3.1239 \(\int \cos ^6(c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=238 \[ -\frac {\left (a^2+3 b^2\right ) \cos ^{11}(c+d x)}{11 d}+\frac {\left (2 a^2+3 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac {\left (a^2+b^2\right ) \cos ^7(c+d x)}{7 d}-\frac {a b \sin ^5(c+d x) \cos ^7(c+d x)}{6 d}-\frac {a b \sin ^3(c+d x) \cos ^7(c+d x)}{12 d}-\frac {a b \sin (c+d x) \cos ^7(c+d x)}{32 d}+\frac {a b \sin (c+d x) \cos ^5(c+d x)}{192 d}+\frac {5 a b \sin (c+d x) \cos ^3(c+d x)}{768 d}+\frac {5 a b \sin (c+d x) \cos (c+d x)}{512 d}+\frac {5 a b x}{512}+\frac {b^2 \cos ^{13}(c+d x)}{13 d} \]

[Out]

5/512*a*b*x-1/7*(a^2+b^2)*cos(d*x+c)^7/d+1/9*(2*a^2+3*b^2)*cos(d*x+c)^9/d-1/11*(a^2+3*b^2)*cos(d*x+c)^11/d+1/1
3*b^2*cos(d*x+c)^13/d+5/512*a*b*cos(d*x+c)*sin(d*x+c)/d+5/768*a*b*cos(d*x+c)^3*sin(d*x+c)/d+1/192*a*b*cos(d*x+
c)^5*sin(d*x+c)/d-1/32*a*b*cos(d*x+c)^7*sin(d*x+c)/d-1/12*a*b*cos(d*x+c)^7*sin(d*x+c)^3/d-1/6*a*b*cos(d*x+c)^7
*sin(d*x+c)^5/d

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Rubi [A]  time = 0.37, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 12, 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} \[ -\frac {\left (a^2+3 b^2\right ) \cos ^{11}(c+d x)}{11 d}+\frac {\left (2 a^2+3 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac {\left (a^2+b^2\right ) \cos ^7(c+d x)}{7 d}-\frac {a b \sin ^5(c+d x) \cos ^7(c+d x)}{6 d}-\frac {a b \sin ^3(c+d x) \cos ^7(c+d x)}{12 d}-\frac {a b \sin (c+d x) \cos ^7(c+d x)}{32 d}+\frac {a b \sin (c+d x) \cos ^5(c+d x)}{192 d}+\frac {5 a b \sin (c+d x) \cos ^3(c+d x)}{768 d}+\frac {5 a b \sin (c+d x) \cos (c+d x)}{512 d}+\frac {5 a b x}{512}+\frac {b^2 \cos ^{13}(c+d x)}{13 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*Sin[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]

[Out]

