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

Optimal. Leaf size=265 \[ -\frac{3 a^2 b \cos ^8(c+d x)}{8 d}+\frac{a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 a^3 x}{128}-\frac{3 a b^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{a b^2 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{5 a b^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{15 a b^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{15}{128} a b^2 x+\frac{b^3 \cos ^8(c+d x)}{8 d}-\frac{b^3 \cos ^6(c+d x)}{6 d} \]

[Out]

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

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Rubi [A]  time = 0.245691, antiderivative size = 265, normalized size of antiderivative = 1., number of steps used = 17, 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} \[ -\frac{3 a^2 b \cos ^8(c+d x)}{8 d}+\frac{a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 a^3 x}{128}-\frac{3 a b^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{a b^2 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{5 a b^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{15 a b^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{15}{128} a b^2 x+\frac{b^3 \cos ^8(c+d x)}{8 d}-\frac{b^3 \cos ^6(c+d x)}{6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.474988, size = 235, normalized size = 0.89 \[ \frac{5 a \left (7 a^2+3 b^2\right ) (c+d x)}{128 d}+\frac{a \left (14 a^2+3 b^2\right ) \sin (2 (c+d x))}{64 d}+\frac{a \left (7 a^2-3 b^2\right ) \sin (4 (c+d x))}{128 d}+\frac{a \left (2 a^2-3 b^2\right ) \sin (6 (c+d x))}{192 d}+\frac{a \left (a^2-3 b^2\right ) \sin (8 (c+d x))}{1024 d}-\frac{3 b \left (7 a^2+b^2\right ) \cos (2 (c+d x))}{128 d}-\frac{b \left (21 a^2+b^2\right ) \cos (4 (c+d x))}{256 d}-\frac{b \left (9 a^2-b^2\right ) \cos (6 (c+d x))}{384 d}-\frac{b \left (3 a^2-b^2\right ) \cos (8 (c+d x))}{1024 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*a*(7*a^2 + 3*b^2)*(c + d*x))/(128*d) - (3*b*(7*a^2 + b^2)*Cos[2*(c + d*x)])/(128*d) - (b*(21*a^2 + b^2)*Cos
[4*(c + d*x)])/(256*d) - (b*(9*a^2 - b^2)*Cos[6*(c + d*x)])/(384*d) - (b*(3*a^2 - b^2)*Cos[8*(c + d*x)])/(1024
*d) + (a*(14*a^2 + 3*b^2)*Sin[2*(c + d*x)])/(64*d) + (a*(7*a^2 - 3*b^2)*Sin[4*(c + d*x)])/(128*d) + (a*(2*a^2
- 3*b^2)*Sin[6*(c + d*x)])/(192*d) + (a*(a^2 - 3*b^2)*Sin[8*(c + d*x)])/(1024*d)

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Maple [A]  time = 0.076, size = 175, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ({b}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{8}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{24}} \right ) +3\,a{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{3\,{a}^{2}b \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{8}}+{a}^{3} \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)^5*(a*cos(d*x+c)+b*sin(d*x+c))^3,x)

[Out]

1/d*(b^3*(-1/8*sin(d*x+c)^2*cos(d*x+c)^6-1/24*cos(d*x+c)^6)+3*a*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)-3/8*a^2*b*cos(d*x+c)^8+a^3*(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 = 1.09768, size = 220, normalized size = 0.83 \begin{align*} -\frac{1152 \, a^{2} 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^{3} - 3 \,{\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 b^{2} - 128 \,{\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 )} b^{3}}{3072 \, d} \end{align*}

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))^3,x, algorithm="maxima")

[Out]

-1/3072*(1152*a^2*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^3 - 3*(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*b^2 - 128*(3*sin(d*x + c)^8 - 8*sin(d*x + c)^6 + 6*sin(d*x + c)^4)*b^3)/d

