\(\int \cos ^4(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx\) [1105]

Optimal result

Integrand size = 29, antiderivative size = 301 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {3 a b x}{64}-\frac {\left (9 a^2+4 b^2\right ) \cos (c+d x)}{105 d}+\frac {\left (9 a^2+4 b^2\right ) \cos ^3(c+d x)}{315 d}-\frac {3 a b \cos (c+d x) \sin (c+d x)}{64 d}-\frac {a b \cos (c+d x) \sin ^3(c+d x)}{32 d}-\frac {\left (15 a^4-44 a^2 b^2+6 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{630 b^2 d}-\frac {a \left (10 a^2-29 b^2\right ) \cos (c+d x) \sin ^5(c+d x)}{504 b d}-\frac {5 \left (3 a^2-8 b^2\right ) \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}+\frac {a \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^3}{9 b d} \]

[Out]

3/64*a*b*x-1/105*(9*a^2+4*b^2)*cos(d*x+c)/d+1/315*(9*a^2+4*b^2)*cos(d*x+c)^3/d-3/64*a*b*cos(d*x+c)*sin(d*x+c)/
d-1/32*a*b*cos(d*x+c)*sin(d*x+c)^3/d-1/630*(15*a^4-44*a^2*b^2+6*b^4)*cos(d*x+c)*sin(d*x+c)^4/b^2/d-1/504*a*(10
*a^2-29*b^2)*cos(d*x+c)*sin(d*x+c)^5/b/d-5/252*(3*a^2-8*b^2)*cos(d*x+c)*sin(d*x+c)^4*(a+b*sin(d*x+c))^2/b^2/d+
1/12*a*cos(d*x+c)*sin(d*x+c)^4*(a+b*sin(d*x+c))^3/b^2/d-1/9*cos(d*x+c)*sin(d*x+c)^5*(a+b*sin(d*x+c))^3/b/d

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 301, 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 = {2974, 3128, 3112, 3102, 2827, 2713, 2715, 8} \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\left (9 a^2+4 b^2\right ) \cos ^3(c+d x)}{315 d}-\frac {\left (9 a^2+4 b^2\right ) \cos (c+d x)}{105 d}-\frac {a \left (10 a^2-29 b^2\right ) \sin ^5(c+d x) \cos (c+d x)}{504 b d}-\frac {5 \left (3 a^2-8 b^2\right ) \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}-\frac {\left (15 a^4-44 a^2 b^2+6 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{630 b^2 d}+\frac {a \sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac {\sin ^5(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{9 b d}-\frac {a b \sin ^3(c+d x) \cos (c+d x)}{32 d}-\frac {3 a b \sin (c+d x) \cos (c+d x)}{64 d}+\frac {3 a b x}{64} \]

[In]

Int[Cos[c + d*x]^4*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]

[Out]

(3*a*b*x)/64 - ((9*a^2 + 4*b^2)*Cos[c + d*x])/(105*d) + ((9*a^2 + 4*b^2)*Cos[c + d*x]^3)/(315*d) - (3*a*b*Cos[
c + d*x]*Sin[c + d*x])/(64*d) - (a*b*Cos[c + d*x]*Sin[c + d*x]^3)/(32*d) - ((15*a^4 - 44*a^2*b^2 + 6*b^4)*Cos[
c + d*x]*Sin[c + d*x]^4)/(630*b^2*d) - (a*(10*a^2 - 29*b^2)*Cos[c + d*x]*Sin[c + d*x]^5)/(504*b*d) - (5*(3*a^2
- 8*b^2)*Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^2)/(252*b^2*d) + (a*Cos[c + d*x]*Sin[c + d*x]^4*(a
+ b*Sin[c + d*x])^3)/(12*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^5*(a + b*Sin[c + d*x])^3)/(9*b*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 2974

