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

Optimal. Leaf size=260 \[ \frac{5 a b \left (10 a^2+11 \left (b^2+c^2\right )\right ) \sin (d+e x)}{24 e}-\frac{5 a c \left (10 a^2+11 \left (b^2+c^2\right )\right ) \cos (d+e x)}{24 e}-\frac{\left (c \left (26 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (26 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right ) (a+b \cos (d+e x)+c \sin (d+e x))}{24 e}+\frac{1}{8} x \left (24 a^2 \left (b^2+c^2\right )+8 a^4+3 \left (b^2+c^2\right )^2\right )-\frac{(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^3}{4 e}-\frac{7 (a c \cos (d+e x)-a b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{12 e} \]

[Out]

((8*a^4 + 24*a^2*(b^2 + c^2) + 3*(b^2 + c^2)^2)*x)/8 - (5*a*c*(10*a^2 + 11*(b^2 + c^2))*Cos[d + e*x])/(24*e) +
(5*a*b*(10*a^2 + 11*(b^2 + c^2))*Sin[d + e*x])/(24*e) - (7*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x])*(a + b*Cos[d
+ e*x] + c*Sin[d + e*x])^2)/(12*e) - ((c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])
^3)/(4*e) - ((a + b*Cos[d + e*x] + c*Sin[d + e*x])*(c*(26*a^2 + 9*(b^2 + c^2))*Cos[d + e*x] - b*(26*a^2 + 9*(b
^2 + c^2))*Sin[d + e*x]))/(24*e)

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Rubi [A]  time = 0.399687, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3120, 3146, 2637, 2638} \[ \frac{5 a b \left (10 a^2+11 \left (b^2+c^2\right )\right ) \sin (d+e x)}{24 e}-\frac{5 a c \left (10 a^2+11 \left (b^2+c^2\right )\right ) \cos (d+e x)}{24 e}-\frac{\left (c \left (26 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (26 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right ) (a+b \cos (d+e x)+c \sin (d+e x))}{24 e}+\frac{1}{8} x \left (24 a^2 \left (b^2+c^2\right )+8 a^4+3 \left (b^2+c^2\right )^2\right )-\frac{(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^3}{4 e}-\frac{7 (a c \cos (d+e x)-a b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{12 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*a^4 + 24*a^2*(b^2 + c^2) + 3*(b^2 + c^2)^2)*x)/8 - (5*a*c*(10*a^2 + 11*(b^2 + c^2))*Cos[d + e*x])/(24*e) +
(5*a*b*(10*a^2 + 11*(b^2 + c^2))*Sin[d + e*x])/(24*e) - (7*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x])*(a + b*Cos[d
+ e*x] + c*Sin[d + e*x])^2)/(12*e) - ((c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])
^3)/(4*e) - ((a + b*Cos[d + e*x] + c*Sin[d + e*x])*(c*(26*a^2 + 9*(b^2 + c^2))*Cos[d + e*x] - b*(26*a^2 + 9*(b
^2 + c^2))*Sin[d + e*x]))/(24*e)

Rule 3120

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_), x_Symbol] :> -Simp[((c*Cos[d
+ e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1))/(e*n), x] + Dist[1/n, Int[Simp[n*a^2 +
(n - 1)*(b^2 + c^2) + a*b*(2*n - 1)*Cos[d + e*x] + a*c*(2*n - 1)*Sin[d + e*x], x]*(a + b*Cos[d + e*x] + c*Sin
[d + e*x])^(n - 2), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && GtQ[n, 1]

