\(\int \cos ^2(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx\) [1059]

Optimal result

Integrand size = 29, antiderivative size = 190 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a b x}{8}-\frac {\left (7 a^2+4 b^2\right ) \cos (c+d x)}{35 d}+\frac {\left (7 a^2+4 b^2\right ) \cos ^3(c+d x)}{105 d}-\frac {a b \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a b \cos (c+d x) \sin ^3(c+d x)}{12 d}+\frac {\left (2 a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{35 d}+\frac {a b \cos (c+d x) \sin ^5(c+d x)}{21 d}+\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{7 d} \]

[Out]

1/8*a*b*x-1/35*(7*a^2+4*b^2)*cos(d*x+c)/d+1/105*(7*a^2+4*b^2)*cos(d*x+c)^3/d-1/8*a*b*cos(d*x+c)*sin(d*x+c)/d-1
/12*a*b*cos(d*x+c)*sin(d*x+c)^3/d+1/35*(2*a^2-b^2)*cos(d*x+c)*sin(d*x+c)^4/d+1/21*a*b*cos(d*x+c)*sin(d*x+c)^5/
d+1/7*cos(d*x+c)*sin(d*x+c)^4*(a+b*sin(d*x+c))^2/d

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2968, 3129, 3112, 3102, 2827, 2713, 2715, 8} \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\left (7 a^2+4 b^2\right ) \cos ^3(c+d x)}{105 d}-\frac {\left (7 a^2+4 b^2\right ) \cos (c+d x)}{35 d}+\frac {\left (2 a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{35 d}+\frac {a b \sin ^5(c+d x) \cos (c+d x)}{21 d}+\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{7 d}-\frac {a b \sin ^3(c+d x) \cos (c+d x)}{12 d}-\frac {a b \sin (c+d x) \cos (c+d x)}{8 d}+\frac {a b x}{8} \]

[In]

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

[Out]

(a*b*x)/8 - ((7*a^2 + 4*b^2)*Cos[c + d*x])/(35*d) + ((7*a^2 + 4*b^2)*Cos[c + d*x]^3)/(105*d) - (a*b*Cos[c + d*
x]*Sin[c + d*x])/(8*d) - (a*b*Cos[c + d*x]*Sin[c + d*x]^3)/(12*d) + ((2*a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4
)/(35*d) + (a*b*Cos[c + d*x]*Sin[c + d*x]^5)/(21*d) + (Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(7*
d)

Rule 8

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

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2715

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] && Integ
erQ[2*n]

Rule 2827

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 2968

Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f,
m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 3112

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a +
b*Sin[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*
c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e
+ f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!LtQ[m, -1]

Rule 3129

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)
*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^
(n + 1)/(d*f*(m + n + 2))), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])
^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e +
f*x] + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a,
0] && NeQ[c, 0])))

Rubi steps \begin{align*} \text {integral}& = \int \sin ^3(c+d x) (a+b \sin (c+d x))^2 \left (1-\sin ^2(c+d x)\right ) \, dx \\ & = \frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{7 d}+\frac {1}{7} \int \sin ^3(c+d x) (a+b \sin (c+d x)) \left (3 a+b \sin (c+d x)-2 a \sin ^2(c+d x)\right ) \, dx \\ & = \frac {a b \cos (c+d x) \sin ^5(c+d x)}{21 d}+\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{7 d}+\frac {1}{42} \int \sin ^3(c+d x) \left (18 a^2+14 a b \sin (c+d x)-6 \left (2 a^2-b^2\right ) \sin ^2(c+d x)\right ) \, dx \\ & = \frac {\left (2 a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{35 d}+\frac {a b \cos (c+d x) \sin ^5(c+d x)}{21 d}+\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{7 d}+\frac {1}{210} \int \sin ^3(c+d x) \left (6 \left (7 a^2+4 b^2\right )+70 a b \sin (c+d x)\right ) \, dx \\ & = \frac {\left (2 a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{35 d}+\frac {a b \cos (c+d x) \sin ^5(c+d x)}{21 d}+\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{7 d}+\frac {1}{3} (a b) \int \sin ^4(c+d x) \, dx+\frac {1}{35} \left (7 a^2+4 b^2\right ) \int \sin ^3(c+d x) \, dx \\ & = -\frac {a b \cos (c+d x) \sin ^3(c+d x)}{12 d}+\frac {\left (2 a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{35 d}+\frac {a b \cos (c+d x) \sin ^5(c+d x)}{21 d}+\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{7 d}+\frac {1}{4} (a b) \int \sin ^2(c+d x) \, dx-\frac {\left (7 a^2+4 b^2\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{35 d} \\ & = -\frac {\left (7 a^2+4 b^2\right ) \cos (c+d x)}{35 d}+\frac {\left (7 a^2+4 b^2\right ) \cos ^3(c+d x)}{105 d}-\frac {a b \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a b \cos (c+d x) \sin ^3(c+d x)}{12 d}+\frac {\left (2 a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{35 d}+\frac {a b \cos (c+d x) \sin ^5(c+d x)}{21 d}+\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{7 d}+\frac {1}{8} (a b) \int 1 \, dx \\ & = \frac {a b x}{8}-\frac {\left (7 a^2+4 b^2\right ) \cos (c+d x)}{35 d}+\frac {\left (7 a^2+4 b^2\right ) \cos ^3(c+d x)}{105 d}-\frac {a b \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a b \cos (c+d x) \sin ^3(c+d x)}{12 d}+\frac {\left (2 a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{35 d}+\frac {a b \cos (c+d x) \sin ^5(c+d x)}{21 d}+\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{7 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.69 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {840 a b c+840 a b d x-105 \left (8 a^2+5 b^2\right ) \cos (c+d x)-35 \left (4 a^2+b^2\right ) \cos (3 (c+d x))+84 a^2 \cos (5 (c+d x))+63 b^2 \cos (5 (c+d x))-15 b^2 \cos (7 (c+d x))-210 a b \sin (2 (c+d x))-210 a b \sin (4 (c+d x))+70 a b \sin (6 (c+d x))}{6720 d} \]

