3.94 \(\int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx\)

Optimal. Leaf size=426 \[ \frac {a^5 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {7 a^5 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {35 a^5 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {35 a^5 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {35 a^5 x}{128}-\frac {5 a^4 b \cos ^8(c+d x)}{8 d}-\frac {5 a^3 b^2 \sin (c+d x) \cos ^7(c+d x)}{4 d}+\frac {5 a^3 b^2 \sin (c+d x) \cos ^5(c+d x)}{24 d}+\frac {25 a^3 b^2 \sin (c+d x) \cos ^3(c+d x)}{96 d}+\frac {25 a^3 b^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac {25}{64} a^3 b^2 x+\frac {5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac {5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac {5 a b^4 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac {5 a b^4 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac {5 a b^4 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {15 a b^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {15}{128} a b^4 x-\frac {b^5 \sin ^8(c+d x)}{8 d}+\frac {b^5 \sin ^6(c+d x)}{6 d} \]

[Out]

35/128*a^5*x+25/64*a^3*b^2*x+15/128*a*b^4*x-5/3*a^2*b^3*cos(d*x+c)^6/d-5/8*a^4*b*cos(d*x+c)^8/d+5/4*a^2*b^3*co
s(d*x+c)^8/d+35/128*a^5*cos(d*x+c)*sin(d*x+c)/d+25/64*a^3*b^2*cos(d*x+c)*sin(d*x+c)/d+15/128*a*b^4*cos(d*x+c)*
sin(d*x+c)/d+35/192*a^5*cos(d*x+c)^3*sin(d*x+c)/d+25/96*a^3*b^2*cos(d*x+c)^3*sin(d*x+c)/d+5/64*a*b^4*cos(d*x+c
)^3*sin(d*x+c)/d+7/48*a^5*cos(d*x+c)^5*sin(d*x+c)/d+5/24*a^3*b^2*cos(d*x+c)^5*sin(d*x+c)/d-5/16*a*b^4*cos(d*x+
c)^5*sin(d*x+c)/d+1/8*a^5*cos(d*x+c)^7*sin(d*x+c)/d-5/4*a^3*b^2*cos(d*x+c)^7*sin(d*x+c)/d-5/8*a*b^4*cos(d*x+c)
^5*sin(d*x+c)^3/d+1/6*b^5*sin(d*x+c)^6/d-1/8*b^5*sin(d*x+c)^8/d

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Rubi [A]  time = 0.41, antiderivative size = 426, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3090, 2635, 8, 2565, 30, 2568, 14, 2564} \[ \frac {5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac {5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac {5 a^3 b^2 \sin (c+d x) \cos ^7(c+d x)}{4 d}+\frac {5 a^3 b^2 \sin (c+d x) \cos ^5(c+d x)}{24 d}+\frac {25 a^3 b^2 \sin (c+d x) \cos ^3(c+d x)}{96 d}+\frac {25 a^3 b^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac {25}{64} a^3 b^2 x-\frac {5 a^4 b \cos ^8(c+d x)}{8 d}+\frac {a^5 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {7 a^5 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {35 a^5 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {35 a^5 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {35 a^5 x}{128}-\frac {5 a b^4 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac {5 a b^4 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac {5 a b^4 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {15 a b^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {15}{128} a b^4 x-\frac {b^5 \sin ^8(c+d x)}{8 d}+\frac {b^5 \sin ^6(c+d x)}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

