\(\int \cos ^6(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2 \, dx\) [1240]

Optimal result

Integrand size = 29, antiderivative size = 250 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\left (12 a^2+5 b^2\right ) x}{1024}-\frac {2 a b \cos ^7(c+d x)}{7 d}+\frac {4 a b \cos ^9(c+d x)}{9 d}-\frac {2 a b \cos ^{11}(c+d x)}{11 d}+\frac {\left (12 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {\left (12 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac {\left (12 a^2+5 b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {\left (44 a^2+45 b^2\right ) \cos ^7(c+d x) \sin (c+d x)}{320 d}+\frac {\left (12 a^2+25 b^2\right ) \cos ^9(c+d x) \sin (c+d x)}{120 d}-\frac {b^2 \cos ^{11}(c+d x) \sin (c+d x)}{12 d} \]

[Out]

1/1024*(12*a^2+5*b^2)*x-2/7*a*b*cos(d*x+c)^7/d+4/9*a*b*cos(d*x+c)^9/d-2/11*a*b*cos(d*x+c)^11/d+1/1024*(12*a^2+
5*b^2)*cos(d*x+c)*sin(d*x+c)/d+1/1536*(12*a^2+5*b^2)*cos(d*x+c)^3*sin(d*x+c)/d+1/1920*(12*a^2+5*b^2)*cos(d*x+c
)^5*sin(d*x+c)/d-1/320*(44*a^2+45*b^2)*cos(d*x+c)^7*sin(d*x+c)/d+1/120*(12*a^2+25*b^2)*cos(d*x+c)^9*sin(d*x+c)
/d-1/12*b^2*cos(d*x+c)^11*sin(d*x+c)/d

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2990, 2645, 276, 3279, 466, 1171, 393, 205, 209} \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\left (12 a^2+25 b^2\right ) \sin (c+d x) \cos ^9(c+d x)}{120 d}-\frac {\left (44 a^2+45 b^2\right ) \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac {\left (12 a^2+5 b^2\right ) \sin (c+d x) \cos ^5(c+d x)}{1920 d}+\frac {\left (12 a^2+5 b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac {\left (12 a^2+5 b^2\right ) \sin (c+d x) \cos (c+d x)}{1024 d}+\frac {x \left (12 a^2+5 b^2\right )}{1024}-\frac {2 a b \cos ^{11}(c+d x)}{11 d}+\frac {4 a b \cos ^9(c+d x)}{9 d}-\frac {2 a b \cos ^7(c+d x)}{7 d}-\frac {b^2 \sin (c+d x) \cos ^{11}(c+d x)}{12 d} \]

[In]

Int[Cos[c + d*x]^6*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]

[Out]

((12*a^2 + 5*b^2)*x)/1024 - (2*a*b*Cos[c + d*x]^7)/(7*d) + (4*a*b*Cos[c + d*x]^9)/(9*d) - (2*a*b*Cos[c + d*x]^
11)/(11*d) + ((12*a^2 + 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + ((12*a^2 + 5*b^2)*Cos[c + d*x]^3*Sin[c +
d*x])/(1536*d) + ((12*a^2 + 5*b^2)*Cos[c + d*x]^5*Sin[c + d*x])/(1920*d) - ((44*a^2 + 45*b^2)*Cos[c + d*x]^7*S
in[c + d*x])/(320*d) + ((12*a^2 + 25*b^2)*Cos[c + d*x]^9*Sin[c + d*x])/(120*d) - (b^2*Cos[c + d*x]^11*Sin[c +
d*x])/(12*d)

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 466

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x
*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1
)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 2645