Rule 3146

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

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int (a+b \cos (d+e x)+c \sin (d+e x))^4 \, dx &=-\frac{(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^3}{4 e}+\frac{1}{4} \int (a+b \cos (d+e x)+c \sin (d+e x))^2 \left (4 a^2+3 \left (b^2+c^2\right )+7 a b \cos (d+e x)+7 a c \sin (d+e x)\right ) \, dx\\ &=-\frac{7 (a c \cos (d+e x)-a b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{12 e}-\frac{(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^3}{4 e}+\frac{\int (a+b \cos (d+e x)+c \sin (d+e x)) \left (a^2 \left (12 a^2+23 \left (b^2+c^2\right )\right )+a b \left (26 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)+a c \left (26 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right ) \, dx}{12 a}\\ &=-\frac{7 (a c \cos (d+e x)-a b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{12 e}-\frac{(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^3}{4 e}-\frac{(a+b \cos (d+e x)+c \sin (d+e x)) \left (c \left (26 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (26 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{24 e}+\frac{\int \left (3 a^2 \left (8 a^4+24 a^2 \left (b^2+c^2\right )+3 \left (b^2+c^2\right )^2\right )+5 a^3 b \left (10 a^2+11 \left (b^2+c^2\right )\right ) \cos (d+e x)+5 a^3 c \left (10 a^2+11 \left (b^2+c^2\right )\right ) \sin (d+e x)\right ) \, dx}{24 a^2}\\ &=\frac{1}{8} \left (8 a^4+24 a^2 \left (b^2+c^2\right )+3 \left (b^2+c^2\right )^2\right ) x-\frac{7 (a c \cos (d+e x)-a b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{12 e}-\frac{(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^3}{4 e}-\frac{(a+b \cos (d+e x)+c \sin (d+e x)) \left (c \left (26 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (26 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{24 e}+\frac{1}{24} \left (5 a b \left (10 a^2+11 \left (b^2+c^2\right )\right )\right ) \int \cos (d+e x) \, dx+\frac{1}{24} \left (5 a c \left (10 a^2+11 \left (b^2+c^2\right )\right )\right ) \int \sin (d+e x) \, dx\\ &=\frac{1}{8} \left (8 a^4+24 a^2 \left (b^2+c^2\right )+3 \left (b^2+c^2\right )^2\right ) x-\frac{5 a c \left (10 a^2+11 \left (b^2+c^2\right )\right ) \cos (d+e x)}{24 e}+\frac{5 a b \left (10 a^2+11 \left (b^2+c^2\right )\right ) \sin (d+e x)}{24 e}-\frac{7 (a c \cos (d+e x)-a b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{12 e}-\frac{(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^3}{4 e}-\frac{(a+b \cos (d+e x)+c \sin (d+e x)) \left (c \left (26 a^2+9 \left (b^2+c^2\right )\right ) \cos (d+e x)-b \left (26 a^2+9 \left (b^2+c^2\right )\right ) \sin (d+e x)\right )}{24 e}\\ \end{align*}

Mathematica [A]  time = 1.08928, size = 237, normalized size = 0.91 \[ \frac{12 \left (24 a^2 \left (b^2+c^2\right )+8 a^4+3 \left (b^2+c^2\right )^2\right ) (d+e x)+96 a b \left (4 a^2+3 \left (b^2+c^2\right )\right ) \sin (d+e x)+24 \left (b^2-c^2\right ) \left (6 a^2+b^2+c^2\right ) \sin (2 (d+e x))-96 a c \left (4 a^2+3 \left (b^2+c^2\right )\right ) \cos (d+e x)-48 b c \left (6 a^2+b^2+c^2\right ) \cos (2 (d+e x))+32 a b \left (b^2-3 c^2\right ) \sin (3 (d+e x))+32 a c \left (c^2-3 b^2\right ) \cos (3 (d+e x))+3 \left (-6 b^2 c^2+b^4+c^4\right ) \sin (4 (d+e x))-12 b c \left (b^2-c^2\right ) \cos (4 (d+e x))}{96 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*(8*a^4 + 24*a^2*(b^2 + c^2) + 3*(b^2 + c^2)^2)*(d + e*x) - 96*a*c*(4*a^2 + 3*(b^2 + c^2))*Cos[d + e*x] - 4
8*b*c*(6*a^2 + b^2 + c^2)*Cos[2*(d + e*x)] + 32*a*c*(-3*b^2 + c^2)*Cos[3*(d + e*x)] - 12*b*c*(b^2 - c^2)*Cos[4
*(d + e*x)] + 96*a*b*(4*a^2 + 3*(b^2 + c^2))*Sin[d + e*x] + 24*(b^2 - c^2)*(6*a^2 + b^2 + c^2)*Sin[2*(d + e*x)
] + 32*a*b*(b^2 - 3*c^2)*Sin[3*(d + e*x)] + 3*(b^4 - 6*b^2*c^2 + c^4)*Sin[4*(d + e*x)])/(96*e)