[In]

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

[Out]

(840*a*b*c + 840*a*b*d*x - 105*(8*a^2 + 5*b^2)*Cos[c + d*x] - 35*(4*a^2 + b^2)*Cos[3*(c + d*x)] + 84*a^2*Cos[5
*(c + d*x)] + 63*b^2*Cos[5*(c + d*x)] - 15*b^2*Cos[7*(c + d*x)] - 210*a*b*Sin[2*(c + d*x)] - 210*a*b*Sin[4*(c
+ d*x)] + 70*a*b*Sin[6*(c + d*x)])/(6720*d)

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.72

method	result	size
parallelrisch	\(\frac {\left (-140 a^{2}-35 b^{2}\right ) \cos \left (3 d x +3 c \right )+\left (84 a^{2}+63 b^{2}\right ) \cos \left (5 d x +5 c \right )-15 b^{2} \cos \left (7 d x +7 c \right )-210 a b \sin \left (2 d x +2 c \right )-210 a b \sin \left (4 d x +4 c \right )+70 a b \sin \left (6 d x +6 c \right )+\left (-840 a^{2}-525 b^{2}\right ) \cos \left (d x +c \right )+840 a b x d -896 a^{2}-512 b^{2}}{6720 d}\)	\(136\)
derivativedivides	\(\frac {a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+2 a b \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+b^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{7}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{35}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{105}\right )}{d}\)	\(150\)
default	\(\frac {a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+2 a b \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+b^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{7}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{35}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{105}\right )}{d}\)	\(150\)
risch	\(\frac {a b x}{8}-\frac {a^{2} \cos \left (d x +c \right )}{8 d}-\frac {5 b^{2} \cos \left (d x +c \right )}{64 d}-\frac {b^{2} \cos \left (7 d x +7 c \right )}{448 d}+\frac {a b \sin \left (6 d x +6 c \right )}{96 d}+\frac {\cos \left (5 d x +5 c \right ) a^{2}}{80 d}+\frac {3 \cos \left (5 d x +5 c \right ) b^{2}}{320 d}-\frac {a b \sin \left (4 d x +4 c \right )}{32 d}-\frac {\cos \left (3 d x +3 c \right ) a^{2}}{48 d}-\frac {\cos \left (3 d x +3 c \right ) b^{2}}{192 d}-\frac {a b \sin \left (2 d x +2 c \right )}{32 d}\)	\(168\)
norman	\(\frac {-\frac {28 a^{2}+16 b^{2}}{105 d}+\frac {a b x}{8}-\frac {4 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (8 a^{2}-16 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {\left (8 a^{2}+16 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {\left (20 a^{2}+32 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {\left (28 a^{2}+16 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {5 a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {97 a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {97 a b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {5 a b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a b \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {7 a b x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {21 a b x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {35 a b x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {35 a b x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {21 a b x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {7 a b x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a b x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\)	\(384\)

[In]

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

[Out]

1/6720*((-140*a^2-35*b^2)*cos(3*d*x+3*c)+(84*a^2+63*b^2)*cos(5*d*x+5*c)-15*b^2*cos(7*d*x+7*c)-210*a*b*sin(2*d*
x+2*c)-210*a*b*sin(4*d*x+4*c)+70*a*b*sin(6*d*x+6*c)+(-840*a^2-525*b^2)*cos(d*x+c)+840*a*b*x*d-896*a^2-512*b^2)
/d

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.55 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {120 \, b^{2} \cos \left (d x + c\right )^{7} - 168 \, {\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 105 \, a b d x + 280 \, {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{3} - 35 \, {\left (8 \, a b \cos \left (d x + c\right )^{5} - 14 \, a b \cos \left (d x + c\right )^{3} + 3 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, d} \]

[In]

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

[Out]