(5*a*b*x)/512 - ((a^2 + b^2)*Cos[c + d*x]^7)/(7*d) + ((2*a^2 + 3*b^2)*Cos[c + d*x]^9)/(9*d) - ((a^2 + 3*b^2)*C
os[c + d*x]^11)/(11*d) + (b^2*Cos[c + d*x]^13)/(13*d) + (5*a*b*Cos[c + d*x]*Sin[c + d*x])/(512*d) + (5*a*b*Cos
[c + d*x]^3*Sin[c + d*x])/(768*d) + (a*b*Cos[c + d*x]^5*Sin[c + d*x])/(192*d) - (a*b*Cos[c + d*x]^7*Sin[c + d*
x])/(32*d) - (a*b*Cos[c + d*x]^7*Sin[c + d*x]^3)/(12*d) - (a*b*Cos[c + d*x]^7*Sin[c + d*x]^5)/(6*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 ^5(c+d x) (a+b \sin (c+d x))^2 \, dx &=(2 a b) \int \cos ^6(c+d x) \sin ^6(c+d x) \, dx+\int \cos ^6(c+d x) \sin ^5(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac {a b \cos ^7(c+d x) \sin ^5(c+d x)}{6 d}+\frac {1}{6} (5 a b) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx+\frac {\left (\sqrt {\cos ^2(c+d x)} \sec (c+d x)\right ) \operatorname {Subst}\left (\int x^5 \left (1-x^2\right )^{5/2} \left (a^2+b^2 x^2\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {a b \cos ^7(c+d x) \sin ^3(c+d x)}{12 d}-\frac {a b \cos ^7(c+d x) \sin ^5(c+d x)}{6 d}+\frac {1}{4} (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 (1-x)^{5/2} x^2 \left (a^2+b^2 x\right ) \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=-\frac {a b \cos ^7(c+d x) \sin (c+d x)}{32 d}-\frac {a b \cos ^7(c+d x) \sin ^3(c+d x)}{12 d}-\frac {a b \cos ^7(c+d x) \sin ^5(c+d x)}{6 d}+\frac {1}{32} (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 \left (\left (a^2+b^2\right ) (1-x)^{5/2}+\left (-2 a^2-3 b^2\right ) (1-x)^{7/2}+\left (a^2+3 b^2\right ) (1-x)^{9/2}-b^2 (1-x)^{11/2}\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 (2 a^2+3 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac {\left (a^2+3 b^2\right ) \cos ^{11}(c+d x)}{11 d}+\frac {b^2 \cos ^{13}(c+d x)}{13 d}+\frac {a b \cos ^5(c+d x) \sin (c+d x)}{192 d}-\frac {a b \cos ^7(c+d x) \sin (c+d x)}{32 d}-\frac {a b \cos ^7(c+d x) \sin ^3(c+d x)}{12 d}-\frac {a b \cos ^7(c+d x) \sin ^5(c+d x)}{6 d}+\frac {1}{192} (5 a b) \int \cos ^4(c+d x) \, dx\\ &=-\frac {\left (a^2+b^2\right ) \cos ^7(c+d x)}{7 d}+\frac {\left (2 a^2+3 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac {\left (a^2+3 b^2\right ) \cos ^{11}(c+d x)}{11 d}+\frac {b^2 \cos ^{13}(c+d x)}{13 d}+\frac {5 a b \cos ^3(c+d x) \sin (c+d x)}{768 d}+\frac {a b \cos ^5(c+d x) \sin (c+d x)}{192 d}-\frac {a b \cos ^7(c+d x) \sin (c+d x)}{32 d}-\frac {a b \cos ^7(c+d x) \sin ^3(c+d x)}{12 d}-\frac {a b \cos ^7(c+d x) \sin ^5(c+d x)}{6 d}+\frac {1}{256} (5 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 (2 a^2+3 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac {\left (a^2+3 b^2\right ) \cos ^{11}(c+d x)}{11 d}+\frac {b^2 \cos ^{13}(c+d x)}{13 d}+\frac {5 a b \cos (c+d x) \sin (c+d x)}{512 d}+\frac {5 a b \cos ^3(c+d x) \sin (c+d x)}{768 d}+\frac {a b \cos ^5(c+d x) \sin (c+d x)}{192 d}-\frac {a b \cos ^7(c+d x) \sin (c+d x)}{32 d}-\frac {a b \cos ^7(c+d x) \sin ^3(c+d x)}{12 d}-\frac {a b \cos ^7(c+d x) \sin ^5(c+d x)}{6 d}+\frac {1}{512} (5 a b) \int 1 \, dx\\ &=\frac {5 a b x}{512}-\frac {\left (a^2+b^2\right ) \cos ^7(c+d x)}{7 d}+\frac {\left (2 a^2+3 b^2\right ) \cos ^9(c+d x)}{9 d}-\frac {\left (a^2+3 b^2\right ) \cos ^{11}(c+d x)}{11 d}+\frac {b^2 \cos ^{13}(c+d x)}{13 d}+\frac {5 a b \cos (c+d x) \sin (c+d x)}{512 d}+\frac {5 a b \cos ^3(c+d x) \sin (c+d x)}{768 d}+\frac {a b \cos ^5(c+d x) \sin (c+d x)}{192 d}-\frac {a b \cos ^7(c+d x) \sin (c+d x)}{32 d}-\frac {a b \cos ^7(c+d x) \sin ^3(c+d x)}{12 d}-\frac {a b \cos ^7(c+d x) \sin ^5(c+d x)}{6 d}\\ \end {align*}

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Mathematica [A]  time = 2.23, size = 210, normalized size = 0.88 \[ \frac {-180180 \left (2 a^2+b^2\right ) \cos (c+d x)-15015 \left (8 a^2+3 b^2\right ) \cos (3 (c+d x))+36036 a^2 \cos (5 (c+d x))+25740 a^2 \cos (7 (c+d x))-4004 a^2 \cos (9 (c+d x))-3276 a^2 \cos (11 (c+d x))-135135 a b \sin (4 (c+d x))+27027 a b \sin (8 (c+d x))-3003 a b \sin (12 (c+d x))+360360 a b c+360360 a b d x+27027 b^2 \cos (5 (c+d x))+7722 b^2 \cos (7 (c+d x))-6006 b^2 \cos (9 (c+d x))-819 b^2 \cos (11 (c+d x))+693 b^2 \cos (13 (c+d x))}{36900864 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*Sin[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]