(35*a^5*x)/128 + (25*a^3*b^2*x)/64 + (15*a*b^4*x)/128 - (5*a^2*b^3*Cos[c + d*x]^6)/(3*d) - (5*a^4*b*Cos[c + d*
x]^8)/(8*d) + (5*a^2*b^3*Cos[c + d*x]^8)/(4*d) + (35*a^5*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (25*a^3*b^2*Cos[
c + d*x]*Sin[c + d*x])/(64*d) + (15*a*b^4*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (35*a^5*Cos[c + d*x]^3*Sin[c +
d*x])/(192*d) + (25*a^3*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(96*d) + (5*a*b^4*Cos[c + d*x]^3*Sin[c + d*x])/(64*d)
+ (7*a^5*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (5*a^3*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(24*d) - (5*a*b^4*Cos[
c + d*x]^5*Sin[c + d*x])/(16*d) + (a^5*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (5*a^3*b^2*Cos[c + d*x]^7*Sin[c +
d*x])/(4*d) - (5*a*b^4*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d) + (b^5*Sin[c + d*x]^6)/(6*d) - (b^5*Sin[c + d*x]^8
)/(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 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 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 ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=\int \left (a^5 \cos ^8(c+d x)+5 a^4 b \cos ^7(c+d x) \sin (c+d x)+10 a^3 b^2 \cos ^6(c+d x) \sin ^2(c+d x)+10 a^2 b^3 \cos ^5(c+d x) \sin ^3(c+d x)+5 a b^4 \cos ^4(c+d x) \sin ^4(c+d x)+b^5 \cos ^3(c+d x) \sin ^5(c+d x)\right ) \, dx\\ &=a^5 \int \cos ^8(c+d x) \, dx+\left (5 a^4 b\right ) \int \cos ^7(c+d x) \sin (c+d x) \, dx+\left (10 a^3 b^2\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+\left (10 a^2 b^3\right ) \int \cos ^5(c+d x) \sin ^3(c+d x) \, dx+\left (5 a b^4\right ) \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx+b^5 \int \cos ^3(c+d x) \sin ^5(c+d x) \, dx\\ &=\frac {a^5 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {5 a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{8} \left (7 a^5\right ) \int \cos ^6(c+d x) \, dx+\frac {1}{4} \left (5 a^3 b^2\right ) \int \cos ^6(c+d x) \, dx+\frac {1}{8} \left (15 a b^4\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx-\frac {\left (5 a^4 b\right ) \operatorname {Subst}\left (\int x^7 \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (10 a^2 b^3\right ) \operatorname {Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {b^5 \operatorname {Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {5 a^4 b \cos ^8(c+d x)}{8 d}+\frac {7 a^5 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {5 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^5 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {5 a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{48} \left (35 a^5\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{24} \left (25 a^3 b^2\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{16} \left (5 a b^4\right ) \int \cos ^4(c+d x) \, dx-\frac {\left (10 a^2 b^3\right ) \operatorname {Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {b^5 \operatorname {Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac {5 a^4 b \cos ^8(c+d x)}{8 d}+\frac {5 a^2 b^3 \cos ^8(c+d x)}{4 d}+\frac {35 a^5 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {25 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac {5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^5 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {5 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^5 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {5 a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {b^5 \sin ^6(c+d x)}{6 d}-\frac {b^5 \sin ^8(c+d x)}{8 d}+\frac {1}{64} \left (35 a^5\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{32} \left (25 a^3 b^2\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{64} \left (15 a b^4\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac {5 a^4 b \cos ^8(c+d x)}{8 d}+\frac {5 a^2 b^3 \cos ^8(c+d x)}{4 d}+\frac {35 a^5 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {25 a^3 b^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {15 a b^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 a^5 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {25 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac {5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^5 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {5 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^5 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {5 a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {b^5 \sin ^6(c+d x)}{6 d}-\frac {b^5 \sin ^8(c+d x)}{8 d}+\frac {1}{128} \left (35 a^5\right ) \int 1 \, dx+\frac {1}{64} \left (25 a^3 b^2\right ) \int 1 \, dx+\frac {1}{128} \left (15 a b^4\right ) \int 1 \, dx\\ &=\frac {35 a^5 x}{128}+\frac {25}{64} a^3 b^2 x+\frac {15}{128} a b^4 x-\frac {5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac {5 a^4 b \cos ^8(c+d x)}{8 d}+\frac {5 a^2 b^3 \cos ^8(c+d x)}{4 d}+\frac {35 a^5 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {25 a^3 b^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {15 a b^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 a^5 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {25 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac {5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^5 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {5 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^5 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {5 a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {b^5 \sin ^6(c+d x)}{6 d}-\frac {b^5 \sin ^8(c+d x)}{8 d}\\ \end {align*}