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 2990

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

Rule 3279

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.),
x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff^(n + 1)/f, Subst[Int[x^n*((a + (a + b)*ff^2*x^2
)^p/(1 + ff^2*x^2)^((m + n)/2 + p + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/
2] && IntegerQ[n/2] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = (2 a b) \int \cos ^6(c+d x) \sin ^5(c+d x) \, dx+\int \cos ^6(c+d x) \sin ^4(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {x^4 \left (a^2+\left (a^2+b^2\right ) x^2\right )}{\left (1+x^2\right )^7} \, dx,x,\tan (c+d x)\right )}{d}-\frac {(2 a b) \text {Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {b^2 \cos ^{11}(c+d x) \sin (c+d x)}{12 d}-\frac {\text {Subst}\left (\int \frac {-b^2+12 b^2 x^2-12 \left (a^2+b^2\right ) x^4}{\left (1+x^2\right )^6} \, dx,x,\tan (c+d x)\right )}{12 d}-\frac {(2 a b) \text {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {2 a b \cos ^7(c+d x)}{7 d}+\frac {4 a b \cos ^9(c+d x)}{9 d}-\frac {2 a b \cos ^{11}(c+d x)}{11 d}+\frac {\left (12 a^2+25 b^2\right ) \cos ^9(c+d x) \sin (c+d x)}{120 d}-\frac {b^2 \cos ^{11}(c+d x) \sin (c+d x)}{12 d}+\frac {\text {Subst}\left (\int \frac {-3 \left (4 a^2+5 b^2\right )+120 \left (a^2+b^2\right ) x^2}{\left (1+x^2\right )^5} \, dx,x,\tan (c+d x)\right )}{120 d} \\ & = -\frac {2 a b \cos ^7(c+d x)}{7 d}+\frac {4 a b \cos ^9(c+d x)}{9 d}-\frac {2 a b \cos ^{11}(c+d x)}{11 d}-\frac {\left (44 a^2+45 b^2\right ) \cos ^7(c+d x) \sin (c+d x)}{320 d}+\frac {\left (12 a^2+25 b^2\right ) \cos ^9(c+d x) \sin (c+d x)}{120 d}-\frac {b^2 \cos ^{11}(c+d x) \sin (c+d x)}{12 d}+\frac {\left (12 a^2+5 b^2\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^4} \, dx,x,\tan (c+d x)\right )}{320 d} \\ & = -\frac {2 a b \cos ^7(c+d x)}{7 d}+\frac {4 a b \cos ^9(c+d x)}{9 d}-\frac {2 a b \cos ^{11}(c+d x)}{11 d}+\frac {\left (12 a^2+5 b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {\left (44 a^2+45 b^2\right ) \cos ^7(c+d x) \sin (c+d x)}{320 d}+\frac {\left (12 a^2+25 b^2\right ) \cos ^9(c+d x) \sin (c+d x)}{120 d}-\frac {b^2 \cos ^{11}(c+d x) \sin (c+d x)}{12 d}+\frac {\left (12 a^2+5 b^2\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{384 d} \\ & = -\frac {2 a b \cos ^7(c+d x)}{7 d}+\frac {4 a b \cos ^9(c+d x)}{9 d}-\frac {2 a b \cos ^{11}(c+d x)}{11 d}+\frac {\left (12 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac {\left (12 a^2+5 b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {\left (44 a^2+45 b^2\right ) \cos ^7(c+d x) \sin (c+d x)}{320 d}+\frac {\left (12 a^2+25 b^2\right ) \cos ^9(c+d x) \sin (c+d x)}{120 d}-\frac {b^2 \cos ^{11}(c+d x) \sin (c+d x)}{12 d}+\frac {\left (12 a^2+5 b^2\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{512 d} \\ & = -\frac {2 a b \cos ^7(c+d x)}{7 d}+\frac {4 a b \cos ^9(c+d x)}{9 d}-\frac {2 a b \cos ^{11}(c+d x)}{11 d}+\frac {\left (12 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {\left (12 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac {\left (12 a^2+5 b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {\left (44 a^2+45 b^2\right ) \cos ^7(c+d x) \sin (c+d x)}{320 d}+\frac {\left (12 a^2+25 b^2\right ) \cos ^9(c+d x) \sin (c+d x)}{120 d}-\frac {b^2 \cos ^{11}(c+d x) \sin (c+d x)}{12 d}+\frac {\left (12 a^2+5 b^2\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{1024 d} \\ & = \frac {\left (12 a^2+5 b^2\right ) x}{1024}-\frac {2 a b \cos ^7(c+d x)}{7 d}+\frac {4 a b \cos ^9(c+d x)}{9 d}-\frac {2 a b \cos ^{11}(c+d x)}{11 d}+\frac {\left (12 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {\left (12 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac {\left (12 a^2+5 b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {\left (44 a^2+45 b^2\right ) \cos ^7(c+d x) \sin (c+d x)}{320 d}+\frac {\left (12 a^2+25 b^2\right ) \cos ^9(c+d x) \sin (c+d x)}{120 d}-\frac {b^2 \cos ^{11}(c+d x) \sin (c+d x)}{12 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.81 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {166320 b^2 c+332640 a^2 d x+138600 b^2 d x-554400 a b \cos (c+d x)-184800 a b \cos (3 (c+d x))+55440 a b \cos (5 (c+d x))+39600 a b \cos (7 (c+d x))-6160 a b \cos (9 (c+d x))-5040 a b \cos (11 (c+d x))+55440 a^2 \sin (2 (c+d x))-110880 a^2 \sin (4 (c+d x))-51975 b^2 \sin (4 (c+d x))-27720 a^2 \sin (6 (c+d x))+13860 a^2 \sin (8 (c+d x))+10395 b^2 \sin (8 (c+d x))+5544 a^2 \sin (10 (c+d x))-1155 b^2 \sin (12 (c+d x))}{28385280 d} \]