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Fricas [A]  time = 0.533803, size = 350, normalized size = 1.32 \begin{align*} -\frac{64 \, b^{3} \cos \left (d x + c\right )^{6} + 48 \,{\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{8} - 15 \,{\left (7 \, a^{3} + 3 \, a b^{2}\right )} d x -{\left (48 \,{\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{7} + 8 \,{\left (7 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 10 \,{\left (7 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 15 \,{\left (7 \, a^{3} + 3 \, a b^{2}\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)^5*(a*cos(d*x+c)+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/384*(64*b^3*cos(d*x + c)^6 + 48*(3*a^2*b - b^3)*cos(d*x + c)^8 - 15*(7*a^3 + 3*a*b^2)*d*x - (48*(a^3 - 3*a*
b^2)*cos(d*x + c)^7 + 8*(7*a^3 + 3*a*b^2)*cos(d*x + c)^5 + 10*(7*a^3 + 3*a*b^2)*cos(d*x + c)^3 + 15*(7*a^3 + 3
*a*b^2)*cos(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 22.5692, size = 508, normalized size = 1.92 \begin{align*} \begin{cases} \frac{35 a^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac{35 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac{105 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac{35 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac{35 a^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac{35 a^{3} \sin ^{7}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{128 d} + \frac{385 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac{511 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} + \frac{93 a^{3} \sin{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac{3 a^{2} b \cos ^{8}{\left (c + d x \right )}}{8 d} + \frac{15 a b^{2} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac{15 a b^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac{45 a b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac{15 a b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac{15 a b^{2} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac{15 a b^{2} \sin ^{7}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{128 d} + \frac{55 a b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac{73 a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac{15 a b^{2} \sin{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac{b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{6 d} - \frac{b^{3} \cos ^{8}{\left (c + d x \right )}}{24 d} & \text{for}\: d \neq 0 \\x \left (a \cos{\left (c \right )} + b \sin{\left (c \right )}\right )^{3} \cos ^{5}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

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))**3,x)

[Out]

Piecewise((35*a**3*x*sin(c + d*x)**8/128 + 35*a**3*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 105*a**3*x*sin(c + d
*x)**4*cos(c + d*x)**4/64 + 35*a**3*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 35*a**3*x*cos(c + d*x)**8/128 + 35*
a**3*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 385*a**3*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) + 511*a**3*sin(c
+ d*x)**3*cos(c + d*x)**5/(384*d) + 93*a**3*sin(c + d*x)*cos(c + d*x)**7/(128*d) - 3*a**2*b*cos(c + d*x)**8/(8
*d) + 15*a*b**2*x*sin(c + d*x)**8/128 + 15*a*b**2*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 45*a*b**2*x*sin(c + d
*x)**4*cos(c + d*x)**4/64 + 15*a*b**2*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 15*a*b**2*x*cos(c + d*x)**8/128 +
15*a*b**2*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 55*a*b**2*sin(c + d*x)**5*cos(c + d*x)**3/(128*d) + 73*a*b**
2*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) - 15*a*b**2*sin(c + d*x)*cos(c + d*x)**7/(128*d) - b**3*sin(c + d*x)
**2*cos(c + d*x)**6/(6*d) - b**3*cos(c + d*x)**8/(24*d), Ne(d, 0)), (x*(a*cos(c) + b*sin(c))**3*cos(c)**5, Tru
e))

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Giac [A]  time = 1.18124, size = 294, normalized size = 1.11 \begin{align*} \frac{5}{128} \,{\left (7 \, a^{3} + 3 \, a b^{2}\right )} x - \frac{{\left (3 \, a^{2} b - b^{3}\right )} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac{{\left (9 \, a^{2} b - b^{3}\right )} \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac{{\left (21 \, a^{2} b + b^{3}\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac{3 \,{\left (7 \, a^{2} b + b^{3}\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac{{\left (a^{3} - 3 \, a b^{2}\right )} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac{{\left (2 \, a^{3} - 3 \, a b^{2}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{{\left (7 \, a^{3} - 3 \, a b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac{{\left (14 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \end{align*}

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))^3,x, algorithm="giac")

[Out]

5/128*(7*a^3 + 3*a*b^2)*x - 1/1024*(3*a^2*b - b^3)*cos(8*d*x + 8*c)/d - 1/384*(9*a^2*b - b^3)*cos(6*d*x + 6*c)
/d - 1/256*(21*a^2*b + b^3)*cos(4*d*x + 4*c)/d - 3/128*(7*a^2*b + b^3)*cos(2*d*x + 2*c)/d + 1/1024*(a^3 - 3*a*
b^2)*sin(8*d*x + 8*c)/d + 1/192*(2*a^3 - 3*a*b^2)*sin(6*d*x + 6*c)/d + 1/128*(7*a^3 - 3*a*b^2)*sin(4*d*x + 4*c
)/d + 1/64*(14*a^3 + 3*a*b^2)*sin(2*d*x + 2*c)/d