3.438 \(\int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx\)

Optimal. Leaf size=247 \[ -\frac{\left (168 a^2 b^2+35 a^4+24 b^4\right ) \sin ^3(c+d x)}{105 d}+\frac{\left (168 a^2 b^2+35 a^4+24 b^4\right ) \sin (c+d x)}{35 d}+\frac{b^2 \left (37 a^2+6 b^2\right ) \sin (c+d x) \cos ^4(c+d x)}{35 d}+\frac{a b \left (6 a^2+5 b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{6 d}+\frac{a b \left (6 a^2+5 b^2\right ) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{1}{4} a b x \left (6 a^2+5 b^2\right )+\frac{8 a b^3 \sin (c+d x) \cos ^5(c+d x)}{21 d}+\frac{b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))^2}{7 d} \]

[Out]

(a*b*(6*a^2 + 5*b^2)*x)/4 + ((35*a^4 + 168*a^2*b^2 + 24*b^4)*Sin[c + d*x])/(35*d) + (a*b*(6*a^2 + 5*b^2)*Cos[c
+ d*x]*Sin[c + d*x])/(4*d) + (a*b*(6*a^2 + 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(6*d) + (b^2*(37*a^2 + 6*b^2)*
Cos[c + d*x]^4*Sin[c + d*x])/(35*d) + (8*a*b^3*Cos[c + d*x]^5*Sin[c + d*x])/(21*d) + (b^2*Cos[c + d*x]^4*(a +
b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) - ((35*a^4 + 168*a^2*b^2 + 24*b^4)*Sin[c + d*x]^3)/(105*d)

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Rubi [A]  time = 0.400815, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2793, 3033, 3023, 2748, 2633, 2635, 8} \[ -\frac{\left (168 a^2 b^2+35 a^4+24 b^4\right ) \sin ^3(c+d x)}{105 d}+\frac{\left (168 a^2 b^2+35 a^4+24 b^4\right ) \sin (c+d x)}{35 d}+\frac{b^2 \left (37 a^2+6 b^2\right ) \sin (c+d x) \cos ^4(c+d x)}{35 d}+\frac{a b \left (6 a^2+5 b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{6 d}+\frac{a b \left (6 a^2+5 b^2\right ) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{1}{4} a b x \left (6 a^2+5 b^2\right )+\frac{8 a b^3 \sin (c+d x) \cos ^5(c+d x)}{21 d}+\frac{b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))^2}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3*(a + b*Cos[c + d*x])^4,x]

[Out]

(a*b*(6*a^2 + 5*b^2)*x)/4 + ((35*a^4 + 168*a^2*b^2 + 24*b^4)*Sin[c + d*x])/(35*d) + (a*b*(6*a^2 + 5*b^2)*Cos[c
+ d*x]*Sin[c + d*x])/(4*d) + (a*b*(6*a^2 + 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(6*d) + (b^2*(37*a^2 + 6*b^2)*
Cos[c + d*x]^4*Sin[c + d*x])/(35*d) + (8*a*b^3*Cos[c + d*x]^5*Sin[c + d*x])/(21*d) + (b^2*Cos[c + d*x]^4*(a +
b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) - ((35*a^4 + 168*a^2*b^2 + 24*b^4)*Sin[c + d*x]^3)/(105*d)

Rule 2793

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -S
imp[(b^2*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n)), x] + Dist[1/(d
*(m + n)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d*(m + n) + b^2*(b*c*(m - 2) + a*d
*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n -
2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &
& NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] ||
(EqQ[a, 0] && NeQ[c, 0])))

Rule 3033

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*d*Cos[e + f*x]*Sin[e + f*x]*(a + b
*Sin[e + f*x])^(m + 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*c*
(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e +
f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!LtQ[m, -1]

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

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 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]