[In]

Integrate[Cos[c + d*x]^6*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]

[Out]

(166320*b^2*c + 332640*a^2*d*x + 138600*b^2*d*x - 554400*a*b*Cos[c + d*x] - 184800*a*b*Cos[3*(c + d*x)] + 5544
0*a*b*Cos[5*(c + d*x)] + 39600*a*b*Cos[7*(c + d*x)] - 6160*a*b*Cos[9*(c + d*x)] - 5040*a*b*Cos[11*(c + d*x)] +
55440*a^2*Sin[2*(c + d*x)] - 110880*a^2*Sin[4*(c + d*x)] - 51975*b^2*Sin[4*(c + d*x)] - 27720*a^2*Sin[6*(c +
d*x)] + 13860*a^2*Sin[8*(c + d*x)] + 10395*b^2*Sin[8*(c + d*x)] + 5544*a^2*Sin[10*(c + d*x)] - 1155*b^2*Sin[12
*(c + d*x)])/(28385280*d)

Maple [A] (verified)

Time = 1.94 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.79

method	result	size
parallelrisch	\(\frac {\left (-110880 a^{2}-51975 b^{2}\right ) \sin \left (4 d x +4 c \right )+\left (13860 a^{2}+10395 b^{2}\right ) \sin \left (8 d x +8 c \right )+332640 a^{2} d x +138600 b^{2} d x +55440 a^{2} \sin \left (2 d x +2 c \right )-27720 a^{2} \sin \left (6 d x +6 c \right )+5544 a^{2} \sin \left (10 d x +10 c \right )-6160 a b \cos \left (9 d x +9 c \right )-5040 a b \cos \left (11 d x +11 c \right )-554400 a b \cos \left (d x +c \right )-184800 a b \cos \left (3 d x +3 c \right )+55440 a b \cos \left (5 d x +5 c \right )+39600 a b \cos \left (7 d x +7 c \right )-1155 b^{2} \sin \left (12 d x +12 c \right )-655360 a b}{28385280 d}\)	\(198\)
derivativedivides	\(\frac {a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+2 a b \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+b^{2} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{24}-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{64}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{384}+\frac {5 d x}{1024}+\frac {5 c}{1024}\right )}{d}\)	\(237\)
default	\(\frac {a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+2 a b \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+b^{2} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{24}-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{64}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{384}+\frac {5 d x}{1024}+\frac {5 c}{1024}\right )}{d}\)	\(237\)
risch	\(\frac {a^{2} \sin \left (10 d x +10 c \right )}{5120 d}+\frac {3 a^{2} x}{256}+\frac {5 b^{2} x}{1024}-\frac {5 a b \cos \left (d x +c \right )}{256 d}-\frac {a b \cos \left (11 d x +11 c \right )}{5632 d}-\frac {b^{2} \sin \left (12 d x +12 c \right )}{24576 d}-\frac {a b \cos \left (9 d x +9 c \right )}{4608 d}+\frac {\sin \left (8 d x +8 c \right ) a^{2}}{2048 d}+\frac {3 \sin \left (8 d x +8 c \right ) b^{2}}{8192 d}+\frac {5 a b \cos \left (7 d x +7 c \right )}{3584 d}-\frac {a^{2} \sin \left (6 d x +6 c \right )}{1024 d}+\frac {a b \cos \left (5 d x +5 c \right )}{512 d}-\frac {\sin \left (4 d x +4 c \right ) a^{2}}{256 d}-\frac {15 \sin \left (4 d x +4 c \right ) b^{2}}{8192 d}-\frac {5 a b \cos \left (3 d x +3 c \right )}{768 d}+\frac {a^{2} \sin \left (2 d x +2 c \right )}{512 d}\)	\(243\)