[Out]

(360360*a*b*c + 360360*a*b*d*x - 180180*(2*a^2 + b^2)*Cos[c + d*x] - 15015*(8*a^2 + 3*b^2)*Cos[3*(c + d*x)] +
36036*a^2*Cos[5*(c + d*x)] + 27027*b^2*Cos[5*(c + d*x)] + 25740*a^2*Cos[7*(c + d*x)] + 7722*b^2*Cos[7*(c + d*x
)] - 4004*a^2*Cos[9*(c + d*x)] - 6006*b^2*Cos[9*(c + d*x)] - 3276*a^2*Cos[11*(c + d*x)] - 819*b^2*Cos[11*(c +
d*x)] + 693*b^2*Cos[13*(c + d*x)] - 135135*a*b*Sin[4*(c + d*x)] + 27027*a*b*Sin[8*(c + d*x)] - 3003*a*b*Sin[12
*(c + d*x)])/(36900864*d)

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fricas [A]  time = 0.89, size = 161, normalized size = 0.68 \[ \frac {354816 \, b^{2} \cos \left (d x + c\right )^{13} - 419328 \, {\left (a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{11} + 512512 \, {\left (2 \, a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{9} - 658944 \, {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{7} + 45045 \, a b d x - 3003 \, {\left (256 \, a b \cos \left (d x + c\right )^{11} - 640 \, a b \cos \left (d x + c\right )^{9} + 432 \, a b \cos \left (d x + c\right )^{7} - 8 \, a b \cos \left (d x + c\right )^{5} - 10 \, a b \cos \left (d x + c\right )^{3} - 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4612608 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^5*(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/4612608*(354816*b^2*cos(d*x + c)^13 - 419328*(a^2 + 3*b^2)*cos(d*x + c)^11 + 512512*(2*a^2 + 3*b^2)*cos(d*x
+ c)^9 - 658944*(a^2 + b^2)*cos(d*x + c)^7 + 45045*a*b*d*x - 3003*(256*a*b*cos(d*x + c)^11 - 640*a*b*cos(d*x +
c)^9 + 432*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 = 0.79, size = 214, normalized size = 0.90 \[ \frac {5}{512} \, a b x + \frac {b^{2} \cos \left (13 \, d x + 13 \, c\right )}{53248 \, d} - \frac {a b \sin \left (12 \, d x + 12 \, c\right )}{12288 \, d} + \frac {3 \, a b \sin \left (8 \, d x + 8 \, c\right )}{4096 \, d} - \frac {15 \, a b \sin \left (4 \, d x + 4 \, c\right )}{4096 \, d} - \frac {{\left (4 \, a^{2} + b^{2}\right )} \cos \left (11 \, d x + 11 \, c\right )}{45056 \, d} - \frac {{\left (2 \, a^{2} + 3 \, b^{2}\right )} \cos \left (9 \, d x + 9 \, c\right )}{18432 \, d} + \frac {{\left (10 \, a^{2} + 3 \, b^{2}\right )} \cos \left (7 \, d x + 7 \, c\right )}{14336 \, d} + \frac {{\left (4 \, a^{2} + 3 \, b^{2}\right )} \cos \left (5 \, d x + 5 \, c\right )}{4096 \, d} - \frac {5 \, {\left (8 \, a^{2} + 3 \, b^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{12288 \, d} - \frac {5 \, {\left (2 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )}{1024 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^5*(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

5/512*a*b*x + 1/53248*b^2*cos(13*d*x + 13*c)/d - 1/12288*a*b*sin(12*d*x + 12*c)/d + 3/4096*a*b*sin(8*d*x + 8*c
)/d - 15/4096*a*b*sin(4*d*x + 4*c)/d - 1/45056*(4*a^2 + b^2)*cos(11*d*x + 11*c)/d - 1/18432*(2*a^2 + 3*b^2)*co
s(9*d*x + 9*c)/d + 1/14336*(10*a^2 + 3*b^2)*cos(7*d*x + 7*c)/d + 1/4096*(4*a^2 + 3*b^2)*cos(5*d*x + 5*c)/d - 5
/12288*(8*a^2 + 3*b^2)*cos(3*d*x + 3*c)/d - 5/1024*(2*a^2 + b^2)*cos(d*x + c)/d