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Maple [A]  time = 0.072, size = 335, normalized size = 1.3 \begin{align*}{\frac{1}{e} \left ({b}^{4} \left ({\frac{\sin \left ( ex+d \right ) }{4} \left ( \left ( \cos \left ( ex+d \right ) \right ) ^{3}+{\frac{3\,\cos \left ( ex+d \right ) }{2}} \right ) }+{\frac{3\,ex}{8}}+{\frac{3\,d}{8}} \right ) +{c}^{4} \left ( -{\frac{\cos \left ( ex+d \right ) }{4} \left ( \left ( \sin \left ( ex+d \right ) \right ) ^{3}+{\frac{3\,\sin \left ( ex+d \right ) }{2}} \right ) }+{\frac{3\,ex}{8}}+{\frac{3\,d}{8}} \right ) +{a}^{4} \left ( ex+d \right ) -6\,{a}^{2}bc \left ( \cos \left ( ex+d \right ) \right ) ^{2}-4\,a{b}^{2}c \left ( \cos \left ( ex+d \right ) \right ) ^{3}+4\,ab{c}^{2} \left ( \sin \left ( ex+d \right ) \right ) ^{3}+4\,{a}^{3}b\sin \left ( ex+d \right ) -4\,\cos \left ( ex+d \right ){a}^{3}c+6\,{a}^{2}{b}^{2} \left ( 1/2\,\sin \left ( ex+d \right ) \cos \left ( ex+d \right ) +1/2\,ex+d/2 \right ) +6\,{a}^{2}{c}^{2} \left ( -1/2\,\sin \left ( ex+d \right ) \cos \left ( ex+d \right ) +1/2\,ex+d/2 \right ) +{\frac{4\,a{b}^{3} \left ( 2+ \left ( \cos \left ( ex+d \right ) \right ) ^{2} \right ) \sin \left ( ex+d \right ) }{3}}-{\frac{4\,a{c}^{3} \left ( 2+ \left ( \sin \left ( ex+d \right ) \right ) ^{2} \right ) \cos \left ( ex+d \right ) }{3}}- \left ( \cos \left ( ex+d \right ) \right ) ^{4}{b}^{3}c+6\,{b}^{2}{c}^{2} \left ( -1/4\,\sin \left ( ex+d \right ) \left ( \cos \left ( ex+d \right ) \right ) ^{3}+1/8\,\sin \left ( ex+d \right ) \cos \left ( ex+d \right ) +1/8\,ex+d/8 \right ) +b{c}^{3} \left ( \sin \left ( ex+d \right ) \right ) ^{4} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/e*(b^4*(1/4*(cos(e*x+d)^3+3/2*cos(e*x+d))*sin(e*x+d)+3/8*e*x+3/8*d)+c^4*(-1/4*(sin(e*x+d)^3+3/2*sin(e*x+d))*
cos(e*x+d)+3/8*e*x+3/8*d)+a^4*(e*x+d)-6*a^2*b*c*cos(e*x+d)^2-4*a*b^2*c*cos(e*x+d)^3+4*a*b*c^2*sin(e*x+d)^3+4*a
^3*b*sin(e*x+d)-4*cos(e*x+d)*a^3*c+6*a^2*b^2*(1/2*sin(e*x+d)*cos(e*x+d)+1/2*e*x+1/2*d)+6*a^2*c^2*(-1/2*sin(e*x
+d)*cos(e*x+d)+1/2*e*x+1/2*d)+4/3*a*b^3*(2+cos(e*x+d)^2)*sin(e*x+d)-4/3*a*c^3*(2+sin(e*x+d)^2)*cos(e*x+d)-cos(
e*x+d)^4*b^3*c+6*b^2*c^2*(-1/4*sin(e*x+d)*cos(e*x+d)^3+1/8*sin(e*x+d)*cos(e*x+d)+1/8*e*x+1/8*d)+b*c^3*sin(e*x+
d)^4)