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Mathematica [C]  time = 0.87, size = 259, normalized size = 0.61 \[ \frac {120 a (a-i b) (a+i b) \left (7 a^2+3 b^2\right ) (c+d x)+24 a \left (7 a^4-10 a^2 b^2-5 b^4\right ) \sin (4 (c+d x))+3 a \left (a^4-10 a^2 b^2+5 b^4\right ) \sin (8 (c+d x))-24 b \left (35 a^4+30 a^2 b^2+3 b^4\right ) \cos (2 (c+d x))+12 b \left (-35 a^4-10 a^2 b^2+b^4\right ) \cos (4 (c+d x))+8 b \left (-15 a^4+10 a^2 b^2+b^4\right ) \cos (6 (c+d x))-3 b \left (5 a^4-10 a^2 b^2+b^4\right ) \cos (8 (c+d x))+96 a^3 \left (7 a^2+5 b^2\right ) \sin (2 (c+d x))+32 a^3 \left (a^2-5 b^2\right ) \sin (6 (c+d x))}{3072 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

(120*a*(a - I*b)*(a + I*b)*(7*a^2 + 3*b^2)*(c + d*x) - 24*b*(35*a^4 + 30*a^2*b^2 + 3*b^4)*Cos[2*(c + d*x)] + 1
2*b*(-35*a^4 - 10*a^2*b^2 + b^4)*Cos[4*(c + d*x)] + 8*b*(-15*a^4 + 10*a^2*b^2 + b^4)*Cos[6*(c + d*x)] - 3*b*(5
*a^4 - 10*a^2*b^2 + b^4)*Cos[8*(c + d*x)] + 96*a^3*(7*a^2 + 5*b^2)*Sin[2*(c + d*x)] + 24*a*(7*a^4 - 10*a^2*b^2
- 5*b^4)*Sin[4*(c + d*x)] + 32*a^3*(a^2 - 5*b^2)*Sin[6*(c + d*x)] + 3*a*(a^4 - 10*a^2*b^2 + 5*b^4)*Sin[8*(c +
d*x)])/(3072*d)

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fricas [A]  time = 0.70, size = 220, normalized size = 0.52 \[ -\frac {96 \, b^{5} \cos \left (d x + c\right )^{4} + 48 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{8} + 128 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{6} - 15 \, {\left (7 \, a^{5} + 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} d x - {\left (48 \, {\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} + 8 \, {\left (7 \, a^{5} + 10 \, a^{3} b^{2} - 45 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 10 \, {\left (7 \, a^{5} + 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (7 \, a^{5} + 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{384 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="fricas")

[Out]

-1/384*(96*b^5*cos(d*x + c)^4 + 48*(5*a^4*b - 10*a^2*b^3 + b^5)*cos(d*x + c)^8 + 128*(5*a^2*b^3 - b^5)*cos(d*x
+ c)^6 - 15*(7*a^5 + 10*a^3*b^2 + 3*a*b^4)*d*x - (48*(a^5 - 10*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^7 + 8*(7*a^5 +
10*a^3*b^2 - 45*a*b^4)*cos(d*x + c)^5 + 10*(7*a^5 + 10*a^3*b^2 + 3*a*b^4)*cos(d*x + c)^3 + 15*(7*a^5 + 10*a^3
*b^2 + 3*a*b^4)*cos(d*x + c))*sin(d*x + c))/d

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giac [A]  time = 0.64, size = 278, normalized size = 0.65 \[ \frac {5}{128} \, {\left (7 \, a^{5} + 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} x - \frac {{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {{\left (15 \, a^{4} b - 10 \, a^{2} b^{3} - b^{5}\right )} \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {{\left (35 \, a^{4} b + 10 \, a^{2} b^{3} - b^{5}\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {{\left (35 \, a^{4} b + 30 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {{\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {{\left (a^{5} - 5 \, a^{3} b^{2}\right )} \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} + \frac {{\left (7 \, a^{5} - 10 \, a^{3} b^{2} - 5 \, a b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (7 \, a^{5} + 5 \, a^{3} b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="giac")

