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

Optimal. Leaf size=381 \[ \frac {a^4 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {7 a^4 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {35 a^4 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {35 a^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {35 a^4 x}{128}-\frac {a^3 b \cos ^8(c+d x)}{2 d}-\frac {3 a^2 b^2 \sin (c+d x) \cos ^7(c+d x)}{4 d}+\frac {a^2 b^2 \sin (c+d x) \cos ^5(c+d x)}{8 d}+\frac {5 a^2 b^2 \sin (c+d x) \cos ^3(c+d x)}{32 d}+\frac {15 a^2 b^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac {15}{64} a^2 b^2 x+\frac {a b^3 \cos ^8(c+d x)}{2 d}-\frac {2 a b^3 \cos ^6(c+d x)}{3 d}-\frac {b^4 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac {b^4 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac {b^4 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {3 b^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {3 b^4 x}{128} \]

[Out]

35/128*a^4*x+15/64*a^2*b^2*x+3/128*b^4*x-2/3*a*b^3*cos(d*x+c)^6/d-1/2*a^3*b*cos(d*x+c)^8/d+1/2*a*b^3*cos(d*x+c
)^8/d+35/128*a^4*cos(d*x+c)*sin(d*x+c)/d+15/64*a^2*b^2*cos(d*x+c)*sin(d*x+c)/d+3/128*b^4*cos(d*x+c)*sin(d*x+c)
/d+35/192*a^4*cos(d*x+c)^3*sin(d*x+c)/d+5/32*a^2*b^2*cos(d*x+c)^3*sin(d*x+c)/d+1/64*b^4*cos(d*x+c)^3*sin(d*x+c
)/d+7/48*a^4*cos(d*x+c)^5*sin(d*x+c)/d+1/8*a^2*b^2*cos(d*x+c)^5*sin(d*x+c)/d-1/16*b^4*cos(d*x+c)^5*sin(d*x+c)/
d+1/8*a^4*cos(d*x+c)^7*sin(d*x+c)/d-3/4*a^2*b^2*cos(d*x+c)^7*sin(d*x+c)/d-1/8*b^4*cos(d*x+c)^5*sin(d*x+c)^3/d

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Rubi [A]  time = 0.39, antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3090, 2635, 8, 2565, 30, 2568, 14} \[ -\frac {3 a^2 b^2 \sin (c+d x) \cos ^7(c+d x)}{4 d}+\frac {a^2 b^2 \sin (c+d x) \cos ^5(c+d x)}{8 d}+\frac {5 a^2 b^2 \sin (c+d x) \cos ^3(c+d x)}{32 d}+\frac {15 a^2 b^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac {15}{64} a^2 b^2 x-\frac {a^3 b \cos ^8(c+d x)}{2 d}+\frac {a^4 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {7 a^4 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {35 a^4 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {35 a^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {35 a^4 x}{128}+\frac {a b^3 \cos ^8(c+d x)}{2 d}-\frac {2 a b^3 \cos ^6(c+d x)}{3 d}-\frac {b^4 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac {b^4 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac {b^4 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {3 b^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {3 b^4 x}{128} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(35*a^4*x)/128 + (15*a^2*b^2*x)/64 + (3*b^4*x)/128 - (2*a*b^3*Cos[c + d*x]^6)/(3*d) - (a^3*b*Cos[c + d*x]^8)/(
2*d) + (a*b^3*Cos[c + d*x]^8)/(2*d) + (35*a^4*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (15*a^2*b^2*Cos[c + d*x]*Si
n[c + d*x])/(64*d) + (3*b^4*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (35*a^4*Cos[c + d*x]^3*Sin[c + d*x])/(192*d)
+ (5*a^2*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(32*d) + (b^4*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) + (7*a^4*Cos[c + d
*x]^5*Sin[c + d*x])/(48*d) + (a^2*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(8*d) - (b^4*Cos[c + d*x]^5*Sin[c + d*x])/(
16*d) + (a^4*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (3*a^2*b^2*Cos[c + d*x]^7*Sin[c + d*x])/(4*d) - (b^4*Cos[c +
d*x]^5*Sin[c + d*x]^3)/(8*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 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 2568

