3.1241 \(\int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=187 \[ \frac {\left (a^2+2 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac {\left (a^2+b^2\right ) \cos ^7(c+d x)}{7 d}-\frac {a b \sin ^3(c+d x) \cos ^7(c+d x)}{5 d}-\frac {3 a b \sin (c+d x) \cos ^7(c+d x)}{40 d}+\frac {a b \sin (c+d x) \cos ^5(c+d x)}{80 d}+\frac {a b \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {3 a b \sin (c+d x) \cos (c+d x)}{128 d}+\frac {3 a b x}{128}-\frac {b^2 \cos ^{11}(c+d x)}{11 d} \]

[Out]

3/128*a*b*x-1/7*(a^2+b^2)*cos(d*x+c)^7/d+1/9*(a^2+2*b^2)*cos(d*x+c)^9/d-1/11*b^2*cos(d*x+c)^11/d+3/128*a*b*cos
(d*x+c)*sin(d*x+c)/d+1/64*a*b*cos(d*x+c)^3*sin(d*x+c)/d+1/80*a*b*cos(d*x+c)^5*sin(d*x+c)/d-3/40*a*b*cos(d*x+c)
^7*sin(d*x+c)/d-1/5*a*b*cos(d*x+c)^7*sin(d*x+c)^3/d

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Rubi [A]  time = 0.31, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2911, 2568, 2635, 8, 3201, 446, 77} \[ \frac {\left (a^2+2 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac {\left (a^2+b^2\right ) \cos ^7(c+d x)}{7 d}-\frac {a b \sin ^3(c+d x) \cos ^7(c+d x)}{5 d}-\frac {3 a b \sin (c+d x) \cos ^7(c+d x)}{40 d}+\frac {a b \sin (c+d x) \cos ^5(c+d x)}{80 d}+\frac {a b \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {3 a b \sin (c+d x) \cos (c+d x)}{128 d}+\frac {3 a b x}{128}-\frac {b^2 \cos ^{11}(c+d x)}{11 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]

[Out]

(3*a*b*x)/128 - ((a^2 + b^2)*Cos[c + d*x]^7)/(7*d) + ((a^2 + 2*b^2)*Cos[c + d*x]^9)/(9*d) - (b^2*Cos[c + d*x]^
11)/(11*d) + (3*a*b*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a*b*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) + (a*b*Cos[c
+ d*x]^5*Sin[c + d*x])/(80*d) - (3*a*b*Cos[c + d*x]^7*Sin[c + d*x])/(40*d) - (a*b*Cos[c + d*x]^7*Sin[c + d*x]
^3)/(5*d)

Rule 8

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

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 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 2911