[Out]

5/128*(7*a^5 + 10*a^3*b^2 + 3*a*b^4)*x - 1/1024*(5*a^4*b - 10*a^2*b^3 + b^5)*cos(8*d*x + 8*c)/d - 1/384*(15*a^
4*b - 10*a^2*b^3 - b^5)*cos(6*d*x + 6*c)/d - 1/256*(35*a^4*b + 10*a^2*b^3 - b^5)*cos(4*d*x + 4*c)/d - 1/128*(3
5*a^4*b + 30*a^2*b^3 + 3*b^5)*cos(2*d*x + 2*c)/d + 1/1024*(a^5 - 10*a^3*b^2 + 5*a*b^4)*sin(8*d*x + 8*c)/d + 1/
96*(a^5 - 5*a^3*b^2)*sin(6*d*x + 6*c)/d + 1/128*(7*a^5 - 10*a^3*b^2 - 5*a*b^4)*sin(4*d*x + 4*c)/d + 1/32*(7*a^
5 + 5*a^3*b^2)*sin(2*d*x + 2*c)/d

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maple [A]  time = 0.18, size = 305, normalized size = 0.72 \[ \frac {b^{5} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{8}-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{12}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{24}\right )+5 a \,b^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+10 a^{2} b^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )+10 a^{3} b^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {5 a^{4} b \left (\cos ^{8}\left (d x +c \right )\right )}{8}+a^{5} \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*(a*cos(d*x+c)+b*sin(d*x+c))^5,x)

[Out]

1/d*(b^5*(-1/8*sin(d*x+c)^4*cos(d*x+c)^4-1/12*sin(d*x+c)^2*cos(d*x+c)^4-1/24*cos(d*x+c)^4)+5*a*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+3/12
8*c)+10*a^2*b^3*(-1/8*sin(d*x+c)^2*cos(d*x+c)^6-1/24*cos(d*x+c)^6)+10*a^3*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)-5/8*a^4*b*cos(d*x+c)^8+a^5*(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+35/128*c))

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maxima [A]  time = 0.33, size = 228, normalized size = 0.54 \[ -\frac {1920 \, a^{4} b \cos \left (d x + c\right )^{8} + {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{5} - 10 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} b^{2} - 1280 \, {\left (3 \, \sin \left (d x + c\right )^{8} - 8 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4}\right )} a^{2} b^{3} - 15 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{4} + 128 \, {\left (3 \, \sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6}\right )} b^{5}}{3072 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="maxima")

[Out]

-1/3072*(1920*a^4*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^5 - 10*(64*sin(2*d*x + 2*c)^3 + 120*d*x + 120*c - 3*sin(8*d*x + 8*c) -
24*sin(4*d*x + 4*c))*a^3*b^2 - 1280*(3*sin(d*x + c)^8 - 8*sin(d*x + c)^6 + 6*sin(d*x + c)^4)*a^2*b^3 - 15*(24*
d*x + 24*c + sin(8*d*x + 8*c) - 8*sin(4*d*x + 4*c))*a*b^4 + 128*(3*sin(d*x + c)^8 - 4*sin(d*x + c)^6)*b^5)/d

