - 420 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )\right )} a^{4} + 6 \, {\left (35 \, \sin \left (d x + c\right )^{9} - 135 \, \sin \left (d x + c\right )^{7} + 189 \, \sin \left (d x + c\right )^{5} - 105 \, \sin \left (d x + c\right )^{3}\right )} a^{2} b^{2} - 20 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a b^{3} - {\left (35 \, \sin \left (d x + c\right )^{9} - 90 \, \sin \left (d x + c\right )^{7} + 63 \, \sin \left (d x + c\right )^{5}\right )} b^{4}}{315 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 2.00, size = 334, normalized size = 1.20 \[ -\frac {\frac {b^4\,\sin \left (3\,c+3\,d\,x\right )}{192}-\frac {3\,b^4\,\sin \left (c+d\,x\right )}{128}-\frac {7\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{64}-\frac {9\,a^4\,\sin \left (5\,c+5\,d\,x\right )}{320}-\frac {9\,a^4\,\sin \left (7\,c+7\,d\,x\right )}{1792}-\frac {a^4\,\sin \left (9\,c+9\,d\,x\right )}{2304}-\frac {63\,a^4\,\sin \left (c+d\,x\right )}{128}+\frac {b^4\,\sin \left (5\,c+5\,d\,x\right )}{320}-\frac {b^4\,\sin \left (7\,c+7\,d\,x\right )}{1792}-\frac {b^4\,\sin \left (9\,c+9\,d\,x\right )}{2304}+\frac {a\,b^3\,\cos \left (3\,c+3\,d\,x\right )}{24}+\frac {7\,a^3\,b\,\cos \left (3\,c+3\,d\,x\right )}{48}+\frac {a^3\,b\,\cos \left (5\,c+5\,d\,x\right )}{16}-\frac {3\,a\,b^3\,\cos \left (7\,c+7\,d\,x\right )}{448}+\frac {a^3\,b\,\cos \left (7\,c+7\,d\,x\right )}{64}-\frac {a\,b^3\,\cos \left (9\,c+9\,d\,x\right )}{576}+\frac {a^3\,b\,\cos \left (9\,c+9\,d\,x\right )}{576}-\frac {21\,a^2\,b^2\,\sin \left (c+d\,x\right )}{64}+\frac {3\,a^2\,b^2\,\sin \left (5\,c+5\,d\,x\right )}{80}+\frac {15\,a^2\,b^2\,\sin \left (7\,c+7\,d\,x\right )}{896}+\frac {a^2\,b^2\,\sin \left (9\,c+9\,d\,x\right )}{384}+\frac {3\,a\,b^3\,\cos \left (c+d\,x\right )}{32}+\frac {7\,a^3\,b\,\cos \left (c+d\,x\right )}{32}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 15.62, size = 367, normalized size = 1.32 \[ \begin {cases} \frac {128 a^{4} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {64 a^{4} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {16 a^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {8 a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac {a^{4} \sin {\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{d} - \frac {4 a^{3} b \cos ^{9}{\left (c + d x \right )}}{9 d} + \frac {32 a^{2} b^{2} \sin ^{9}{\left (c + d x \right )}}{105 d} + \frac {48 a^{2} b^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {12 a^{2} b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {2 a^{2} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {4 a b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {8 a b^{3} \cos ^{9}{\left (c + d x \right )}}{63 d} + \frac {8 b^{4} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {4 b^{4} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\relax (c )} + b \sin {\relax (c )}\right )^{4} \cos ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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