3.220 \(\int (a \cos (c+d x)+b \sin (c+d x))^6 \, dx\)

Optimal. Leaf size=161 \[ -\frac {5 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{24 d}-\frac {5 \left (a^2+b^2\right )^2 (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{16 d}+\frac {5}{16} x \left (a^2+b^2\right )^3-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d} \]

[Out]

5/16*(a^2+b^2)^3*x-5/16*(a^2+b^2)^2*(b*cos(d*x+c)-a*sin(d*x+c))*(a*cos(d*x+c)+b*sin(d*x+c))/d-5/24*(a^2+b^2)*(
b*cos(d*x+c)-a*sin(d*x+c))*(a*cos(d*x+c)+b*sin(d*x+c))^3/d-1/6*(b*cos(d*x+c)-a*sin(d*x+c))*(a*cos(d*x+c)+b*sin
(d*x+c))^5/d

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Rubi [A]  time = 0.08, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3073, 8} \[ -\frac {5 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{24 d}-\frac {5 \left (a^2+b^2\right )^2 (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{16 d}+\frac {5}{16} x \left (a^2+b^2\right )^3-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^6,x]

[Out]

(5*(a^2 + b^2)^3*x)/16 - (5*(a^2 + b^2)^2*(b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x]))
/(16*d) - (5*(a^2 + b^2)*(b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(24*d) - ((b*C
os[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^5)/(6*d)

Rule 8

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

Rule 3073

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

Rubi steps

\begin {align*} \int (a \cos (c+d x)+b \sin (c+d x))^6 \, dx &=-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d}+\frac {1}{6} \left (5 \left (a^2+b^2\right )\right ) \int (a \cos (c+d x)+b \sin (c+d x))^4 \, dx\\ &=-\frac {5 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{24 d}-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d}+\frac {1}{8} \left (5 \left (a^2+b^2\right )^2\right ) \int (a \cos (c+d x)+b \sin (c+d x))^2 \, dx\\ &=-\frac {5 \left (a^2+b^2\right )^2 (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{16 d}-\frac {5 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{24 d}-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d}+\frac {1}{16} \left (5 \left (a^2+b^2\right )^3\right ) \int 1 \, dx\\ &=\frac {5}{16} \left (a^2+b^2\right )^3 x-\frac {5 \left (a^2+b^2\right )^2 (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{16 d}-\frac {5 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{24 d}-\frac {(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d}\\ \end {align*}

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Mathematica [A]  time = 0.69, size = 192, normalized size = 1.19 \[ \frac {-36 a b \left (a^4-b^4\right ) \cos (4 (c+d x))+60 \left (a^2+b^2\right )^3 (c+d x)+45 \left (a^2-b^2\right ) \left (a^2+b^2\right )^2 \sin (2 (c+d x))-90 a b \left (a^2+b^2\right )^2 \cos (2 (c+d x))-2 a b \left (3 a^4-10 a^2 b^2+3 b^4\right ) \cos (6 (c+d x))+9 \left (a^6-5 a^4 b^2-5 a^2 b^4+b^6\right ) \sin (4 (c+d x))+\left (a^6-15 a^4 b^2+15 a^2 b^4-b^6\right ) \sin (6 (c+d x))}{192 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^6,x]

[Out]

(60*(a^2 + b^2)^3*(c + d*x) - 90*a*b*(a^2 + b^2)^2*Cos[2*(c + d*x)] - 36*a*b*(a^4 - b^4)*Cos[4*(c + d*x)] - 2*
a*b*(3*a^4 - 10*a^2*b^2 + 3*b^4)*Cos[6*(c + d*x)] + 45*(a^2 - b^2)*(a^2 + b^2)^2*Sin[2*(c + d*x)] + 9*(a^6 - 5
*a^4*b^2 - 5*a^2*b^4 + b^6)*Sin[4*(c + d*x)] + (a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*Sin[6*(c + d*x)])/(192*d)

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fricas [A]  time = 1.18, size = 219, normalized size = 1.36 \[ -\frac {144 \, a b^{5} \cos \left (d x + c\right )^{2} + 16 \, {\left (3 \, a^{5} b - 10 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{6} + 48 \, {\left (5 \, a^{3} b^{3} - 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{4} - 15 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d x - {\left (8 \, {\left (a^{6} - 15 \, a^{4} b^{2} + 15 \, a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (5 \, a^{6} + 15 \, a^{4} b^{2} - 105 \, a^{2} b^{4} + 13 \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (5 \, a^{6} + 15 \, a^{4} b^{2} + 15 \, a^{2} b^{4} - 11 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cos(d*x+c)+b*sin(d*x+c))^6,x, algorithm="fricas")