[In]

int(cos(d*x+c)^6*sin(d*x+c)^4*(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/28385280*((-110880*a^2-51975*b^2)*sin(4*d*x+4*c)+(13860*a^2+10395*b^2)*sin(8*d*x+8*c)+332640*a^2*d*x+138600*
b^2*d*x+55440*a^2*sin(2*d*x+2*c)-27720*a^2*sin(6*d*x+6*c)+5544*a^2*sin(10*d*x+10*c)-6160*a*b*cos(9*d*x+9*c)-50
40*a*b*cos(11*d*x+11*c)-554400*a*b*cos(d*x+c)-184800*a*b*cos(3*d*x+3*c)+55440*a*b*cos(5*d*x+5*c)+39600*a*b*cos
(7*d*x+7*c)-1155*b^2*sin(12*d*x+12*c)-655360*a*b)/d

Fricas [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.73 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {645120 \, a b \cos \left (d x + c\right )^{11} - 1576960 \, a b \cos \left (d x + c\right )^{9} + 1013760 \, a b \cos \left (d x + c\right )^{7} - 3465 \, {\left (12 \, a^{2} + 5 \, b^{2}\right )} d x + 231 \, {\left (1280 \, b^{2} \cos \left (d x + c\right )^{11} - 128 \, {\left (12 \, a^{2} + 25 \, b^{2}\right )} \cos \left (d x + c\right )^{9} + 48 \, {\left (44 \, a^{2} + 45 \, b^{2}\right )} \cos \left (d x + c\right )^{7} - 8 \, {\left (12 \, a^{2} + 5 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 10 \, {\left (12 \, a^{2} + 5 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (12 \, a^{2} + 5 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3548160 \, d} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^4*(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/3548160*(645120*a*b*cos(d*x + c)^11 - 1576960*a*b*cos(d*x + c)^9 + 1013760*a*b*cos(d*x + c)^7 - 3465*(12*a^
2 + 5*b^2)*d*x + 231*(1280*b^2*cos(d*x + c)^11 - 128*(12*a^2 + 25*b^2)*cos(d*x + c)^9 + 48*(44*a^2 + 45*b^2)*c
os(d*x + c)^7 - 8*(12*a^2 + 5*b^2)*cos(d*x + c)^5 - 10*(12*a^2 + 5*b^2)*cos(d*x + c)^3 - 15*(12*a^2 + 5*b^2)*c
os(d*x + c))*sin(d*x + c))/d

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (231) = 462\).

