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

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

[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)

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Rubi [A]  time = 0.388978, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 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]

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 8

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

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 30

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

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 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]]

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*}

Mathematica [A]  time = 0.600107, size = 222, normalized size = 0.58 \[ \frac{24 \left (30 a^2 b^2+35 a^4+3 b^4\right ) (c+d x)+96 a^2 \left (7 a^2+3 b^2\right ) \sin (2 (c+d x))+32 a^2 \left (a^2-3 b^2\right ) \sin (6 (c+d x))+24 \left (-6 a^2 b^2+7 a^4-b^4\right ) \sin (4 (c+d x))+3 \left (-6 a^2 b^2+a^4+b^4\right ) \sin (8 (c+d x))-96 a b \left (7 a^2+3 b^2\right ) \cos (2 (c+d x))-48 a b \left (7 a^2+b^2\right ) \cos (4 (c+d x))-32 a b \left (3 a^2-b^2\right ) \cos (6 (c+d x))-12 a b \left (a^2-b^2\right ) \cos (8 (c+d x))}{3072 d} \]

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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Maple [A]  time = 0.086, size = 250, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ({b}^{4} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16}}+{\frac{\sin \left ( dx+c \right ) }{64} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{128}}+{\frac{3\,c}{128}} \right ) +4\,a{b}^{3} \left ( -1/8\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}-1/24\, \left ( \cos \left ( dx+c \right ) \right ) ^{6} \right ) +6\,{a}^{2}{b}^{2} \left ( -1/8\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}+1/48\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+5/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) \sin \left ( dx+c \right ) +{\frac{5\,dx}{128}}+{\frac{5\,c}{128}} \right ) -{\frac{{a}^{3}b \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{2}}+{a}^{4} \left ({\frac{\sin \left ( dx+c \right ) }{8} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{7}+{\frac{7\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{6}}+{\frac{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{24}}+{\frac{35\,\cos \left ( dx+c \right ) }{16}} \right ) }+{\frac{35\,dx}{128}}+{\frac{35\,c}{128}} \right ) \right ) } \end{align*}

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*sin(d*x+c)*cos(d*x+c)^5+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 = 1.14786, size = 269, normalized size = 0.71 \begin{align*} -\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} \end{align*}

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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Fricas [A]  time = 0.539061, size = 424, normalized size = 1.11 \begin{align*} -\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} \end{align*}

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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Sympy [A]  time = 15.7511, size = 736, normalized size = 1.93 \begin{align*} \text{result too large to display} \end{align*}

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)
- 2*a*b**3*sin(c + d*x)**2*cos(c + d*x)**6/(3*d) - a*b**3*cos(c + d*x)**8/(6*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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Giac [A]  time = 1.2205, size = 331, normalized size = 0.87 \begin{align*} \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} \end{align*}

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