\(\int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx\) [1106]

Optimal result

Integrand size = 29, antiderivative size = 278 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {1}{128} \left (8 a^2+3 b^2\right ) x-\frac {6 a b \cos (c+d x)}{35 d}+\frac {2 a b \cos ^3(c+d x)}{35 d}-\frac {\left (8 a^2+3 b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}-\frac {\left (40 a^4-140 a^2 b^2+21 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{1344 b^2 d}-\frac {a \left (20 a^2-69 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{840 b d}-\frac {\left (20 a^2-63 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{336 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{56 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{8 b d} \]

[Out]

1/128*(8*a^2+3*b^2)*x-6/35*a*b*cos(d*x+c)/d+2/35*a*b*cos(d*x+c)^3/d-1/128*(8*a^2+3*b^2)*cos(d*x+c)*sin(d*x+c)/
d-1/1344*(40*a^4-140*a^2*b^2+21*b^4)*cos(d*x+c)*sin(d*x+c)^3/b^2/d-1/840*a*(20*a^2-69*b^2)*cos(d*x+c)*sin(d*x+
c)^4/b/d-1/336*(20*a^2-63*b^2)*cos(d*x+c)*sin(d*x+c)^3*(a+b*sin(d*x+c))^2/b^2/d+5/56*a*cos(d*x+c)*sin(d*x+c)^3
*(a+b*sin(d*x+c))^3/b^2/d-1/8*cos(d*x+c)*sin(d*x+c)^4*(a+b*sin(d*x+c))^3/b/d

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2974, 3128, 3112, 3102, 2827, 2715, 8, 2713} \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {a \left (20 a^2-69 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{840 b d}-\frac {\left (20 a^2-63 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{336 b^2 d}-\frac {\left (8 a^2+3 b^2\right ) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {1}{128} x \left (8 a^2+3 b^2\right )-\frac {\left (40 a^4-140 a^2 b^2+21 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{1344 b^2 d}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{56 b^2 d}+\frac {2 a b \cos ^3(c+d x)}{35 d}-\frac {6 a b \cos (c+d x)}{35 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{8 b d} \]

[In]

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

[Out]

((8*a^2 + 3*b^2)*x)/128 - (6*a*b*Cos[c + d*x])/(35*d) + (2*a*b*Cos[c + d*x]^3)/(35*d) - ((8*a^2 + 3*b^2)*Cos[c
+ d*x]*Sin[c + d*x])/(128*d) - ((40*a^4 - 140*a^2*b^2 + 21*b^4)*Cos[c + d*x]*Sin[c + d*x]^3)/(1344*b^2*d) - (
a*(20*a^2 - 69*b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(840*b*d) - ((20*a^2 - 63*b^2)*Cos[c + d*x]*Sin[c + d*x]^3*(a
+ b*Sin[c + d*x])^2)/(336*b^2*d) + (5*a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(56*b^2*d) - (Cos
[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^3)/(8*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 {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{56 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{8 b d}-\frac {\int \sin ^2(c+d x) (a+b \sin (c+d x))^2 \left (15 a^2-56 b^2+2 a b \sin (c+d x)-\left (20 a^2-63 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{56 b^2} \\ & = -\frac {\left (20 a^2-63 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{336 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{56 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{8 b d}-\frac {\int \sin ^2(c+d x) (a+b \sin (c+d x)) \left (3 a \left (10 a^2-49 b^2\right )+b \left (2 a^2-21 b^2\right ) \sin (c+d x)-2 a \left (20 a^2-69 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{336 b^2} \\ & = -\frac {a \left (20 a^2-69 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{840 b d}-\frac {\left (20 a^2-63 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{336 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{56 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{8 b d}-\frac {\int \sin ^2(c+d x) \left (15 a^2 \left (10 a^2-49 b^2\right )-288 a b^3 \sin (c+d x)-5 \left (40 a^4-140 a^2 b^2+21 b^4\right ) \sin ^2(c+d x)\right ) \, dx}{1680 b^2} \\ & = -\frac {\left (40 a^4-140 a^2 b^2+21 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{1344 b^2 d}-\frac {a \left (20 a^2-69 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{840 b d}-\frac {\left (20 a^2-63 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{336 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{56 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{8 b d}-\frac {\int \sin ^2(c+d x) \left (-105 b^2 \left (8 a^2+3 b^2\right )-1152 a b^3 \sin (c+d x)\right ) \, dx}{6720 b^2} \\ & = -\frac {\left (40 a^4-140 a^2 b^2+21 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{1344 b^2 d}-\frac {a \left (20 a^2-69 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{840 b d}-\frac {\left (20 a^2-63 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{336 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{56 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{8 b d}+\frac {1}{35} (6 a b) \int \sin ^3(c+d x) \, dx-\frac {1}{64} \left (-8 a^2-3 b^2\right ) \int \sin ^2(c+d x) \, dx \\ & = -\frac {\left (8 a^2+3 b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}-\frac {\left (40 a^4-140 a^2 b^2+21 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{1344 b^2 d}-\frac {a \left (20 a^2-69 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{840 b d}-\frac {\left (20 a^2-63 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{336 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{56 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{8 b d}-\frac {1}{128} \left (-8 a^2-3 b^2\right ) \int 1 \, dx-\frac {(6 a b) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{35 d} \\ & = \frac {1}{128} \left (8 a^2+3 b^2\right ) x-\frac {6 a b \cos (c+d x)}{35 d}+\frac {2 a b \cos ^3(c+d x)}{35 d}-\frac {\left (8 a^2+3 b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}-\frac {\left (40 a^4-140 a^2 b^2+21 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{1344 b^2 d}-\frac {a \left (20 a^2-69 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{840 b d}-\frac {\left (20 a^2-63 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{336 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{56 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^3}{8 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.51 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {1680 b^2 c+3360 a^2 d x+1260 b^2 d x-5040 a b \cos (c+d x)-1680 a b \cos (3 (c+d x))+336 a b \cos (5 (c+d x))+240 a b \cos (7 (c+d x))+840 a^2 \sin (2 (c+d x))-840 a^2 \sin (4 (c+d x))-420 b^2 \sin (4 (c+d x))-280 a^2 \sin (6 (c+d x))+\frac {105}{2} b^2 \sin (8 (c+d x))}{53760 d} \]