[Out]

-1/48*(144*a*b^5*cos(d*x + c)^2 + 16*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*cos(d*x + c)^6 + 48*(5*a^3*b^3 - 3*a*b^5
)*cos(d*x + c)^4 - 15*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d*x - (8*(a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*cos(d
*x + c)^5 + 2*(5*a^6 + 15*a^4*b^2 - 105*a^2*b^4 + 13*b^6)*cos(d*x + c)^3 + 3*(5*a^6 + 15*a^4*b^2 + 15*a^2*b^4
- 11*b^6)*cos(d*x + c))*sin(d*x + c))/d

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giac [A]  time = 0.32, size = 235, normalized size = 1.46 \[ \frac {5}{16} \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} x - \frac {{\left (3 \, a^{5} b - 10 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac {3 \, {\left (a^{5} b - a b^{5}\right )} \cos \left (4 \, d x + 4 \, c\right )}{16 \, d} - \frac {15 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (2 \, d x + 2 \, c\right )}{32 \, d} + \frac {{\left (a^{6} - 15 \, a^{4} b^{2} + 15 \, a^{2} b^{4} - b^{6}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {3 \, {\left (a^{6} - 5 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + b^{6}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {15 \, {\left (a^{6} + a^{4} b^{2} - a^{2} b^{4} - b^{6}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cos(d*x+c)+b*sin(d*x+c))^6,x, algorithm="giac")

[Out]

5/16*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*x - 1/96*(3*a^5*b - 10*a^3*b^3 + 3*a*b^5)*cos(6*d*x + 6*c)/d - 3/16*(
a^5*b - a*b^5)*cos(4*d*x + 4*c)/d - 15/32*(a^5*b + 2*a^3*b^3 + a*b^5)*cos(2*d*x + 2*c)/d + 1/192*(a^6 - 15*a^4
*b^2 + 15*a^2*b^4 - b^6)*sin(6*d*x + 6*c)/d + 3/64*(a^6 - 5*a^4*b^2 - 5*a^2*b^4 + b^6)*sin(4*d*x + 4*c)/d + 15
/64*(a^6 + a^4*b^2 - a^2*b^4 - b^6)*sin(2*d*x + 2*c)/d

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maple [A]  time = 0.34, size = 285, normalized size = 1.77 \[ \frac {b^{6} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+a \,b^{5} \left (\sin ^{6}\left (d x +c \right )\right )+15 a^{2} b^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+20 a^{3} b^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{12}\right )+15 a^{4} b^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-a^{5} b \left (\cos ^{6}\left (d x +c \right )\right )+a^{6} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*cos(d*x+c)+b*sin(d*x+c))^6,x)

[Out]

1/d*(b^6*(-1/6*(sin(d*x+c)^5+5/4*sin(d*x+c)^3+15/8*sin(d*x+c))*cos(d*x+c)+5/16*d*x+5/16*c)+a*b^5*sin(d*x+c)^6+
15*a^2*b^4*(-1/6*sin(d*x+c)^3*cos(d*x+c)^3-1/8*sin(d*x+c)*cos(d*x+c)^3+1/16*sin(d*x+c)*cos(d*x+c)+1/16*d*x+1/1
6*c)+20*a^3*b^3*(-1/6*sin(d*x+c)^2*cos(d*x+c)^4-1/12*cos(d*x+c)^4)+15*a^4*b^2*(-1/6*sin(d*x+c)*cos(d*x+c)^5+1/
24*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+1/16*d*x+1/16*c)-a^5*b*cos(d*x+c)^6+a^6*(1/6*(cos(d*x+c)^5+5/4*cos
(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c))

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maxima [A]  time = 0.33, size = 238, normalized size = 1.48 \[ -\frac {192 \, a^{5} b \cos \left (d x + c\right )^{6} - 192 \, a b^{5} \sin \left (d x + c\right )^{6} + {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{6} - 15 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4} b^{2} + 320 \, {\left (2 \, \sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4}\right )} a^{3} b^{3} + 15 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} b^{4} - {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 60 \, d x + 60 \, c + 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b^{6}}{192 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cos(d*x+c)+b*sin(d*x+c))^6,x, algorithm="maxima")