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 2635

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] && Integer
Q[2*n]

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 ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx &=\int \left (a^4 \cos ^8(c+d x)+4 a^3 b \cos ^7(c+d x) \sin (c+d x)+6 a^2 b^2 \cos ^6(c+d x) \sin ^2(c+d x)+4 a b^3 \cos ^5(c+d x) \sin ^3(c+d x)+b^4 \cos ^4(c+d x) \sin ^4(c+d x)\right ) \, dx\\ &=a^4 \int \cos ^8(c+d x) \, dx+\left (4 a^3 b\right ) \int \cos ^7(c+d x) \sin (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \cos ^5(c+d x) \sin ^3(c+d x) \, dx+b^4 \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx\\ &=\frac {a^4 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a^2 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{8} \left (7 a^4\right ) \int \cos ^6(c+d x) \, dx+\frac {1}{4} \left (3 a^2 b^2\right ) \int \cos ^6(c+d x) \, dx+\frac {1}{8} \left (3 b^4\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx-\frac {\left (4 a^3 b\right ) \operatorname {Subst}\left (\int x^7 \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (4 a b^3\right ) \operatorname {Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^3 b \cos ^8(c+d x)}{2 d}+\frac {7 a^4 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^2 b^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac {b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^4 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a^2 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{48} \left (35 a^4\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{8} \left (5 a^2 b^2\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{16} b^4 \int \cos ^4(c+d x) \, dx-\frac {\left (4 a b^3\right ) \operatorname {Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {2 a b^3 \cos ^6(c+d x)}{3 d}-\frac {a^3 b \cos ^8(c+d x)}{2 d}+\frac {a b^3 \cos ^8(c+d x)}{2 d}+\frac {35 a^4 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {5 a^2 b^2 \cos ^3(c+d x) \sin (c+d x)}{32 d}+\frac {b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^4 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^2 b^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac {b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^4 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a^2 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{64} \left (35 a^4\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{32} \left (15 a^2 b^2\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{64} \left (3 b^4\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {2 a b^3 \cos ^6(c+d x)}{3 d}-\frac {a^3 b \cos ^8(c+d x)}{2 d}+\frac {a b^3 \cos ^8(c+d x)}{2 d}+\frac {35 a^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {15 a^2 b^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {3 b^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 a^4 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {5 a^2 b^2 \cos ^3(c+d x) \sin (c+d x)}{32 d}+\frac {b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^4 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^2 b^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac {b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^4 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a^2 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{128} \left (35 a^4\right ) \int 1 \, dx+\frac {1}{64} \left (15 a^2 b^2\right ) \int 1 \, dx+\frac {1}{128} \left (3 b^4\right ) \int 1 \, dx\\ &=\frac {35 a^4 x}{128}+\frac {15}{64} a^2 b^2 x+\frac {3 b^4 x}{128}-\frac {2 a b^3 \cos ^6(c+d x)}{3 d}-\frac {a^3 b \cos ^8(c+d x)}{2 d}+\frac {a b^3 \cos ^8(c+d x)}{2 d}+\frac {35 a^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {15 a^2 b^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {3 b^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 a^4 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {5 a^2 b^2 \cos ^3(c+d x) \sin (c+d x)}{32 d}+\frac {b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^4 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a^2 b^2 \cos ^5(c+d x) \sin (c+d x)}{8 d}-\frac {b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^4 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a^2 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}\\ \end {align*}