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 3201

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

Rubi steps

\begin {align*} \int \cos ^6(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx &=(2 a b) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx+\int \cos ^6(c+d x) \sin ^3(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac {a b \cos ^7(c+d x) \sin ^3(c+d x)}{5 d}+\frac {1}{5} (3 a b) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+\frac {\left (\sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \operatorname {Subst}\left (\int x^3 \left (1-x^2\right )^{5/2} \left (a^2+b^2 x^2\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {3 a b \cos ^7(c+d x) \sin (c+d x)}{40 d}-\frac {a b \cos ^7(c+d x) \sin ^3(c+d x)}{5 d}+\frac {1}{40} (3 a b) \int \cos ^6(c+d x) \, dx+\frac {\left (\sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \operatorname {Subst}\left (\int (1-x)^{5/2} x \left (a^2+b^2 x\right ) \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac {a b \cos ^5(c+d x) \sin (c+d x)}{80 d}-\frac {3 a b \cos ^7(c+d x) \sin (c+d x)}{40 d}-\frac {a b \cos ^7(c+d x) \sin ^3(c+d x)}{5 d}+\frac {1}{16} (a b) \int \cos ^4(c+d x) \, dx+\frac {\left (\sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \operatorname {Subst}\left (\int \left (\left (a^2+b^2\right ) (1-x)^{5/2}+\left (-a^2-2 b^2\right ) (1-x)^{7/2}+b^2 (1-x)^{9/2}\right ) \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=-\frac {\left (a^2+b^2\right ) \cos ^7(c+d x)}{7 d}+\frac {\left (a^2+2 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac {b^2 \cos ^{11}(c+d x)}{11 d}+\frac {a b \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {a b \cos ^5(c+d x) \sin (c+d x)}{80 d}-\frac {3 a b \cos ^7(c+d x) \sin (c+d x)}{40 d}-\frac {a b \cos ^7(c+d x) \sin ^3(c+d x)}{5 d}+\frac {1}{64} (3 a b) \int \cos ^2(c+d x) \, dx\\ &=-\frac {\left (a^2+b^2\right ) \cos ^7(c+d x)}{7 d}+\frac {\left (a^2+2 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac {b^2 \cos ^{11}(c+d x)}{11 d}+\frac {3 a b \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a b \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {a b \cos ^5(c+d x) \sin (c+d x)}{80 d}-\frac {3 a b \cos ^7(c+d x) \sin (c+d x)}{40 d}-\frac {a b \cos ^7(c+d x) \sin ^3(c+d x)}{5 d}+\frac {1}{128} (3 a b) \int 1 \, dx\\ &=\frac {3 a b x}{128}-\frac {\left (a^2+b^2\right ) \cos ^7(c+d x)}{7 d}+\frac {\left (a^2+2 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac {b^2 \cos ^{11}(c+d x)}{11 d}+\frac {3 a b \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a b \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {a b \cos ^5(c+d x) \sin (c+d x)}{80 d}-\frac {3 a b \cos ^7(c+d x) \sin (c+d x)}{40 d}-\frac {a b \cos ^7(c+d x) \sin ^3(c+d x)}{5 d}\\ \end {align*}

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Mathematica [A]  time = 1.13, size = 197, normalized size = 1.05 \[ \frac {-6930 \left (12 a^2+5 b^2\right ) \cos (c+d x)-2310 \left (16 a^2+5 b^2\right ) \cos (3 (c+d x))+5940 a^2 \cos (7 (c+d x))+1540 a^2 \cos (9 (c+d x))+13860 a b \sin (2 (c+d x))-27720 a b \sin (4 (c+d x))-6930 a b \sin (6 (c+d x))+3465 a b \sin (8 (c+d x))+1386 a b \sin (10 (c+d x))+83160 a b c+83160 a b d x+3465 b^2 \cos (5 (c+d x))+2475 b^2 \cos (7 (c+d x))-385 b^2 \cos (9 (c+d x))-315 b^2 \cos (11 (c+d x))}{3548160 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]

[Out]

(83160*a*b*c + 83160*a*b*d*x - 6930*(12*a^2 + 5*b^2)*Cos[c + d*x] - 2310*(16*a^2 + 5*b^2)*Cos[3*(c + d*x)] + 3
465*b^2*Cos[5*(c + d*x)] + 5940*a^2*Cos[7*(c + d*x)] + 2475*b^2*Cos[7*(c + d*x)] + 1540*a^2*Cos[9*(c + d*x)] -
385*b^2*Cos[9*(c + d*x)] - 315*b^2*Cos[11*(c + d*x)] + 13860*a*b*Sin[2*(c + d*x)] - 27720*a*b*Sin[4*(c + d*x)
] - 6930*a*b*Sin[6*(c + d*x)] + 3465*a*b*Sin[8*(c + d*x)] + 1386*a*b*Sin[10*(c + d*x)])/(3548160*d)

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fricas [A]  time = 0.78, size = 128, normalized size = 0.68 \[ -\frac {40320 \, b^{2} \cos \left (d x + c\right )^{11} - 49280 \, {\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{9} + 63360 \, {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{7} - 10395 \, a b d x - 693 \, {\left (128 \, a b \cos \left (d x + c\right )^{9} - 176 \, a b \cos \left (d x + c\right )^{7} + 8 \, a b \cos \left (d x + c\right )^{5} + 10 \, a b \cos \left (d x + c\right )^{3} + 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{443520 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3*(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/443520*(40320*b^2*cos(d*x + c)^11 - 49280*(a^2 + 2*b^2)*cos(d*x + c)^9 + 63360*(a^2 + b^2)*cos(d*x + c)^7 -
10395*a*b*d*x - 693*(128*a*b*cos(d*x + c)^9 - 176*a*b*cos(d*x + c)^7 + 8*a*b*cos(d*x + c)^5 + 10*a*b*cos(d*x
+ c)^3 + 15*a*b*cos(d*x + c))*sin(d*x + c))/d