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Maxima [A]  time = 1.02385, size = 446, normalized size = 1.72 \begin{align*} -\frac{b^{3} c \cos \left (e x + d\right )^{4}}{e} + \frac{b c^{3} \sin \left (e x + d\right )^{4}}{e} + a^{4} x + \frac{{\left (12 \, e x + 12 \, d + \sin \left (4 \, e x + 4 \, d\right ) + 8 \, \sin \left (2 \, e x + 2 \, d\right )\right )} b^{4}}{32 \, e} + \frac{3 \,{\left (4 \, e x + 4 \, d - \sin \left (4 \, e x + 4 \, d\right )\right )} b^{2} c^{2}}{16 \, e} + \frac{{\left (12 \, e x + 12 \, d + \sin \left (4 \, e x + 4 \, d\right ) - 8 \, \sin \left (2 \, e x + 2 \, d\right )\right )} c^{4}}{32 \, e} - 4 \, a^{3}{\left (\frac{c \cos \left (e x + d\right )}{e} - \frac{b \sin \left (e x + d\right )}{e}\right )} - \frac{3}{2} \,{\left (\frac{4 \, b c \cos \left (e x + d\right )^{2}}{e} - \frac{{\left (2 \, e x + 2 \, d + \sin \left (2 \, e x + 2 \, d\right )\right )} b^{2}}{e} - \frac{{\left (2 \, e x + 2 \, d - \sin \left (2 \, e x + 2 \, d\right )\right )} c^{2}}{e}\right )} a^{2} - \frac{4}{3} \,{\left (\frac{3 \, b^{2} c \cos \left (e x + d\right )^{3}}{e} - \frac{3 \, b c^{2} \sin \left (e x + d\right )^{3}}{e} + \frac{{\left (\sin \left (e x + d\right )^{3} - 3 \, \sin \left (e x + d\right )\right )} b^{3}}{e} - \frac{{\left (\cos \left (e x + d\right )^{3} - 3 \, \cos \left (e x + d\right )\right )} c^{3}}{e}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^3*c*cos(e*x + d)^4/e + b*c^3*sin(e*x + d)^4/e + a^4*x + 1/32*(12*e*x + 12*d + sin(4*e*x + 4*d) + 8*sin(2*e*
x + 2*d))*b^4/e + 3/16*(4*e*x + 4*d - sin(4*e*x + 4*d))*b^2*c^2/e + 1/32*(12*e*x + 12*d + sin(4*e*x + 4*d) - 8
*sin(2*e*x + 2*d))*c^4/e - 4*a^3*(c*cos(e*x + d)/e - b*sin(e*x + d)/e) - 3/2*(4*b*c*cos(e*x + d)^2/e - (2*e*x
+ 2*d + sin(2*e*x + 2*d))*b^2/e - (2*e*x + 2*d - sin(2*e*x + 2*d))*c^2/e)*a^2 - 4/3*(3*b^2*c*cos(e*x + d)^3/e
- 3*b*c^2*sin(e*x + d)^3/e + (sin(e*x + d)^3 - 3*sin(e*x + d))*b^3/e - (cos(e*x + d)^3 - 3*cos(e*x + d))*c^3/e
)*a