[Out]

-1/192*(192*a^5*b*cos(d*x + c)^6 - 192*a*b^5*sin(d*x + c)^6 + (4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*
d*x + 4*c) - 48*sin(2*d*x + 2*c))*a^6 - 15*(4*sin(2*d*x + 2*c)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c))*a^4*b^2
+ 320*(2*sin(d*x + c)^6 - 3*sin(d*x + c)^4)*a^3*b^3 + 15*(4*sin(2*d*x + 2*c)^3 - 12*d*x - 12*c + 3*sin(4*d*x
+ 4*c))*a^2*b^4 - (4*sin(2*d*x + 2*c)^3 + 60*d*x + 60*c + 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*b^6)/d

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mupad [B]  time = 4.12, size = 519, normalized size = 3.22 \[ \frac {5\,\mathrm {atan}\left (\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\left (a^2+b^2\right )}^3}{8\,\left (\frac {5\,a^6}{8}+\frac {15\,a^4\,b^2}{8}+\frac {15\,a^2\,b^4}{8}+\frac {5\,b^6}{8}\right )}\right )\,{\left (a^2+b^2\right )}^3}{8\,d}-\frac {5\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,{\left (a^2+b^2\right )}^3}{8\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (40\,a^5\,b-\frac {160\,a^3\,b^3}{3}+64\,a\,b^5\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-\frac {11\,a^6}{8}+\frac {15\,a^4\,b^2}{8}+\frac {15\,a^2\,b^4}{8}+\frac {5\,b^6}{8}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (-\frac {11\,a^6}{8}+\frac {15\,a^4\,b^2}{8}+\frac {15\,a^2\,b^4}{8}+\frac {5\,b^6}{8}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {5\,a^6}{24}-\frac {235\,a^4\,b^2}{8}+\frac {85\,a^2\,b^4}{8}+\frac {85\,b^6}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {5\,a^6}{24}-\frac {235\,a^4\,b^2}{8}+\frac {85\,a^2\,b^4}{8}+\frac {85\,b^6}{24}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {15\,a^6}{4}-\frac {195\,a^4\,b^2}{4}+\frac {285\,a^2\,b^4}{4}-\frac {33\,b^6}{4}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {15\,a^6}{4}-\frac {195\,a^4\,b^2}{4}+\frac {285\,a^2\,b^4}{4}-\frac {33\,b^6}{4}\right )+80\,a^3\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+80\,a^3\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+12\,a^5\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+12\,a^5\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*cos(c + d*x) + b*sin(c + d*x))^6,x)

[Out]

(5*atan((5*tan(c/2 + (d*x)/2)*(a^2 + b^2)^3)/(8*((5*a^6)/8 + (5*b^6)/8 + (15*a^2*b^4)/8 + (15*a^4*b^2)/8)))*(a
^2 + b^2)^3)/(8*d) - (5*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2)*(a^2 + b^2)^3)/(8*d) + (tan(c/2 + (d*x)/2)^6*(64*
a*b^5 + 40*a^5*b - (160*a^3*b^3)/3) - tan(c/2 + (d*x)/2)*((5*b^6)/8 - (11*a^6)/8 + (15*a^2*b^4)/8 + (15*a^4*b^
2)/8) + tan(c/2 + (d*x)/2)^11*((5*b^6)/8 - (11*a^6)/8 + (15*a^2*b^4)/8 + (15*a^4*b^2)/8) - tan(c/2 + (d*x)/2)^
3*((5*a^6)/24 + (85*b^6)/24 + (85*a^2*b^4)/8 - (235*a^4*b^2)/8) + tan(c/2 + (d*x)/2)^9*((5*a^6)/24 + (85*b^6)/
24 + (85*a^2*b^4)/8 - (235*a^4*b^2)/8) + tan(c/2 + (d*x)/2)^5*((15*a^6)/4 - (33*b^6)/4 + (285*a^2*b^4)/4 - (19
5*a^4*b^2)/4) - tan(c/2 + (d*x)/2)^7*((15*a^6)/4 - (33*b^6)/4 + (285*a^2*b^4)/4 - (195*a^4*b^2)/4) + 80*a^3*b^
3*tan(c/2 + (d*x)/2)^4 + 80*a^3*b^3*tan(c/2 + (d*x)/2)^8 + 12*a^5*b*tan(c/2 + (d*x)/2)^2 + 12*a^5*b*tan(c/2 +
(d*x)/2)^10)/(d*(6*tan(c/2 + (d*x)/2)^2 + 15*tan(c/2 + (d*x)/2)^4 + 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d*
x)/2)^8 + 6*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1))