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giac [A]  time = 0.57, size = 217, normalized size = 1.16 \[ \frac {3}{128} \, a b x - \frac {b^{2} \cos \left (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac {b^{2} \cos \left (5 \, d x + 5 \, c\right )}{1024 \, d} + \frac {a b \sin \left (10 \, d x + 10 \, c\right )}{2560 \, d} + \frac {a b \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a b \sin \left (6 \, d x + 6 \, c\right )}{512 \, d} - \frac {a b \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {a b \sin \left (2 \, d x + 2 \, c\right )}{256 \, d} + \frac {{\left (4 \, a^{2} - b^{2}\right )} \cos \left (9 \, d x + 9 \, c\right )}{9216 \, d} + \frac {{\left (12 \, a^{2} + 5 \, b^{2}\right )} \cos \left (7 \, d x + 7 \, c\right )}{7168 \, d} - \frac {{\left (16 \, a^{2} + 5 \, b^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{1536 \, d} - \frac {{\left (12 \, a^{2} + 5 \, b^{2}\right )} \cos \left (d x + c\right )}{512 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3*(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

3/128*a*b*x - 1/11264*b^2*cos(11*d*x + 11*c)/d + 1/1024*b^2*cos(5*d*x + 5*c)/d + 1/2560*a*b*sin(10*d*x + 10*c)
/d + 1/1024*a*b*sin(8*d*x + 8*c)/d - 1/512*a*b*sin(6*d*x + 6*c)/d - 1/128*a*b*sin(4*d*x + 4*c)/d + 1/256*a*b*s
in(2*d*x + 2*c)/d + 1/9216*(4*a^2 - b^2)*cos(9*d*x + 9*c)/d + 1/7168*(12*a^2 + 5*b^2)*cos(7*d*x + 7*c)/d - 1/1
536*(16*a^2 + 5*b^2)*cos(3*d*x + 3*c)/d - 1/512*(12*a^2 + 5*b^2)*cos(d*x + c)/d

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maple [A]  time = 0.33, size = 171, normalized size = 0.91 \[ \frac {a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+2 a b \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+b^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*sin(d*x+c)^3*(a+b*sin(d*x+c))^2,x)

[Out]

1/d*(a^2*(-1/9*sin(d*x+c)^2*cos(d*x+c)^7-2/63*cos(d*x+c)^7)+2*a*b*(-1/10*sin(d*x+c)^3*cos(d*x+c)^7-3/80*sin(d*
x+c)*cos(d*x+c)^7+1/160*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+3/256*d*x+3/256*c)+b^2*(-1/
11*sin(d*x+c)^4*cos(d*x+c)^7-4/99*sin(d*x+c)^2*cos(d*x+c)^7-8/693*cos(d*x+c)^7))

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maxima [A]  time = 0.34, size = 115, normalized size = 0.61 \[ \frac {56320 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{2} + 693 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b - 5120 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} b^{2}}{3548160 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^3*(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/3548160*(56320*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a^2 + 693*(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*b - 5120*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^
7)*b^2)/d

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mupad [B]  time = 14.94, size = 386, normalized size = 2.06 \[ \frac {3\,a\,b\,x}{128}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (\frac {4\,a^2}{3}+\frac {32\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (8\,a^2-48\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {64\,a^2}{7}-\frac {48\,b^2}{7}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {32\,a^2}{3}-\frac {80\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {44\,a^2}{63}+\frac {16\,b^2}{63}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {32\,a^2}{63}-\frac {80\,b^2}{63}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {64\,a^2}{3}+\frac {176\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {72\,a^2}{7}+\frac {240\,b^2}{7}\right )+4\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+\frac {4\,a^2}{63}+\frac {16\,b^2}{693}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-\frac {3323\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320}+\frac {108\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}-\frac {841\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{32}+\frac {841\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}-\frac {108\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{5}+\frac {3323\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{320}-\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{2}-\frac {3\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{64}+\frac {3\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6*sin(c + d*x)^3*(a + b*sin(c + d*x))^2,x)