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[a*(n + 3)*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*((a + b*Sin[e + f*x])^(m + 1)/(b^2*d*f*(m
+ n + 3)*(m + n + 4))), x] + (-Dist[1/(b^2*(m + n + 3)*(m + n + 4)), Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x
])^m*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 3)*(m + n + 4) + a*b*m*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*
(m + n + 3)*(m + n + 5))*Sin[e + f*x]^2, x], x], x] - Simp[Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*((a + b*Sin[e
+ f*x])^(m + 1)/(b*d^2*f*(m + n + 4))), x]) /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[
m, 0] || IntegersQ[2*m, 2*n]) &&  !m < -1 &&  !LtQ[n, -1] && NeQ[m + n + 3, 0] && NeQ[m + n + 4, 0]

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 3128

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((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*((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)) + (d*(A*b + a*B)*(m + n +
2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, 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}& = \frac {a \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^3}{9 b d}-\frac {\int \sin ^3(c+d x) (a+b \sin (c+d x))^2 \left (24 \left (a^2-3 b^2\right )+2 a b \sin (c+d x)-10 \left (3 a^2-8 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{72 b^2} \\ & = -\frac {5 \left (3 a^2-8 b^2\right ) \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}+\frac {a \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^3}{9 b d}-\frac {\int \sin ^3(c+d x) (a+b \sin (c+d x)) \left (8 a \left (6 a^2-23 b^2\right )+2 b \left (a^2-12 b^2\right ) \sin (c+d x)-6 a \left (10 a^2-29 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{504 b^2} \\ & = -\frac {a \left (10 a^2-29 b^2\right ) \cos (c+d x) \sin ^5(c+d x)}{504 b d}-\frac {5 \left (3 a^2-8 b^2\right ) \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}+\frac {a \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^3}{9 b d}-\frac {\int \sin ^3(c+d x) \left (48 a^2 \left (6 a^2-23 b^2\right )-378 a b^3 \sin (c+d x)-24 \left (15 a^4-44 a^2 b^2+6 b^4\right ) \sin ^2(c+d x)\right ) \, dx}{3024 b^2} \\ & = -\frac {\left (15 a^4-44 a^2 b^2+6 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{630 b^2 d}-\frac {a \left (10 a^2-29 b^2\right ) \cos (c+d x) \sin ^5(c+d x)}{504 b d}-\frac {5 \left (3 a^2-8 b^2\right ) \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}+\frac {a \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^3}{9 b d}-\frac {\int \sin ^3(c+d x) \left (-144 b^2 \left (9 a^2+4 b^2\right )-1890 a b^3 \sin (c+d x)\right ) \, dx}{15120 b^2} \\ & = -\frac {\left (15 a^4-44 a^2 b^2+6 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{630 b^2 d}-\frac {a \left (10 a^2-29 b^2\right ) \cos (c+d x) \sin ^5(c+d x)}{504 b d}-\frac {5 \left (3 a^2-8 b^2\right ) \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}+\frac {a \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^3}{9 b d}+\frac {1}{8} (a b) \int \sin ^4(c+d x) \, dx-\frac {1}{105} \left (-9 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)}{32 d}-\frac {\left (15 a^4-44 a^2 b^2+6 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{630 b^2 d}-\frac {a \left (10 a^2-29 b^2\right ) \cos (c+d x) \sin ^5(c+d x)}{504 b d}-\frac {5 \left (3 a^2-8 b^2\right ) \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}+\frac {a \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^3}{9 b d}+\frac {1}{32} (3 a b) \int \sin ^2(c+d x) \, dx-\frac {\left (9 a^2+4 b^2\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{105 d} \\ & = -\frac {\left (9 a^2+4 b^2\right ) \cos (c+d x)}{105 d}+\frac {\left (9 a^2+4 b^2\right ) \cos ^3(c+d x)}{315 d}-\frac {3 a b \cos (c+d x) \sin (c+d x)}{64 d}-\frac {a b \cos (c+d x) \sin ^3(c+d x)}{32 d}-\frac {\left (15 a^4-44 a^2 b^2+6 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{630 b^2 d}-\frac {a \left (10 a^2-29 b^2\right ) \cos (c+d x) \sin ^5(c+d x)}{504 b d}-\frac {5 \left (3 a^2-8 b^2\right ) \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}+\frac {a \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^3}{9 b d}+\frac {1}{64} (3 a b) \int 1 \, dx \\ & = \frac {3 a b x}{64}-\frac {\left (9 a^2+4 b^2\right ) \cos (c+d x)}{105 d}+\frac {\left (9 a^2+4 b^2\right ) \cos ^3(c+d x)}{315 d}-\frac {3 a b \cos (c+d x) \sin (c+d x)}{64 d}-\frac {a b \cos (c+d x) \sin ^3(c+d x)}{32 d}-\frac {\left (15 a^4-44 a^2 b^2+6 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{630 b^2 d}-\frac {a \left (10 a^2-29 b^2\right ) \cos (c+d x) \sin ^5(c+d x)}{504 b d}-\frac {5 \left (3 a^2-8 b^2\right ) \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^2}{252 b^2 d}+\frac {a \cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{12 b^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x) (a+b \sin (c+d x))^3}{9 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.48 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {7560 a b c+7560 a b d x-3780 \left (2 a^2+b^2\right ) \cos (c+d x)-840 \left (3 a^2+b^2\right ) \cos (3 (c+d x))+504 a^2 \cos (5 (c+d x))+504 b^2 \cos (5 (c+d x))+360 a^2 \cos (7 (c+d x))+90 b^2 \cos (7 (c+d x))-70 b^2 \cos (9 (c+d x))-2520 a b \sin (4 (c+d x))+315 a b \sin (8 (c+d x))}{161280 d} \]