Time = 2.68 (sec) , antiderivative size = 656, normalized size of antiderivative = 2.62 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\begin {cases} \frac {3 a^{2} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {15 a^{2} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {15 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {15 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {3 a^{2} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {3 a^{2} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {7 a^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} - \frac {7 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {3 a^{2} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {2 a b \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {8 a b \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {16 a b \cos ^{11}{\left (c + d x \right )}}{693 d} + \frac {5 b^{2} x \sin ^{12}{\left (c + d x \right )}}{1024} + \frac {15 b^{2} x \sin ^{10}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{512} + \frac {75 b^{2} x \sin ^{8}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{1024} + \frac {25 b^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{256} + \frac {75 b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{1024} + \frac {15 b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{512} + \frac {5 b^{2} x \cos ^{12}{\left (c + d x \right )}}{1024} + \frac {5 b^{2} \sin ^{11}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{1024 d} + \frac {85 b^{2} \sin ^{9}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3072 d} + \frac {33 b^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{512 d} - \frac {33 b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{512 d} - \frac {85 b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{3072 d} - \frac {5 b^{2} \sin {\left (c + d x \right )} \cos ^{11}{\left (c + d x \right )}}{1024 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{2} \sin ^{4}{\left (c \right )} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**6*sin(d*x+c)**4*(a+b*sin(d*x+c))**2,x)

[Out]

Piecewise((3*a**2*x*sin(c + d*x)**10/256 + 15*a**2*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 15*a**2*x*sin(c + d
*x)**6*cos(c + d*x)**4/128 + 15*a**2*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 15*a**2*x*sin(c + d*x)**2*cos(c +
d*x)**8/256 + 3*a**2*x*cos(c + d*x)**10/256 + 3*a**2*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 7*a**2*sin(c + d*
x)**7*cos(c + d*x)**3/(128*d) + a**2*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) - 7*a**2*sin(c + d*x)**3*cos(c + d
*x)**7/(128*d) - 3*a**2*sin(c + d*x)*cos(c + d*x)**9/(256*d) - 2*a*b*sin(c + d*x)**4*cos(c + d*x)**7/(7*d) - 8
*a*b*sin(c + d*x)**2*cos(c + d*x)**9/(63*d) - 16*a*b*cos(c + d*x)**11/(693*d) + 5*b**2*x*sin(c + d*x)**12/1024
+ 15*b**2*x*sin(c + d*x)**10*cos(c + d*x)**2/512 + 75*b**2*x*sin(c + d*x)**8*cos(c + d*x)**4/1024 + 25*b**2*x
*sin(c + d*x)**6*cos(c + d*x)**6/256 + 75*b**2*x*sin(c + d*x)**4*cos(c + d*x)**8/1024 + 15*b**2*x*sin(c + d*x)
**2*cos(c + d*x)**10/512 + 5*b**2*x*cos(c + d*x)**12/1024 + 5*b**2*sin(c + d*x)**11*cos(c + d*x)/(1024*d) + 85
*b**2*sin(c + d*x)**9*cos(c + d*x)**3/(3072*d) + 33*b**2*sin(c + d*x)**7*cos(c + d*x)**5/(512*d) - 33*b**2*sin
(c + d*x)**5*cos(c + d*x)**7/(512*d) - 85*b**2*sin(c + d*x)**3*cos(c + d*x)**9/(3072*d) - 5*b**2*sin(c + d*x)*
cos(c + d*x)**11/(1024*d), Ne(d, 0)), (x*(a + b*sin(c))**2*sin(c)**4*cos(c)**6, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.55 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {2772 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} - 81920 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a b + 1155 \, {\left (4 \, \sin \left (4 \, d x + 4 \, c\right )^{3} + 120 \, d x + 120 \, c + 9 \, \sin \left (8 \, d x + 8 \, c\right ) - 48 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{2}}{28385280 \, d} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^4*(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/28385280*(2772*(32*sin(2*d*x + 2*c)^5 + 120*d*x + 120*c + 5*sin(8*d*x + 8*c) - 40*sin(4*d*x + 4*c))*a^2 - 81
920*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^7)*a*b + 1155*(4*sin(4*d*x + 4*c)^3 + 120*d*x +
120*c + 9*sin(8*d*x + 8*c) - 48*sin(4*d*x + 4*c))*b^2)/d