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maple [A]  time = 0.33, size = 225, normalized size = 0.95 \[ \frac {a^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+2 a b \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{24}-\frac {\sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{64}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{384}+\frac {5 d x}{1024}+\frac {5 c}{1024}\right )+b^{2} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{13}-\frac {6 \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{143}-\frac {8 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{429}-\frac {16 \left (\cos ^{7}\left (d x +c \right )\right )}{3003}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*sin(d*x+c)^5*(a+b*sin(d*x+c))^2,x)

[Out]

1/d*(a^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)+2*a*b*(-1/12*sin(
d*x+c)^5*cos(d*x+c)^7-1/24*sin(d*x+c)^3*cos(d*x+c)^7-1/64*sin(d*x+c)*cos(d*x+c)^7+1/384*(cos(d*x+c)^5+5/4*cos(
d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/1024*d*x+5/1024*c)+b^2*(-1/13*sin(d*x+c)^6*cos(d*x+c)^7-6/143*sin(d*x+c
)^4*cos(d*x+c)^7-8/429*sin(d*x+c)^2*cos(d*x+c)^7-16/3003*cos(d*x+c)^7))

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maxima [A]  time = 0.34, size = 135, normalized size = 0.57 \[ -\frac {53248 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{2} - 3003 \, {\left (4 \, \sin \left (4 \, d x + 4 \, c\right )^{3} + 120 \, d x + 120 \, c + 9 \, \sin \left (8 \, d x + 8 \, c\right ) - 48 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b - 12288 \, {\left (231 \, \cos \left (d x + c\right )^{13} - 819 \, \cos \left (d x + c\right )^{11} + 1001 \, \cos \left (d x + c\right )^{9} - 429 \, \cos \left (d x + c\right )^{7}\right )} b^{2}}{36900864 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^5*(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/36900864*(53248*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^7)*a^2 - 3003*(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))*a*b - 12288*(231*cos(d*x + c)^13 - 819*cos(d
*x + c)^11 + 1001*cos(d*x + c)^9 - 429*cos(d*x + c)^7)*b^2)/d

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mupad [B]  time = 15.01, size = 441, normalized size = 1.85 \[ \frac {5\,a\,b\,x}{512}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (16\,a^2-96\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,\left (\frac {16\,a^2}{3}-32\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {128\,a^2}{3}+192\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {256\,a^2}{63}-\frac {64\,b^2}{21}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {416\,a^2}{231}+\frac {64\,b^2}{77}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {96\,a^2}{7}+\frac {768\,b^2}{7}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {208\,a^2}{693}+\frac {32\,b^2}{231}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {64\,a^2}{21}+\frac {1216\,b^2}{7}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {1376\,a^2}{63}-\frac {512\,b^2}{21}\right )+\frac {32\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}}{3}+\frac {16\,a^2}{693}+\frac {32\,b^2}{3003}+\frac {95\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384}+\frac {277\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}-\frac {4025\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{128}+\frac {59435\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{768}-\frac {16813\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{192}+\frac {16813\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{192}-\frac {59435\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{768}+\frac {4025\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}-\frac {277\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{192}-\frac {95\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{23}}{384}-\frac {5\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{25}}{256}+\frac {5\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{256}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6*sin(c + d*x)^5*(a + b*sin(c + d*x))^2,x)

[Out]