Rubi steps

\begin{align*} \int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx &=\frac{b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac{1}{7} \int \cos ^3(c+d x) (a+b \cos (c+d x)) \left (a \left (7 a^2+4 b^2\right )+3 b \left (7 a^2+2 b^2\right ) \cos (c+d x)+16 a b^2 \cos ^2(c+d x)\right ) \, dx\\ &=\frac{8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac{b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac{1}{42} \int \cos ^3(c+d x) \left (6 a^2 \left (7 a^2+4 b^2\right )+28 a b \left (6 a^2+5 b^2\right ) \cos (c+d x)+6 b^2 \left (37 a^2+6 b^2\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{b^2 \left (37 a^2+6 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{35 d}+\frac{8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac{b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac{1}{210} \int \cos ^3(c+d x) \left (6 \left (35 a^4+168 a^2 b^2+24 b^4\right )+140 a b \left (6 a^2+5 b^2\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{b^2 \left (37 a^2+6 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{35 d}+\frac{8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac{b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac{1}{3} \left (2 a b \left (6 a^2+5 b^2\right )\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{35} \left (35 a^4+168 a^2 b^2+24 b^4\right ) \int \cos ^3(c+d x) \, dx\\ &=\frac{a b \left (6 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{6 d}+\frac{b^2 \left (37 a^2+6 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{35 d}+\frac{8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac{b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac{1}{2} \left (a b \left (6 a^2+5 b^2\right )\right ) \int \cos ^2(c+d x) \, dx-\frac{\left (35 a^4+168 a^2 b^2+24 b^4\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d}\\ &=\frac{\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin (c+d x)}{35 d}+\frac{a b \left (6 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 d}+\frac{a b \left (6 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{6 d}+\frac{b^2 \left (37 a^2+6 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{35 d}+\frac{8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac{b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}-\frac{\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin ^3(c+d x)}{105 d}+\frac{1}{4} \left (a b \left (6 a^2+5 b^2\right )\right ) \int 1 \, dx\\ &=\frac{1}{4} a b \left (6 a^2+5 b^2\right ) x+\frac{\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin (c+d x)}{35 d}+\frac{a b \left (6 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{4 d}+\frac{a b \left (6 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{6 d}+\frac{b^2 \left (37 a^2+6 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{35 d}+\frac{8 a b^3 \cos ^5(c+d x) \sin (c+d x)}{21 d}+\frac{b^2 \cos ^4(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}-\frac{\left (35 a^4+168 a^2 b^2+24 b^4\right ) \sin ^3(c+d x)}{105 d}\\ \end{align*}

Mathematica [A]  time = 0.408403, size = 181, normalized size = 0.73 \[ \frac{1680 a b \left (6 a^2+5 b^2\right ) (c+d x)+21 b^2 \left (24 a^2+7 b^2\right ) \sin (5 (c+d x))+420 a b \left (16 a^2+15 b^2\right ) \sin (2 (c+d x))+420 a b \left (2 a^2+3 b^2\right ) \sin (4 (c+d x))+105 \left (240 a^2 b^2+48 a^4+35 b^4\right ) \sin (c+d x)+35 \left (120 a^2 b^2+16 a^4+21 b^4\right ) \sin (3 (c+d x))+140 a b^3 \sin (6 (c+d x))+15 b^4 \sin (7 (c+d x))}{6720 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3*(a + b*Cos[c + d*x])^4,x]

[Out]

(1680*a*b*(6*a^2 + 5*b^2)*(c + d*x) + 105*(48*a^4 + 240*a^2*b^2 + 35*b^4)*Sin[c + d*x] + 420*a*b*(16*a^2 + 15*
b^2)*Sin[2*(c + d*x)] + 35*(16*a^4 + 120*a^2*b^2 + 21*b^4)*Sin[3*(c + d*x)] + 420*a*b*(2*a^2 + 3*b^2)*Sin[4*(c
+ d*x)] + 21*b^2*(24*a^2 + 7*b^2)*Sin[5*(c + d*x)] + 140*a*b^3*Sin[6*(c + d*x)] + 15*b^4*Sin[7*(c + d*x)])/(6
720*d)

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Maple [A]  time = 0.052, size = 190, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({\frac{{b}^{4}\sin \left ( dx+c \right ) }{7} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) }+4\,a{b}^{3} \left ( 1/6\, \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}{16}}+{\frac{5\,c}{16}} \right ) +{\frac{6\,{a}^{2}{b}^{2}\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+4\,{a}^{3}b \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{\frac{{a}^{4} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*(a+b*cos(d*x+c))^4,x)

[Out]

1/d*(1/7*b^4*(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*a*b^3*(1/6*(cos(d*x+c)^5+5/4*c
os(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c)+6/5*a^2*b^2*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d
*x+c)+4*a^3*b*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+1/3*a^4*(2+cos(d*x+c)^2)*sin(d*x+c)
)

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Maxima [A]  time = 0.978239, size = 259, normalized size = 1.05 \begin{align*} -\frac{560 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{4} - 210 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} b - 672 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{2} b^{2} + 35 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a b^{3} + 48 \,{\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} b^{4}}{1680 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/1680*(560*(sin(d*x + c)^3 - 3*sin(d*x + c))*a^4 - 210*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c
))*a^3*b - 672*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*a^2*b^2 + 35*(4*sin(2*d*x + 2*c)^3 - 6
0*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*a*b^3 + 48*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35
*sin(d*x + c)^3 - 35*sin(d*x + c))*b^4)/d

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Fricas [A]  time = 1.97599, size = 416, normalized size = 1.68 \begin{align*} \frac{105 \,{\left (6 \, a^{3} b + 5 \, a b^{3}\right )} d x +{\left (60 \, b^{4} \cos \left (d x + c\right )^{6} + 280 \, a b^{3} \cos \left (d x + c\right )^{5} + 72 \,{\left (7 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + 280 \, a^{4} + 1344 \, a^{2} b^{2} + 192 \, b^{4} + 70 \,{\left (6 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} + 4 \,{\left (35 \, a^{4} + 168 \, a^{2} b^{2} + 24 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + 105 \,{\left (6 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{420 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