[In]

Integrate[Cos[c + d*x]^4*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]

[Out]

(7560*a*b*c + 7560*a*b*d*x - 3780*(2*a^2 + b^2)*Cos[c + d*x] - 840*(3*a^2 + b^2)*Cos[3*(c + d*x)] + 504*a^2*Co
s[5*(c + d*x)] + 504*b^2*Cos[5*(c + d*x)] + 360*a^2*Cos[7*(c + d*x)] + 90*b^2*Cos[7*(c + d*x)] - 70*b^2*Cos[9*
(c + d*x)] - 2520*a*b*Sin[4*(c + d*x)] + 315*a*b*Sin[8*(c + d*x)])/(161280*d)

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.47

method	result	size
parallelrisch	\(\frac {840 \left (-3 a^{2}-b^{2}\right ) \cos \left (3 d x +3 c \right )+504 \left (a^{2}+b^{2}\right ) \cos \left (5 d x +5 c \right )+90 \left (4 a^{2}+b^{2}\right ) \cos \left (7 d x +7 c \right )-70 b^{2} \cos \left (9 d x +9 c \right )-2520 a b \sin \left (4 d x +4 c \right )+315 a b \sin \left (8 d x +8 c \right )+3780 \left (-2 a^{2}-b^{2}\right ) \cos \left (d x +c \right )+7560 a b x d -9216 a^{2}-4096 b^{2}}{161280 d}\)	\(142\)
derivativedivides	\(\frac {a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+2 a b \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+b^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )}{d}\)	\(161\)
default	\(\frac {a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+2 a b \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+b^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )}{d}\)	\(161\)
risch	\(\frac {3 a b x}{64}-\frac {3 a^{2} \cos \left (d x +c \right )}{64 d}-\frac {3 b^{2} \cos \left (d x +c \right )}{128 d}-\frac {b^{2} \cos \left (9 d x +9 c \right )}{2304 d}+\frac {a b \sin \left (8 d x +8 c \right )}{512 d}+\frac {\cos \left (7 d x +7 c \right ) a^{2}}{448 d}+\frac {\cos \left (7 d x +7 c \right ) b^{2}}{1792 d}+\frac {\cos \left (5 d x +5 c \right ) a^{2}}{320 d}+\frac {\cos \left (5 d x +5 c \right ) b^{2}}{320 d}-\frac {a b \sin \left (4 d x +4 c \right )}{64 d}-\frac {\cos \left (3 d x +3 c \right ) a^{2}}{64 d}-\frac {\cos \left (3 d x +3 c \right ) b^{2}}{192 d}\)	\(186\)
norman	\(\frac {-\frac {2 \left (26 a^{2}+56 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {13 a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {3 a b x}{64}+\frac {63 a b x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {155 a b \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {169 a b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {3 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 d}-\frac {36 a^{2}+16 b^{2}}{315 d}+\frac {169 a b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {13 a b \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {3 a b \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {27 a b x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {27 a b x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {189 a b x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {4 a^{2} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {27 a b x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {155 a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}-\frac {2 \left (2 a^{2}-8 b^{2}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (36 a^{2}+16 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}-\frac {4 \left (3 a^{2}+8 b^{2}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {4 \left (a^{2}+16 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}-\frac {4 \left (7 a^{2}-8 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {63 a b x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {27 a b x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {189 a b x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {3 a b x \left (\tan ^{18}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}\)	\(504\)