-1/840*(120*b^2*cos(d*x + c)^7 - 168*(a^2 + 2*b^2)*cos(d*x + c)^5 - 105*a*b*d*x + 280*(a^2 + b^2)*cos(d*x + c)
^3 - 35*(8*a*b*cos(d*x + c)^5 - 14*a*b*cos(d*x + c)^3 + 3*a*b*cos(d*x + c))*sin(d*x + c))/d

Sympy [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.45 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\begin {cases} - \frac {a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {2 a^{2} \cos ^{5}{\left (c + d x \right )}}{15 d} + \frac {a b x \sin ^{6}{\left (c + d x \right )}}{8} + \frac {3 a b x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{8} + \frac {3 a b x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{8} + \frac {a b x \cos ^{6}{\left (c + d x \right )}}{8} + \frac {a b \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {a b \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {a b \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{8 d} - \frac {b^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {4 b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{15 d} - \frac {8 b^{2} \cos ^{7}{\left (c + d x \right )}}{105 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{2} \sin ^{3}{\left (c \right )} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**2*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)**3/(3*d) - 2*a**2*cos(c + d*x)**5/(15*d) + a*b*x*sin(c + d*x)**6
/8 + 3*a*b*x*sin(c + d*x)**4*cos(c + d*x)**2/8 + 3*a*b*x*sin(c + d*x)**2*cos(c + d*x)**4/8 + a*b*x*cos(c + d*x
)**6/8 + a*b*sin(c + d*x)**5*cos(c + d*x)/(8*d) - a*b*sin(c + d*x)**3*cos(c + d*x)**3/(3*d) - a*b*sin(c + d*x)
*cos(c + d*x)**5/(8*d) - b**2*sin(c + d*x)**4*cos(c + d*x)**3/(3*d) - 4*b**2*sin(c + d*x)**2*cos(c + d*x)**5/(
15*d) - 8*b**2*cos(c + d*x)**7/(105*d), Ne(d, 0)), (x*(a + b*sin(c))**2*sin(c)**3*cos(c)**2, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.55 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {224 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{2} - 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b - 32 \, {\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} b^{2}}{3360 \, d} \]

[In]

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

[Out]

1/3360*(224*(3*cos(d*x + c)^5 - 5*cos(d*x + c)^3)*a^2 - 35*(4*sin(2*d*x + 2*c)^3 - 12*d*x - 12*c + 3*sin(4*d*x
+ 4*c))*a*b - 32*(15*cos(d*x + c)^7 - 42*cos(d*x + c)^5 + 35*cos(d*x + c)^3)*b^2)/d

Giac [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.74 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {1}{8} \, a b x - \frac {b^{2} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {a b \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac {a b \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac {a b \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} + \frac {{\left (4 \, a^{2} + 3 \, b^{2}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {{\left (4 \, a^{2} + b^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {{\left (8 \, a^{2} + 5 \, b^{2}\right )} \cos \left (d x + c\right )}{64 \, d} \]

[In]

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

[Out]

1/8*a*b*x - 1/448*b^2*cos(7*d*x + 7*c)/d + 1/96*a*b*sin(6*d*x + 6*c)/d - 1/32*a*b*sin(4*d*x + 4*c)/d - 1/32*a*
b*sin(2*d*x + 2*c)/d + 1/320*(4*a^2 + 3*b^2)*cos(5*d*x + 5*c)/d - 1/192*(4*a^2 + b^2)*cos(3*d*x + 3*c)/d - 1/6
4*(8*a^2 + 5*b^2)*cos(d*x + c)/d

Mupad [B] (verification not implemented)

Time = 14.29 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.23 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a\,b\,x}{8}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {8\,a^2}{3}-\frac {16\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {8\,a^2}{5}+\frac {16\,b^2}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {20\,a^2}{3}+\frac {32\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {28\,a^2}{15}+\frac {16\,b^2}{15}\right )+4\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {4\,a^2}{15}+\frac {16\,b^2}{105}+\frac {5\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-\frac {97\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}+\frac {97\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{12}-\frac {5\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}-\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{4}+\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]

[In]

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

[Out]

(a*b*x)/8 - (tan(c/2 + (d*x)/2)^6*((8*a^2)/3 - (16*b^2)/3) + tan(c/2 + (d*x)/2)^4*((8*a^2)/5 + (16*b^2)/5) + t
an(c/2 + (d*x)/2)^8*((20*a^2)/3 + (32*b^2)/3) + tan(c/2 + (d*x)/2)^2*((28*a^2)/15 + (16*b^2)/15) + 4*a^2*tan(c
/2 + (d*x)/2)^10 + (4*a^2)/15 + (16*b^2)/105 + (5*a*b*tan(c/2 + (d*x)/2)^3)/3 - (97*a*b*tan(c/2 + (d*x)/2)^5)/
12 + (97*a*b*tan(c/2 + (d*x)/2)^9)/12 - (5*a*b*tan(c/2 + (d*x)/2)^11)/3 - (a*b*tan(c/2 + (d*x)/2)^13)/4 + (a*b
*tan(c/2 + (d*x)/2))/4)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^7)