[Out]

(3*a*b*x)/128 - (tan(c/2 + (d*x)/2)^16*((4*a^2)/3 + (32*b^2)/3) + tan(c/2 + (d*x)/2)^10*(8*a^2 - 48*b^2) + tan
(c/2 + (d*x)/2)^6*((64*a^2)/7 - (48*b^2)/7) + tan(c/2 + (d*x)/2)^14*((32*a^2)/3 - (80*b^2)/3) + tan(c/2 + (d*x
)/2)^2*((44*a^2)/63 + (16*b^2)/63) - tan(c/2 + (d*x)/2)^4*((32*a^2)/63 - (80*b^2)/63) + tan(c/2 + (d*x)/2)^12*
((64*a^2)/3 + (176*b^2)/3) + tan(c/2 + (d*x)/2)^8*((72*a^2)/7 + (240*b^2)/7) + 4*a^2*tan(c/2 + (d*x)/2)^18 + (
4*a^2)/63 + (16*b^2)/693 + (a*b*tan(c/2 + (d*x)/2)^3)/2 - (3323*a*b*tan(c/2 + (d*x)/2)^5)/320 + (108*a*b*tan(c
/2 + (d*x)/2)^7)/5 - (841*a*b*tan(c/2 + (d*x)/2)^9)/32 + (841*a*b*tan(c/2 + (d*x)/2)^13)/32 - (108*a*b*tan(c/2
+ (d*x)/2)^15)/5 + (3323*a*b*tan(c/2 + (d*x)/2)^17)/320 - (a*b*tan(c/2 + (d*x)/2)^19)/2 - (3*a*b*tan(c/2 + (d
*x)/2)^21)/64 + (3*a*b*tan(c/2 + (d*x)/2))/64)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^11)

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sympy [A]  time = 35.40, size = 384, normalized size = 2.05 \[ \begin {cases} - \frac {a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {2 a^{2} \cos ^{9}{\left (c + d x \right )}}{63 d} + \frac {3 a b x \sin ^{10}{\left (c + d x \right )}}{128} + \frac {15 a b x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{128} + \frac {15 a b x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {15 a b x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{64} + \frac {15 a b x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 a b x \cos ^{10}{\left (c + d x \right )}}{128} + \frac {3 a b \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {7 a b \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{64 d} + \frac {a b \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {7 a b \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} - \frac {3 a b \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{128 d} - \frac {b^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {4 b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {8 b^{2} \cos ^{11}{\left (c + d x \right )}}{693 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right )^{2} \sin ^{3}{\relax (c )} \cos ^{6}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*sin(d*x+c)**3*(a+b*sin(d*x+c))**2,x)

[Out]

Piecewise((-a**2*sin(c + d*x)**2*cos(c + d*x)**7/(7*d) - 2*a**2*cos(c + d*x)**9/(63*d) + 3*a*b*x*sin(c + d*x)*
*10/128 + 15*a*b*x*sin(c + d*x)**8*cos(c + d*x)**2/128 + 15*a*b*x*sin(c + d*x)**6*cos(c + d*x)**4/64 + 15*a*b*
x*sin(c + d*x)**4*cos(c + d*x)**6/64 + 15*a*b*x*sin(c + d*x)**2*cos(c + d*x)**8/128 + 3*a*b*x*cos(c + d*x)**10
/128 + 3*a*b*sin(c + d*x)**9*cos(c + d*x)/(128*d) + 7*a*b*sin(c + d*x)**7*cos(c + d*x)**3/(64*d) + a*b*sin(c +
d*x)**5*cos(c + d*x)**5/(5*d) - 7*a*b*sin(c + d*x)**3*cos(c + d*x)**7/(64*d) - 3*a*b*sin(c + d*x)*cos(c + d*x
)**9/(128*d) - b**2*sin(c + d*x)**4*cos(c + d*x)**7/(7*d) - 4*b**2*sin(c + d*x)**2*cos(c + d*x)**9/(63*d) - 8*
b**2*cos(c + d*x)**11/(693*d), Ne(d, 0)), (x*(a + b*sin(c))**2*sin(c)**3*cos(c)**6, True))

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