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Mathematica [A]  time = 0.61, size = 222, normalized size = 0.58 \[ \frac {96 a^2 \left (7 a^2+3 b^2\right ) \sin (2 (c+d x))+32 a^2 \left (a^2-3 b^2\right ) \sin (6 (c+d x))-96 a b \left (7 a^2+3 b^2\right ) \cos (2 (c+d x))-48 a b \left (7 a^2+b^2\right ) \cos (4 (c+d x))-32 a b \left (3 a^2-b^2\right ) \cos (6 (c+d x))-12 a b \left (a^2-b^2\right ) \cos (8 (c+d x))+24 \left (35 a^4+30 a^2 b^2+3 b^4\right ) (c+d x)+24 \left (7 a^4-6 a^2 b^2-b^4\right ) \sin (4 (c+d x))+3 \left (a^4-6 a^2 b^2+b^4\right ) \sin (8 (c+d x))}{3072 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(24*(35*a^4 + 30*a^2*b^2 + 3*b^4)*(c + d*x) - 96*a*b*(7*a^2 + 3*b^2)*Cos[2*(c + d*x)] - 48*a*b*(7*a^2 + b^2)*C
os[4*(c + d*x)] - 32*a*b*(3*a^2 - b^2)*Cos[6*(c + d*x)] - 12*a*b*(a^2 - b^2)*Cos[8*(c + d*x)] + 96*a^2*(7*a^2
+ 3*b^2)*Sin[2*(c + d*x)] + 24*(7*a^4 - 6*a^2*b^2 - b^4)*Sin[4*(c + d*x)] + 32*a^2*(a^2 - 3*b^2)*Sin[6*(c + d*
x)] + 3*(a^4 - 6*a^2*b^2 + b^4)*Sin[8*(c + d*x)])/(3072*d)

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fricas [A]  time = 0.76, size = 184, normalized size = 0.48 \[ -\frac {256 \, a b^{3} \cos \left (d x + c\right )^{6} + 192 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{8} - 3 \, {\left (35 \, a^{4} + 30 \, a^{2} b^{2} + 3 \, b^{4}\right )} d x - {\left (48 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{7} + 8 \, {\left (7 \, a^{4} + 6 \, a^{2} b^{2} - 9 \, b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (35 \, a^{4} + 30 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (35 \, a^{4} + 30 \, a^{2} b^{2} + 3 \, b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{384 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(256*a*b^3*cos(d*x + c)^6 + 192*(a^3*b - a*b^3)*cos(d*x + c)^8 - 3*(35*a^4 + 30*a^2*b^2 + 3*b^4)*d*x -
(48*(a^4 - 6*a^2*b^2 + b^4)*cos(d*x + c)^7 + 8*(7*a^4 + 6*a^2*b^2 - 9*b^4)*cos(d*x + c)^5 + 2*(35*a^4 + 30*a^2
*b^2 + 3*b^4)*cos(d*x + c)^3 + 3*(35*a^4 + 30*a^2*b^2 + 3*b^4)*cos(d*x + c))*sin(d*x + c))/d

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giac [A]  time = 0.67, size = 245, normalized size = 0.64 \[ \frac {1}{128} \, {\left (35 \, a^{4} + 30 \, a^{2} b^{2} + 3 \, b^{4}\right )} x - \frac {{\left (a^{3} b - a b^{3}\right )} \cos \left (8 \, d x + 8 \, c\right )}{256 \, d} - \frac {{\left (3 \, a^{3} b - a b^{3}\right )} \cos \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac {{\left (7 \, a^{3} b + a b^{3}\right )} \cos \left (4 \, d x + 4 \, c\right )}{64 \, d} - \frac {{\left (7 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (2 \, d x + 2 \, c\right )}{32 \, d} + \frac {{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} + \frac {{\left (7 \, a^{4} - 6 \, a^{2} b^{2} - b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (7 \, a^{4} + 3 \, a^{2} b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(35*a^4 + 30*a^2*b^2 + 3*b^4)*x - 1/256*(a^3*b - a*b^3)*cos(8*d*x + 8*c)/d - 1/96*(3*a^3*b - a*b^3)*cos(
6*d*x + 6*c)/d - 1/64*(7*a^3*b + a*b^3)*cos(4*d*x + 4*c)/d - 1/32*(7*a^3*b + 3*a*b^3)*cos(2*d*x + 2*c)/d + 1/1
024*(a^4 - 6*a^2*b^2 + b^4)*sin(8*d*x + 8*c)/d + 1/96*(a^4 - 3*a^2*b^2)*sin(6*d*x + 6*c)/d + 1/128*(7*a^4 - 6*
a^2*b^2 - b^4)*sin(4*d*x + 4*c)/d + 1/32*(7*a^4 + 3*a^2*b^2)*sin(2*d*x + 2*c)/d