[In]

int(cos(d*x+c)^4*sin(d*x+c)^3*(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/161280*(840*(-3*a^2-b^2)*cos(3*d*x+3*c)+504*(a^2+b^2)*cos(5*d*x+5*c)+90*(4*a^2+b^2)*cos(7*d*x+7*c)-70*b^2*co
s(9*d*x+9*c)-2520*a*b*sin(4*d*x+4*c)+315*a*b*sin(8*d*x+8*c)+3780*(-2*a^2-b^2)*cos(d*x+c)+7560*a*b*x*d-9216*a^2
-4096*b^2)/d

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.39 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {2240 \, b^{2} \cos \left (d x + c\right )^{9} - 2880 \, {\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{7} + 4032 \, {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{5} - 945 \, a b d x - 315 \, {\left (16 \, a b \cos \left (d x + c\right )^{7} - 24 \, a b \cos \left (d x + c\right )^{5} + 2 \, a b \cos \left (d x + c\right )^{3} + 3 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{20160 \, d} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^3*(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/20160*(2240*b^2*cos(d*x + c)^9 - 2880*(a^2 + 2*b^2)*cos(d*x + c)^7 + 4032*(a^2 + b^2)*cos(d*x + c)^5 - 945*
a*b*d*x - 315*(16*a*b*cos(d*x + c)^7 - 24*a*b*cos(d*x + c)^5 + 2*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 = 1.01 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.11 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\begin {cases} - \frac {a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {2 a^{2} \cos ^{7}{\left (c + d x \right )}}{35 d} + \frac {3 a b x \sin ^{8}{\left (c + d x \right )}}{64} + \frac {3 a b x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {9 a b x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{32} + \frac {3 a b x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a b x \cos ^{8}{\left (c + d x \right )}}{64} + \frac {3 a b \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{64 d} + \frac {11 a b \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{64 d} - \frac {11 a b \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{64 d} - \frac {3 a b \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} - \frac {b^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {4 b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {8 b^{2} \cos ^{9}{\left (c + d x \right )}}{315 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{2} \sin ^{3}{\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**4*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)**5/(5*d) - 2*a**2*cos(c + d*x)**7/(35*d) + 3*a*b*x*sin(c + d*x)*
*8/64 + 3*a*b*x*sin(c + d*x)**6*cos(c + d*x)**2/16 + 9*a*b*x*sin(c + d*x)**4*cos(c + d*x)**4/32 + 3*a*b*x*sin(
c + d*x)**2*cos(c + d*x)**6/16 + 3*a*b*x*cos(c + d*x)**8/64 + 3*a*b*sin(c + d*x)**7*cos(c + d*x)/(64*d) + 11*a
*b*sin(c + d*x)**5*cos(c + d*x)**3/(64*d) - 11*a*b*sin(c + d*x)**3*cos(c + d*x)**5/(64*d) - 3*a*b*sin(c + d*x)
*cos(c + d*x)**7/(64*d) - b**2*sin(c + d*x)**4*cos(c + d*x)**5/(5*d) - 4*b**2*sin(c + d*x)**2*cos(c + d*x)**7/
(35*d) - 8*b**2*cos(c + d*x)**9/(315*d), Ne(d, 0)), (x*(a + b*sin(c))**2*sin(c)**3*cos(c)**4, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.33 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {4608 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{2} + 315 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b - 512 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} b^{2}}{161280 \, d} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^3*(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/161280*(4608*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*a^2 + 315*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*sin(4*d*x
+ 4*c))*a*b - 512*(35*cos(d*x + c)^9 - 90*cos(d*x + c)^7 + 63*cos(d*x + c)^5)*b^2)/d