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mupad [B]  time = 2.48, size = 650, normalized size = 1.53 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (-\frac {93\,a^5}{64}+\frac {25\,a^3\,b^2}{32}+\frac {15\,a\,b^4}{64}\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-\frac {93\,a^5}{64}+\frac {25\,a^3\,b^2}{32}+\frac {15\,a\,b^4}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {91\,a^5}{192}+\frac {1985\,a^3\,b^2}{96}-\frac {115\,a\,b^4}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {91\,a^5}{192}+\frac {1985\,a^3\,b^2}{96}-\frac {115\,a\,b^4}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {1799\,a^5}{192}-\frac {4475\,a^3\,b^2}{96}+\frac {1665\,a\,b^4}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {1799\,a^5}{192}-\frac {4475\,a^3\,b^2}{96}+\frac {1665\,a\,b^4}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {1085\,a^5}{192}-\frac {8825\,a^3\,b^2}{96}+\frac {3355\,a\,b^4}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {1085\,a^5}{192}-\frac {8825\,a^3\,b^2}{96}+\frac {3355\,a\,b^4}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (70\,a^4\,b-\frac {160\,a^2\,b^3}{3}+\frac {32\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (70\,a^4\,b-\frac {160\,a^2\,b^3}{3}+\frac {32\,b^5}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {32\,b^5}{3}-\frac {400\,a^2\,b^3}{3}\right )+40\,a^2\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+40\,a^2\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+10\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+10\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {5\,a\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (7\,a^4+10\,a^2\,b^2+3\,b^4\right )}{64\,d}+\frac {5\,a\,\mathrm {atan}\left (\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,a^2+3\,b^2\right )\,\left (a^2+b^2\right )}{64\,\left (\frac {35\,a^5}{64}+\frac {25\,a^3\,b^2}{32}+\frac {15\,a\,b^4}{64}\right )}\right )\,\left (7\,a^2+3\,b^2\right )\,\left (a^2+b^2\right )}{64\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^3*(a*cos(c + d*x) + b*sin(c + d*x))^5,x)

[Out]

(tan(c/2 + (d*x)/2)^15*((15*a*b^4)/64 - (93*a^5)/64 + (25*a^3*b^2)/32) - tan(c/2 + (d*x)/2)*((15*a*b^4)/64 - (
93*a^5)/64 + (25*a^3*b^2)/32) + tan(c/2 + (d*x)/2)^3*((91*a^5)/192 - (115*a*b^4)/64 + (1985*a^3*b^2)/96) - tan
(c/2 + (d*x)/2)^13*((91*a^5)/192 - (115*a*b^4)/64 + (1985*a^3*b^2)/96) + tan(c/2 + (d*x)/2)^5*((1665*a*b^4)/64
+ (1799*a^5)/192 - (4475*a^3*b^2)/96) - tan(c/2 + (d*x)/2)^11*((1665*a*b^4)/64 + (1799*a^5)/192 - (4475*a^3*b
^2)/96) - tan(c/2 + (d*x)/2)^7*((3355*a*b^4)/64 + (1085*a^5)/192 - (8825*a^3*b^2)/96) + tan(c/2 + (d*x)/2)^9*(
(3355*a*b^4)/64 + (1085*a^5)/192 - (8825*a^3*b^2)/96) + tan(c/2 + (d*x)/2)^6*(70*a^4*b + (32*b^5)/3 - (160*a^2
*b^3)/3) + tan(c/2 + (d*x)/2)^10*(70*a^4*b + (32*b^5)/3 - (160*a^2*b^3)/3) - tan(c/2 + (d*x)/2)^8*((32*b^5)/3
- (400*a^2*b^3)/3) + 40*a^2*b^3*tan(c/2 + (d*x)/2)^4 + 40*a^2*b^3*tan(c/2 + (d*x)/2)^12 + 10*a^4*b*tan(c/2 + (
d*x)/2)^2 + 10*a^4*b*tan(c/2 + (d*x)/2)^14)/(d*(8*tan(c/2 + (d*x)/2)^2 + 28*tan(c/2 + (d*x)/2)^4 + 56*tan(c/2
+ (d*x)/2)^6 + 70*tan(c/2 + (d*x)/2)^8 + 56*tan(c/2 + (d*x)/2)^10 + 28*tan(c/2 + (d*x)/2)^12 + 8*tan(c/2 + (d*
x)/2)^14 + tan(c/2 + (d*x)/2)^16 + 1)) - (5*a*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2)*(7*a^4 + 3*b^4 + 10*a^2*b^2
))/(64*d) + (5*a*atan((5*a*tan(c/2 + (d*x)/2)*(7*a^2 + 3*b^2)*(a^2 + b^2))/(64*((15*a*b^4)/64 + (35*a^5)/64 +
(25*a^3*b^2)/32)))*(7*a^2 + 3*b^2)*(a^2 + b^2))/(64*d)