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maple [A]  time = 21.24, size = 250, normalized size = 0.66 \[ \frac {b^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+4 a \,b^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )+6 a^{2} b^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {a^{3} b \left (\cos ^{8}\left (d x +c \right )\right )}{2}+a^{4} \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b^4*(-1/8*sin(d*x+c)^3*cos(d*x+c)^5-1/16*cos(d*x+c)^5*sin(d*x+c)+1/64*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d
*x+c)+3/128*d*x+3/128*c)+4*a*b^3*(-1/8*sin(d*x+c)^2*cos(d*x+c)^6-1/24*cos(d*x+c)^6)+6*a^2*b^2*(-1/8*sin(d*x+c)
*cos(d*x+c)^7+1/48*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/128*d*x+5/128*c)-1/2*a^3*b*cos
(d*x+c)^8+a^4*(1/8*(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)+35/128*d*x+3
5/128*c))

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maxima [A]  time = 0.33, size = 199, normalized size = 0.52 \[ -\frac {1536 \, a^{3} b \cos \left (d x + c\right )^{8} + {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} - 6 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} b^{2} - 512 \, {\left (3 \, \sin \left (d x + c\right )^{8} - 8 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4}\right )} a b^{3} - 3 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{4}}{3072 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3072*(1536*a^3*b*cos(d*x + c)^8 + (128*sin(2*d*x + 2*c)^3 - 840*d*x - 840*c - 3*sin(8*d*x + 8*c) - 168*sin(
4*d*x + 4*c) - 768*sin(2*d*x + 2*c))*a^4 - 6*(64*sin(2*d*x + 2*c)^3 + 120*d*x + 120*c - 3*sin(8*d*x + 8*c) - 2
4*sin(4*d*x + 4*c))*a^2*b^2 - 512*(3*sin(d*x + c)^8 - 8*sin(d*x + c)^6 + 6*sin(d*x + c)^4)*a*b^3 - 3*(24*d*x +
24*c + sin(8*d*x + 8*c) - 8*sin(4*d*x + 4*c))*b^4)/d

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mupad [B]  time = 1.69, size = 343, normalized size = 0.90 \[ \frac {35\,a^4\,x}{128}+\frac {3\,b^4\,x}{128}+\frac {15\,a^2\,b^2\,x}{64}-\frac {2\,a\,b^3\,{\cos \left (c+d\,x\right )}^6}{3\,d}+\frac {a\,b^3\,{\cos \left (c+d\,x\right )}^8}{2\,d}-\frac {a^3\,b\,{\cos \left (c+d\,x\right )}^8}{2\,d}+\frac {35\,a^4\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{192\,d}+\frac {7\,a^4\,{\cos \left (c+d\,x\right )}^5\,\sin \left (c+d\,x\right )}{48\,d}+\frac {a^4\,{\cos \left (c+d\,x\right )}^7\,\sin \left (c+d\,x\right )}{8\,d}+\frac {b^4\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{64\,d}-\frac {3\,b^4\,{\cos \left (c+d\,x\right )}^5\,\sin \left (c+d\,x\right )}{16\,d}+\frac {b^4\,{\cos \left (c+d\,x\right )}^7\,\sin \left (c+d\,x\right )}{8\,d}+\frac {35\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{128\,d}+\frac {3\,b^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{128\,d}+\frac {15\,a^2\,b^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{64\,d}+\frac {5\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{32\,d}+\frac {a^2\,b^2\,{\cos \left (c+d\,x\right )}^5\,\sin \left (c+d\,x\right )}{8\,d}-\frac {3\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^7\,\sin \left (c+d\,x\right )}{4\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(35*a^4*x)/128 + (3*b^4*x)/128 + (15*a^2*b^2*x)/64 - (2*a*b^3*cos(c + d*x)^6)/(3*d) + (a*b^3*cos(c + d*x)^8)/(
2*d) - (a^3*b*cos(c + d*x)^8)/(2*d) + (35*a^4*cos(c + d*x)^3*sin(c + d*x))/(192*d) + (7*a^4*cos(c + d*x)^5*sin
(c + d*x))/(48*d) + (a^4*cos(c + d*x)^7*sin(c + d*x))/(8*d) + (b^4*cos(c + d*x)^3*sin(c + d*x))/(64*d) - (3*b^
4*cos(c + d*x)^5*sin(c + d*x))/(16*d) + (b^4*cos(c + d*x)^7*sin(c + d*x))/(8*d) + (35*a^4*cos(c + d*x)*sin(c +
d*x))/(128*d) + (3*b^4*cos(c + d*x)*sin(c + d*x))/(128*d) + (15*a^2*b^2*cos(c + d*x)*sin(c + d*x))/(64*d) + (
5*a^2*b^2*cos(c + d*x)^3*sin(c + d*x))/(32*d) + (a^2*b^2*cos(c + d*x)^5*sin(c + d*x))/(8*d) - (3*a^2*b^2*cos(c
+ d*x)^7*sin(c + d*x))/(4*d)