Giac [A] (verification not implemented)

none

Time = 0.59 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.90 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {1}{1024} \, {\left (12 \, a^{2} + 5 \, b^{2}\right )} x - \frac {a b \cos \left (11 \, d x + 11 \, c\right )}{5632 \, d} - \frac {a b \cos \left (9 \, d x + 9 \, c\right )}{4608 \, d} + \frac {5 \, a b \cos \left (7 \, d x + 7 \, c\right )}{3584 \, d} + \frac {a b \cos \left (5 \, d x + 5 \, c\right )}{512 \, d} - \frac {5 \, a b \cos \left (3 \, d x + 3 \, c\right )}{768 \, d} - \frac {5 \, a b \cos \left (d x + c\right )}{256 \, d} - \frac {b^{2} \sin \left (12 \, d x + 12 \, c\right )}{24576 \, d} + \frac {a^{2} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} - \frac {a^{2} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} + \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} + \frac {{\left (4 \, a^{2} + 3 \, b^{2}\right )} \sin \left (8 \, d x + 8 \, c\right )}{8192 \, d} - \frac {{\left (32 \, a^{2} + 15 \, b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{8192 \, d} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^4*(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/1024*(12*a^2 + 5*b^2)*x - 1/5632*a*b*cos(11*d*x + 11*c)/d - 1/4608*a*b*cos(9*d*x + 9*c)/d + 5/3584*a*b*cos(7
*d*x + 7*c)/d + 1/512*a*b*cos(5*d*x + 5*c)/d - 5/768*a*b*cos(3*d*x + 3*c)/d - 5/256*a*b*cos(d*x + c)/d - 1/245
76*b^2*sin(12*d*x + 12*c)/d + 1/5120*a^2*sin(10*d*x + 10*c)/d - 1/1024*a^2*sin(6*d*x + 6*c)/d + 1/512*a^2*sin(
2*d*x + 2*c)/d + 1/8192*(4*a^2 + 3*b^2)*sin(8*d*x + 8*c)/d - 1/8192*(32*a^2 + 15*b^2)*sin(4*d*x + 4*c)/d

Mupad [B] (verification not implemented)

Time = 13.41 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.83 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {6930\,a^2\,\sin \left (2\,c+2\,d\,x\right )-13860\,a^2\,\sin \left (4\,c+4\,d\,x\right )-3465\,a^2\,\sin \left (6\,c+6\,d\,x\right )+\frac {3465\,a^2\,\sin \left (8\,c+8\,d\,x\right )}{2}+693\,a^2\,\sin \left (10\,c+10\,d\,x\right )-\frac {51975\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{8}+\frac {10395\,b^2\,\sin \left (8\,c+8\,d\,x\right )}{8}-\frac {1155\,b^2\,\sin \left (12\,c+12\,d\,x\right )}{8}-69300\,a\,b\,\cos \left (c+d\,x\right )-23100\,a\,b\,\cos \left (3\,c+3\,d\,x\right )+6930\,a\,b\,\cos \left (5\,c+5\,d\,x\right )+4950\,a\,b\,\cos \left (7\,c+7\,d\,x\right )-770\,a\,b\,\cos \left (9\,c+9\,d\,x\right )-630\,a\,b\,\cos \left (11\,c+11\,d\,x\right )+41580\,a^2\,d\,x+17325\,b^2\,d\,x}{3548160\,d} \]

[In]

int(cos(c + d*x)^6*sin(c + d*x)^4*(a + b*sin(c + d*x))^2,x)

[Out]

(6930*a^2*sin(2*c + 2*d*x) - 13860*a^2*sin(4*c + 4*d*x) - 3465*a^2*sin(6*c + 6*d*x) + (3465*a^2*sin(8*c + 8*d*
x))/2 + 693*a^2*sin(10*c + 10*d*x) - (51975*b^2*sin(4*c + 4*d*x))/8 + (10395*b^2*sin(8*c + 8*d*x))/8 - (1155*b
^2*sin(12*c + 12*d*x))/8 - 69300*a*b*cos(c + d*x) - 23100*a*b*cos(3*c + 3*d*x) + 6930*a*b*cos(5*c + 5*d*x) + 4
950*a*b*cos(7*c + 7*d*x) - 770*a*b*cos(9*c + 9*d*x) - 630*a*b*cos(11*c + 11*d*x) + 41580*a^2*d*x + 17325*b^2*d
*x)/(3548160*d)