1/420*(105*(6*a^3*b + 5*a*b^3)*d*x + (60*b^4*cos(d*x + c)^6 + 280*a*b^3*cos(d*x + c)^5 + 72*(7*a^2*b^2 + b^4)*
cos(d*x + c)^4 + 280*a^4 + 1344*a^2*b^2 + 192*b^4 + 70*(6*a^3*b + 5*a*b^3)*cos(d*x + c)^3 + 4*(35*a^4 + 168*a^
2*b^2 + 24*b^4)*cos(d*x + c)^2 + 105*(6*a^3*b + 5*a*b^3)*cos(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 8.5724, size = 495, normalized size = 2. \begin{align*} \begin{cases} \frac{2 a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{a^{4} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{3 a^{3} b x \sin ^{4}{\left (c + d x \right )}}{2} + 3 a^{3} b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + \frac{3 a^{3} b x \cos ^{4}{\left (c + d x \right )}}{2} + \frac{3 a^{3} b \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{5 a^{3} b \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac{16 a^{2} b^{2} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac{8 a^{2} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{6 a^{2} b^{2} \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{5 a b^{3} x \sin ^{6}{\left (c + d x \right )}}{4} + \frac{15 a b^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{15 a b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4} + \frac{5 a b^{3} x \cos ^{6}{\left (c + d x \right )}}{4} + \frac{5 a b^{3} \sin ^{5}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{4 d} + \frac{10 a b^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac{11 a b^{3} \sin{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{4 d} + \frac{16 b^{4} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac{8 b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac{2 b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{b^{4} \sin{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \cos{\left (c \right )}\right )^{4} \cos ^{3}{\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)**3*(a+b*cos(d*x+c))**4,x)

[Out]

Piecewise((2*a**4*sin(c + d*x)**3/(3*d) + a**4*sin(c + d*x)*cos(c + d*x)**2/d + 3*a**3*b*x*sin(c + d*x)**4/2 +
3*a**3*b*x*sin(c + d*x)**2*cos(c + d*x)**2 + 3*a**3*b*x*cos(c + d*x)**4/2 + 3*a**3*b*sin(c + d*x)**3*cos(c +
d*x)/(2*d) + 5*a**3*b*sin(c + d*x)*cos(c + d*x)**3/(2*d) + 16*a**2*b**2*sin(c + d*x)**5/(5*d) + 8*a**2*b**2*si
n(c + d*x)**3*cos(c + d*x)**2/d + 6*a**2*b**2*sin(c + d*x)*cos(c + d*x)**4/d + 5*a*b**3*x*sin(c + d*x)**6/4 +
15*a*b**3*x*sin(c + d*x)**4*cos(c + d*x)**2/4 + 15*a*b**3*x*sin(c + d*x)**2*cos(c + d*x)**4/4 + 5*a*b**3*x*cos
(c + d*x)**6/4 + 5*a*b**3*sin(c + d*x)**5*cos(c + d*x)/(4*d) + 10*a*b**3*sin(c + d*x)**3*cos(c + d*x)**3/(3*d)
+ 11*a*b**3*sin(c + d*x)*cos(c + d*x)**5/(4*d) + 16*b**4*sin(c + d*x)**7/(35*d) + 8*b**4*sin(c + d*x)**5*cos(
c + d*x)**2/(5*d) + 2*b**4*sin(c + d*x)**3*cos(c + d*x)**4/d + b**4*sin(c + d*x)*cos(c + d*x)**6/d, Ne(d, 0)),
(x*(a + b*cos(c))**4*cos(c)**3, True))

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Giac [A]  time = 1.36568, size = 266, normalized size = 1.08 \begin{align*} \frac{b^{4} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac{a b^{3} \sin \left (6 \, d x + 6 \, c\right )}{48 \, d} + \frac{1}{4} \,{\left (6 \, a^{3} b + 5 \, a b^{3}\right )} x + \frac{{\left (24 \, a^{2} b^{2} + 7 \, b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac{{\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{16 \, d} + \frac{{\left (16 \, a^{4} + 120 \, a^{2} b^{2} + 21 \, b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac{{\left (16 \, a^{3} b + 15 \, a b^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{16 \, d} + \frac{{\left (48 \, a^{4} + 240 \, a^{2} b^{2} + 35 \, b^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+b*cos(d*x+c))^4,x, algorithm="giac")

[Out]

1/448*b^4*sin(7*d*x + 7*c)/d + 1/48*a*b^3*sin(6*d*x + 6*c)/d + 1/4*(6*a^3*b + 5*a*b^3)*x + 1/320*(24*a^2*b^2 +
7*b^4)*sin(5*d*x + 5*c)/d + 1/16*(2*a^3*b + 3*a*b^3)*sin(4*d*x + 4*c)/d + 1/192*(16*a^4 + 120*a^2*b^2 + 21*b^
4)*sin(3*d*x + 3*c)/d + 1/16*(16*a^3*b + 15*a*b^3)*sin(2*d*x + 2*c)/d + 1/64*(48*a^4 + 240*a^2*b^2 + 35*b^4)*s
in(d*x + c)/d