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sympy [A]  time = 11.26, size = 760, normalized size = 1.99 \[ \begin {cases} \frac {35 a^{4} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {35 a^{4} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {105 a^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {35 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {35 a^{4} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {35 a^{4} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {385 a^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {511 a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} + \frac {93 a^{4} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {a^{3} b \cos ^{8}{\left (c + d x \right )}}{2 d} + \frac {15 a^{2} b^{2} x \sin ^{8}{\left (c + d x \right )}}{64} + \frac {15 a^{2} b^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {45 a^{2} b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{32} + \frac {15 a^{2} b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{2} b^{2} x \cos ^{8}{\left (c + d x \right )}}{64} + \frac {15 a^{2} b^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{64 d} + \frac {55 a^{2} b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{64 d} + \frac {73 a^{2} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{64 d} - \frac {15 a^{2} b^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} + \frac {a b^{3} \sin ^{8}{\left (c + d x \right )}}{6 d} + \frac {2 a b^{3} \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {a b^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {3 b^{4} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {3 b^{4} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {9 b^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {3 b^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {3 b^{4} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 b^{4} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {11 b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {11 b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {3 b^{4} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\relax (c )} + b \sin {\relax (c )}\right )^{4} \cos ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((35*a**4*x*sin(c + d*x)**8/128 + 35*a**4*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 105*a**4*x*sin(c + d
*x)**4*cos(c + d*x)**4/64 + 35*a**4*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 35*a**4*x*cos(c + d*x)**8/128 + 35*
a**4*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 385*a**4*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) + 511*a**4*sin(c
+ d*x)**3*cos(c + d*x)**5/(384*d) + 93*a**4*sin(c + d*x)*cos(c + d*x)**7/(128*d) - a**3*b*cos(c + d*x)**8/(2*d
) + 15*a**2*b**2*x*sin(c + d*x)**8/64 + 15*a**2*b**2*x*sin(c + d*x)**6*cos(c + d*x)**2/16 + 45*a**2*b**2*x*sin
(c + d*x)**4*cos(c + d*x)**4/32 + 15*a**2*b**2*x*sin(c + d*x)**2*cos(c + d*x)**6/16 + 15*a**2*b**2*x*cos(c + d
*x)**8/64 + 15*a**2*b**2*sin(c + d*x)**7*cos(c + d*x)/(64*d) + 55*a**2*b**2*sin(c + d*x)**5*cos(c + d*x)**3/(6
4*d) + 73*a**2*b**2*sin(c + d*x)**3*cos(c + d*x)**5/(64*d) - 15*a**2*b**2*sin(c + d*x)*cos(c + d*x)**7/(64*d)
+ a*b**3*sin(c + d*x)**8/(6*d) + 2*a*b**3*sin(c + d*x)**6*cos(c + d*x)**2/(3*d) + a*b**3*sin(c + d*x)**4*cos(c
+ d*x)**4/d + 3*b**4*x*sin(c + d*x)**8/128 + 3*b**4*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 9*b**4*x*sin(c + d
*x)**4*cos(c + d*x)**4/64 + 3*b**4*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 3*b**4*x*cos(c + d*x)**8/128 + 3*b**
4*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 11*b**4*sin(c + d*x)**5*cos(c + d*x)**3/(128*d) - 11*b**4*sin(c + d*x
)**3*cos(c + d*x)**5/(128*d) - 3*b**4*sin(c + d*x)*cos(c + d*x)**7/(128*d), Ne(d, 0)), (x*(a*cos(c) + b*sin(c)
)**4*cos(c)**4, True))

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