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sympy [A]  time = 4.62, size = 770, normalized size = 4.78 \[ \begin {cases} \frac {5 a^{6} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{6} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 a^{6} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 a^{6} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 a^{6} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 a^{6} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 a^{6} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {a^{5} b \cos ^{6}{\left (c + d x \right )}}{d} + \frac {15 a^{4} b^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {45 a^{4} b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {45 a^{4} b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {15 a^{4} b^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{4} b^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 a^{4} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} - \frac {15 a^{4} b^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {5 a^{3} b^{3} \sin ^{6}{\left (c + d x \right )}}{3 d} + \frac {5 a^{3} b^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {15 a^{2} b^{4} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {45 a^{2} b^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {45 a^{2} b^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {15 a^{2} b^{4} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{2} b^{4} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {5 a^{2} b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} - \frac {15 a^{2} b^{4} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {a b^{5} \sin ^{6}{\left (c + d x \right )}}{d} + \frac {5 b^{6} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 b^{6} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 b^{6} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 b^{6} x \cos ^{6}{\left (c + d x \right )}}{16} - \frac {11 b^{6} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {5 b^{6} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {5 b^{6} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\relax (c )} + b \sin {\relax (c )}\right )^{6} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cos(d*x+c)+b*sin(d*x+c))**6,x)

[Out]

Piecewise((5*a**6*x*sin(c + d*x)**6/16 + 15*a**6*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 15*a**6*x*sin(c + d*x)
**2*cos(c + d*x)**4/16 + 5*a**6*x*cos(c + d*x)**6/16 + 5*a**6*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 5*a**6*sin
(c + d*x)**3*cos(c + d*x)**3/(6*d) + 11*a**6*sin(c + d*x)*cos(c + d*x)**5/(16*d) - a**5*b*cos(c + d*x)**6/d +
15*a**4*b**2*x*sin(c + d*x)**6/16 + 45*a**4*b**2*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 45*a**4*b**2*x*sin(c +
d*x)**2*cos(c + d*x)**4/16 + 15*a**4*b**2*x*cos(c + d*x)**6/16 + 15*a**4*b**2*sin(c + d*x)**5*cos(c + d*x)/(1
6*d) + 5*a**4*b**2*sin(c + d*x)**3*cos(c + d*x)**3/(2*d) - 15*a**4*b**2*sin(c + d*x)*cos(c + d*x)**5/(16*d) +
5*a**3*b**3*sin(c + d*x)**6/(3*d) + 5*a**3*b**3*sin(c + d*x)**4*cos(c + d*x)**2/d + 15*a**2*b**4*x*sin(c + d*x
)**6/16 + 45*a**2*b**4*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 45*a**2*b**4*x*sin(c + d*x)**2*cos(c + d*x)**4/1
6 + 15*a**2*b**4*x*cos(c + d*x)**6/16 + 15*a**2*b**4*sin(c + d*x)**5*cos(c + d*x)/(16*d) - 5*a**2*b**4*sin(c +
d*x)**3*cos(c + d*x)**3/(2*d) - 15*a**2*b**4*sin(c + d*x)*cos(c + d*x)**5/(16*d) + a*b**5*sin(c + d*x)**6/d +
5*b**6*x*sin(c + d*x)**6/16 + 15*b**6*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 15*b**6*x*sin(c + d*x)**2*cos(c
+ d*x)**4/16 + 5*b**6*x*cos(c + d*x)**6/16 - 11*b**6*sin(c + d*x)**5*cos(c + d*x)/(16*d) - 5*b**6*sin(c + d*x)
**3*cos(c + d*x)**3/(6*d) - 5*b**6*sin(c + d*x)*cos(c + d*x)**5/(16*d), Ne(d, 0)), (x*(a*cos(c) + b*sin(c))**6
, True))

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