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Fricas [A]  time = 2.25867, size = 576, normalized size = 2.22 \begin{align*} -\frac{24 \,{\left (b^{3} c - b c^{3}\right )} \cos \left (e x + d\right )^{4} + 32 \,{\left (3 \, a b^{2} c - a c^{3}\right )} \cos \left (e x + d\right )^{3} - 3 \,{\left (8 \, a^{4} + 24 \, a^{2} b^{2} + 3 \, b^{4} + 3 \, c^{4} + 6 \,{\left (4 \, a^{2} + b^{2}\right )} c^{2}\right )} e x + 48 \,{\left (3 \, a^{2} b c + b c^{3}\right )} \cos \left (e x + d\right )^{2} + 96 \,{\left (a^{3} c + a c^{3}\right )} \cos \left (e x + d\right ) -{\left (96 \, a^{3} b + 64 \, a b^{3} + 96 \, a b c^{2} + 6 \,{\left (b^{4} - 6 \, b^{2} c^{2} + c^{4}\right )} \cos \left (e x + d\right )^{3} + 32 \,{\left (a b^{3} - 3 \, a b c^{2}\right )} \cos \left (e x + d\right )^{2} + 3 \,{\left (24 \, a^{2} b^{2} + 3 \, b^{4} - 5 \, c^{4} - 6 \,{\left (4 \, a^{2} - b^{2}\right )} c^{2}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{24 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(24*(b^3*c - b*c^3)*cos(e*x + d)^4 + 32*(3*a*b^2*c - a*c^3)*cos(e*x + d)^3 - 3*(8*a^4 + 24*a^2*b^2 + 3*b
^4 + 3*c^4 + 6*(4*a^2 + b^2)*c^2)*e*x + 48*(3*a^2*b*c + b*c^3)*cos(e*x + d)^2 + 96*(a^3*c + a*c^3)*cos(e*x + d
) - (96*a^3*b + 64*a*b^3 + 96*a*b*c^2 + 6*(b^4 - 6*b^2*c^2 + c^4)*cos(e*x + d)^3 + 32*(a*b^3 - 3*a*b*c^2)*cos(
e*x + d)^2 + 3*(24*a^2*b^2 + 3*b^4 - 5*c^4 - 6*(4*a^2 - b^2)*c^2)*cos(e*x + d))*sin(e*x + d))/e

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Sympy [A]  time = 2.27997, size = 707, normalized size = 2.72 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**4*x + 4*a**3*b*sin(d + e*x)/e - 4*a**3*c*cos(d + e*x)/e + 3*a**2*b**2*x*sin(d + e*x)**2 + 3*a**2
*b**2*x*cos(d + e*x)**2 + 3*a**2*b**2*sin(d + e*x)*cos(d + e*x)/e + 6*a**2*b*c*sin(d + e*x)**2/e + 3*a**2*c**2
*x*sin(d + e*x)**2 + 3*a**2*c**2*x*cos(d + e*x)**2 - 3*a**2*c**2*sin(d + e*x)*cos(d + e*x)/e + 8*a*b**3*sin(d
+ e*x)**3/(3*e) + 4*a*b**3*sin(d + e*x)*cos(d + e*x)**2/e - 4*a*b**2*c*cos(d + e*x)**3/e + 4*a*b*c**2*sin(d +
e*x)**3/e - 4*a*c**3*sin(d + e*x)**2*cos(d + e*x)/e - 8*a*c**3*cos(d + e*x)**3/(3*e) + 3*b**4*x*sin(d + e*x)**
4/8 + 3*b**4*x*sin(d + e*x)**2*cos(d + e*x)**2/4 + 3*b**4*x*cos(d + e*x)**4/8 + 3*b**4*sin(d + e*x)**3*cos(d +
e*x)/(8*e) + 5*b**4*sin(d + e*x)*cos(d + e*x)**3/(8*e) - b**3*c*cos(d + e*x)**4/e + 3*b**2*c**2*x*sin(d + e*x
)**4/4 + 3*b**2*c**2*x*sin(d + e*x)**2*cos(d + e*x)**2/2 + 3*b**2*c**2*x*cos(d + e*x)**4/4 + 3*b**2*c**2*sin(d
+ e*x)**3*cos(d + e*x)/(4*e) - 3*b**2*c**2*sin(d + e*x)*cos(d + e*x)**3/(4*e) - 2*b*c**3*sin(d + e*x)**2*cos(
d + e*x)**2/e - b*c**3*cos(d + e*x)**4/e + 3*c**4*x*sin(d + e*x)**4/8 + 3*c**4*x*sin(d + e*x)**2*cos(d + e*x)*
*2/4 + 3*c**4*x*cos(d + e*x)**4/8 - 5*c**4*sin(d + e*x)**3*cos(d + e*x)/(8*e) - 3*c**4*sin(d + e*x)*cos(d + e*
x)**3/(8*e), Ne(e, 0)), (x*(a + b*cos(d) + c*sin(d))**4, True))