Giac [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.47 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {3}{64} \, a b x - \frac {b^{2} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {a b \sin \left (8 \, d x + 8 \, c\right )}{512 \, d} - \frac {a b \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (4 \, a^{2} + b^{2}\right )} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {{\left (a^{2} + b^{2}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {{\left (3 \, a^{2} + b^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {3 \, {\left (2 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )}{128 \, d} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^3*(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

3/64*a*b*x - 1/2304*b^2*cos(9*d*x + 9*c)/d + 1/512*a*b*sin(8*d*x + 8*c)/d - 1/64*a*b*sin(4*d*x + 4*c)/d + 1/17
92*(4*a^2 + b^2)*cos(7*d*x + 7*c)/d + 1/320*(a^2 + b^2)*cos(5*d*x + 5*c)/d - 1/192*(3*a^2 + b^2)*cos(3*d*x + 3
*c)/d - 3/128*(2*a^2 + b^2)*cos(d*x + c)/d

Mupad [B] (verification not implemented)

Time = 15.39 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.03 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {3\,a\,b\,x}{64}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (4\,a^2-16\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (4\,a^2+\frac {32\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {28\,a^2}{5}-\frac {32\,b^2}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {36\,a^2}{35}+\frac {16\,b^2}{35}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {4\,a^2}{35}+\frac {64\,b^2}{35}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {52\,a^2}{5}+\frac {112\,b^2}{5}\right )+4\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+\frac {4\,a^2}{35}+\frac {16\,b^2}{315}+\frac {13\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{16}-\frac {155\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{16}+\frac {169\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{16}-\frac {169\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{16}+\frac {155\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{16}-\frac {13\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{16}-\frac {3\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{32}+\frac {3\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{32}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9} \]

[In]

int(cos(c + d*x)^4*sin(c + d*x)^3*(a + b*sin(c + d*x))^2,x)

[Out]

(3*a*b*x)/64 - (tan(c/2 + (d*x)/2)^10*(4*a^2 - 16*b^2) + tan(c/2 + (d*x)/2)^12*(4*a^2 + (32*b^2)/3) + tan(c/2
+ (d*x)/2)^6*((28*a^2)/5 - (32*b^2)/5) + tan(c/2 + (d*x)/2)^2*((36*a^2)/35 + (16*b^2)/35) + tan(c/2 + (d*x)/2)
^4*((4*a^2)/35 + (64*b^2)/35) + tan(c/2 + (d*x)/2)^8*((52*a^2)/5 + (112*b^2)/5) + 4*a^2*tan(c/2 + (d*x)/2)^14
+ (4*a^2)/35 + (16*b^2)/315 + (13*a*b*tan(c/2 + (d*x)/2)^3)/16 - (155*a*b*tan(c/2 + (d*x)/2)^5)/16 + (169*a*b*
tan(c/2 + (d*x)/2)^7)/16 - (169*a*b*tan(c/2 + (d*x)/2)^11)/16 + (155*a*b*tan(c/2 + (d*x)/2)^13)/16 - (13*a*b*t
an(c/2 + (d*x)/2)^15)/16 - (3*a*b*tan(c/2 + (d*x)/2)^17)/32 + (3*a*b*tan(c/2 + (d*x)/2))/32)/(d*(tan(c/2 + (d*
x)/2)^2 + 1)^9)