(5*a*b*x)/512 - (tan(c/2 + (d*x)/2)^16*(16*a^2 - 96*b^2) - tan(c/2 + (d*x)/2)^18*((16*a^2)/3 - 32*b^2) + tan(c
/2 + (d*x)/2)^14*((128*a^2)/3 + 192*b^2) - tan(c/2 + (d*x)/2)^6*((256*a^2)/63 - (64*b^2)/21) + tan(c/2 + (d*x)
/2)^4*((416*a^2)/231 + (64*b^2)/77) + tan(c/2 + (d*x)/2)^10*((96*a^2)/7 + (768*b^2)/7) + tan(c/2 + (d*x)/2)^2*
((208*a^2)/693 + (32*b^2)/231) - tan(c/2 + (d*x)/2)^12*((64*a^2)/21 + (1216*b^2)/7) + tan(c/2 + (d*x)/2)^8*((1
376*a^2)/63 - (512*b^2)/21) + (32*a^2*tan(c/2 + (d*x)/2)^20)/3 + (16*a^2)/693 + (32*b^2)/3003 + (95*a*b*tan(c/
2 + (d*x)/2)^3)/384 + (277*a*b*tan(c/2 + (d*x)/2)^5)/192 - (4025*a*b*tan(c/2 + (d*x)/2)^7)/128 + (59435*a*b*ta
n(c/2 + (d*x)/2)^9)/768 - (16813*a*b*tan(c/2 + (d*x)/2)^11)/192 + (16813*a*b*tan(c/2 + (d*x)/2)^15)/192 - (594
35*a*b*tan(c/2 + (d*x)/2)^17)/768 + (4025*a*b*tan(c/2 + (d*x)/2)^19)/128 - (277*a*b*tan(c/2 + (d*x)/2)^21)/192
- (95*a*b*tan(c/2 + (d*x)/2)^23)/384 - (5*a*b*tan(c/2 + (d*x)/2)^25)/256 + (5*a*b*tan(c/2 + (d*x)/2))/256)/(d
*(tan(c/2 + (d*x)/2)^2 + 1)^13)

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sympy [A]  time = 77.82, size = 488, normalized size = 2.05 \[ \begin {cases} - \frac {a^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {4 a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {8 a^{2} \cos ^{11}{\left (c + d x \right )}}{693 d} + \frac {5 a b x \sin ^{12}{\left (c + d x \right )}}{512} + \frac {15 a b x \sin ^{10}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {75 a b x \sin ^{8}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{512} + \frac {25 a b x \sin ^{6}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {75 a b x \sin ^{4}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{512} + \frac {15 a b x \sin ^{2}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{256} + \frac {5 a b x \cos ^{12}{\left (c + d x \right )}}{512} + \frac {5 a b \sin ^{11}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{512 d} + \frac {85 a b \sin ^{9}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{1536 d} + \frac {33 a b \sin ^{7}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{256 d} - \frac {33 a b \sin ^{5}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{256 d} - \frac {85 a b \sin ^{3}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{1536 d} - \frac {5 a b \sin {\left (c + d x \right )} \cos ^{11}{\left (c + d x \right )}}{512 d} - \frac {b^{2} \sin ^{6}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {2 b^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{21 d} - \frac {8 b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{11}{\left (c + d x \right )}}{231 d} - \frac {16 b^{2} \cos ^{13}{\left (c + d x \right )}}{3003 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right )^{2} \sin ^{5}{\relax (c )} \cos ^{6}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*sin(d*x+c)**5*(a+b*sin(d*x+c))**2,x)

[Out]

Piecewise((-a**2*sin(c + d*x)**4*cos(c + d*x)**7/(7*d) - 4*a**2*sin(c + d*x)**2*cos(c + d*x)**9/(63*d) - 8*a**
2*cos(c + d*x)**11/(693*d) + 5*a*b*x*sin(c + d*x)**12/512 + 15*a*b*x*sin(c + d*x)**10*cos(c + d*x)**2/256 + 75
*a*b*x*sin(c + d*x)**8*cos(c + d*x)**4/512 + 25*a*b*x*sin(c + d*x)**6*cos(c + d*x)**6/128 + 75*a*b*x*sin(c + d
*x)**4*cos(c + d*x)**8/512 + 15*a*b*x*sin(c + d*x)**2*cos(c + d*x)**10/256 + 5*a*b*x*cos(c + d*x)**12/512 + 5*
a*b*sin(c + d*x)**11*cos(c + d*x)/(512*d) + 85*a*b*sin(c + d*x)**9*cos(c + d*x)**3/(1536*d) + 33*a*b*sin(c + d
*x)**7*cos(c + d*x)**5/(256*d) - 33*a*b*sin(c + d*x)**5*cos(c + d*x)**7/(256*d) - 85*a*b*sin(c + d*x)**3*cos(c
+ d*x)**9/(1536*d) - 5*a*b*sin(c + d*x)*cos(c + d*x)**11/(512*d) - b**2*sin(c + d*x)**6*cos(c + d*x)**7/(7*d)
- 2*b**2*sin(c + d*x)**4*cos(c + d*x)**9/(21*d) - 8*b**2*sin(c + d*x)**2*cos(c + d*x)**11/(231*d) - 16*b**2*c
os(c + d*x)**13/(3003*d), Ne(d, 0)), (x*(a + b*sin(c))**2*sin(c)**5*cos(c)**6, True))

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