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sympy [A]  time = 11.54, size = 826, normalized size = 1.94 \[ \begin {cases} \frac {35 a^{5} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {35 a^{5} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {105 a^{5} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {35 a^{5} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {35 a^{5} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {35 a^{5} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {385 a^{5} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {511 a^{5} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} + \frac {93 a^{5} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {5 a^{4} b \cos ^{8}{\left (c + d x \right )}}{8 d} + \frac {25 a^{3} b^{2} x \sin ^{8}{\left (c + d x \right )}}{64} + \frac {25 a^{3} b^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {75 a^{3} b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{32} + \frac {25 a^{3} b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{16} + \frac {25 a^{3} b^{2} x \cos ^{8}{\left (c + d x \right )}}{64} + \frac {25 a^{3} b^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{64 d} + \frac {275 a^{3} b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{192 d} + \frac {365 a^{3} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{192 d} - \frac {25 a^{3} b^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} + \frac {5 a^{2} b^{3} \sin ^{8}{\left (c + d x \right )}}{12 d} + \frac {5 a^{2} b^{3} \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {5 a^{2} b^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{2 d} + \frac {15 a b^{4} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {15 a b^{4} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {45 a b^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {15 a b^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {15 a b^{4} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {15 a b^{4} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {55 a b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {55 a b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {15 a b^{4} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} + \frac {b^{5} \sin ^{8}{\left (c + d x \right )}}{24 d} + \frac {b^{5} \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{6 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\relax (c )} + b \sin {\relax (c )}\right )^{5} \cos ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**3*(a*cos(d*x+c)+b*sin(d*x+c))**5,x)

[Out]

Piecewise((35*a**5*x*sin(c + d*x)**8/128 + 35*a**5*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 105*a**5*x*sin(c + d
*x)**4*cos(c + d*x)**4/64 + 35*a**5*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 35*a**5*x*cos(c + d*x)**8/128 + 35*
a**5*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 385*a**5*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) + 511*a**5*sin(c
+ d*x)**3*cos(c + d*x)**5/(384*d) + 93*a**5*sin(c + d*x)*cos(c + d*x)**7/(128*d) - 5*a**4*b*cos(c + d*x)**8/(8
*d) + 25*a**3*b**2*x*sin(c + d*x)**8/64 + 25*a**3*b**2*x*sin(c + d*x)**6*cos(c + d*x)**2/16 + 75*a**3*b**2*x*s
in(c + d*x)**4*cos(c + d*x)**4/32 + 25*a**3*b**2*x*sin(c + d*x)**2*cos(c + d*x)**6/16 + 25*a**3*b**2*x*cos(c +
d*x)**8/64 + 25*a**3*b**2*sin(c + d*x)**7*cos(c + d*x)/(64*d) + 275*a**3*b**2*sin(c + d*x)**5*cos(c + d*x)**3
/(192*d) + 365*a**3*b**2*sin(c + d*x)**3*cos(c + d*x)**5/(192*d) - 25*a**3*b**2*sin(c + d*x)*cos(c + d*x)**7/(
64*d) + 5*a**2*b**3*sin(c + d*x)**8/(12*d) + 5*a**2*b**3*sin(c + d*x)**6*cos(c + d*x)**2/(3*d) + 5*a**2*b**3*s
in(c + d*x)**4*cos(c + d*x)**4/(2*d) + 15*a*b**4*x*sin(c + d*x)**8/128 + 15*a*b**4*x*sin(c + d*x)**6*cos(c + d
*x)**2/32 + 45*a*b**4*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 15*a*b**4*x*sin(c + d*x)**2*cos(c + d*x)**6/32 +
15*a*b**4*x*cos(c + d*x)**8/128 + 15*a*b**4*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 55*a*b**4*sin(c + d*x)**5*c
os(c + d*x)**3/(128*d) - 55*a*b**4*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) - 15*a*b**4*sin(c + d*x)*cos(c + d*
x)**7/(128*d) + b**5*sin(c + d*x)**8/(24*d) + b**5*sin(c + d*x)**6*cos(c + d*x)**2/(6*d), Ne(d, 0)), (x*(a*cos
(c) + b*sin(c))**5*cos(c)**3, True))

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