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Giac [A]  time = 1.12332, size = 386, normalized size = 1.48 \begin{align*} -\frac{1}{8} \,{\left (b^{3} c - b c^{3}\right )} \cos \left (4 \, x e + 4 \, d\right ) e^{\left (-1\right )} - \frac{1}{3} \,{\left (3 \, a b^{2} c - a c^{3}\right )} \cos \left (3 \, x e + 3 \, d\right ) e^{\left (-1\right )} - \frac{1}{2} \,{\left (6 \, a^{2} b c + b^{3} c + b c^{3}\right )} \cos \left (2 \, x e + 2 \, d\right ) e^{\left (-1\right )} -{\left (4 \, a^{3} c + 3 \, a b^{2} c + 3 \, a c^{3}\right )} \cos \left (x e + d\right ) e^{\left (-1\right )} + \frac{1}{32} \,{\left (b^{4} - 6 \, b^{2} c^{2} + c^{4}\right )} e^{\left (-1\right )} \sin \left (4 \, x e + 4 \, d\right ) + \frac{1}{3} \,{\left (a b^{3} - 3 \, a b c^{2}\right )} e^{\left (-1\right )} \sin \left (3 \, x e + 3 \, d\right ) + \frac{1}{4} \,{\left (6 \, a^{2} b^{2} + b^{4} - 6 \, a^{2} c^{2} - c^{4}\right )} e^{\left (-1\right )} \sin \left (2 \, x e + 2 \, d\right ) +{\left (4 \, a^{3} b + 3 \, a b^{3} + 3 \, a b c^{2}\right )} e^{\left (-1\right )} \sin \left (x e + d\right ) + \frac{1}{8} \,{\left (8 \, a^{4} + 24 \, a^{2} b^{2} + 3 \, b^{4} + 24 \, a^{2} c^{2} + 6 \, b^{2} c^{2} + 3 \, c^{4}\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(b^3*c - b*c^3)*cos(4*x*e + 4*d)*e^(-1) - 1/3*(3*a*b^2*c - a*c^3)*cos(3*x*e + 3*d)*e^(-1) - 1/2*(6*a^2*b*
c + b^3*c + b*c^3)*cos(2*x*e + 2*d)*e^(-1) - (4*a^3*c + 3*a*b^2*c + 3*a*c^3)*cos(x*e + d)*e^(-1) + 1/32*(b^4 -
6*b^2*c^2 + c^4)*e^(-1)*sin(4*x*e + 4*d) + 1/3*(a*b^3 - 3*a*b*c^2)*e^(-1)*sin(3*x*e + 3*d) + 1/4*(6*a^2*b^2 +
b^4 - 6*a^2*c^2 - c^4)*e^(-1)*sin(2*x*e + 2*d) + (4*a^3*b + 3*a*b^3 + 3*a*b*c^2)*e^(-1)*sin(x*e + d) + 1/8*(8
*a^4 + 24*a^2*b^2 + 3*b^4 + 24*a^2*c^2 + 6*b^2*c^2 + 3*c^4)*x