[In]

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

[Out]

(1680*b^2*c + 3360*a^2*d*x + 1260*b^2*d*x - 5040*a*b*Cos[c + d*x] - 1680*a*b*Cos[3*(c + d*x)] + 336*a*b*Cos[5*
(c + d*x)] + 240*a*b*Cos[7*(c + d*x)] + 840*a^2*Sin[2*(c + d*x)] - 840*a^2*Sin[4*(c + d*x)] - 420*b^2*Sin[4*(c
+ d*x)] - 280*a^2*Sin[6*(c + d*x)] + (105*b^2*Sin[8*(c + d*x)])/2)/(53760*d)

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.52

method	result	size
parallelrisch	\(\frac {6720 a^{2} d x +2520 b^{2} d x +105 b^{2} \sin \left (8 d x +8 c \right )+480 a b \cos \left (7 d x +7 c \right )-560 a^{2} \sin \left (6 d x +6 c \right )+672 a b \cos \left (5 d x +5 c \right )-1680 a^{2} \sin \left (4 d x +4 c \right )-840 b^{2} \sin \left (4 d x +4 c \right )+1680 a^{2} \sin \left (2 d x +2 c \right )-3360 a b \cos \left (3 d x +3 c \right )-10080 a b \cos \left (d x +c \right )-12288 a b}{107520 d}\)	\(144\)
risch	\(\frac {a^{2} x}{16}+\frac {3 b^{2} x}{128}-\frac {3 a b \cos \left (d x +c \right )}{32 d}+\frac {b^{2} \sin \left (8 d x +8 c \right )}{1024 d}+\frac {a b \cos \left (7 d x +7 c \right )}{224 d}-\frac {a^{2} \sin \left (6 d x +6 c \right )}{192 d}+\frac {a b \cos \left (5 d x +5 c \right )}{160 d}-\frac {\sin \left (4 d x +4 c \right ) a^{2}}{64 d}-\frac {\sin \left (4 d x +4 c \right ) b^{2}}{128 d}-\frac {a b \cos \left (3 d x +3 c \right )}{32 d}+\frac {a^{2} \sin \left (2 d x +2 c \right )}{64 d}\)	\(160\)
derivativedivides	\(\frac {a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+2 a b \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+b^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d}\)	\(163\)
default	\(\frac {a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+2 a b \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+b^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d}\)	\(163\)
norman	\(\frac {-\frac {8 a b}{35 d}+\left (\frac {7 a^{2}}{4}+\frac {21 b^{2}}{32}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {7 a^{2}}{2}+\frac {21 b^{2}}{16}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {35 a^{2}}{8}+\frac {105 b^{2}}{64}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {7 a^{2}}{2}+\frac {21 b^{2}}{16}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {7 a^{2}}{4}+\frac {21 b^{2}}{32}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {a^{2}}{2}+\frac {3 b^{2}}{16}\right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {a^{2}}{16}+\frac {3 b^{2}}{128}\right ) x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {8 a b \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (248 a^{2}+2013 b^{2}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {\left (104 a^{2}+999 b^{2}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {8 a b \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (328 a^{2}-69 b^{2}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {\left (8 a^{2}+3 b^{2}\right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\left (\frac {a^{2}}{2}+\frac {3 b^{2}}{16}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\left (8 a^{2}+3 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}-\frac {64 a b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {\left (328 a^{2}-69 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {8 a b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {\left (104 a^{2}+999 b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {64 a b \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {\left (248 a^{2}+2013 b^{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\left (\frac {a^{2}}{16}+\frac {3 b^{2}}{128}\right ) x}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}\)	\(534\)

[In]

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

[Out]

1/107520*(6720*a^2*d*x+2520*b^2*d*x+105*b^2*sin(8*d*x+8*c)+480*a*b*cos(7*d*x+7*c)-560*a^2*sin(6*d*x+6*c)+672*a
*b*cos(5*d*x+5*c)-1680*a^2*sin(4*d*x+4*c)-840*b^2*sin(4*d*x+4*c)+1680*a^2*sin(2*d*x+2*c)-3360*a*b*cos(3*d*x+3*
c)-10080*a*b*cos(d*x+c)-12288*a*b)/d

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.46 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {3840 \, a b \cos \left (d x + c\right )^{7} - 5376 \, a b \cos \left (d x + c\right )^{5} + 105 \, {\left (8 \, a^{2} + 3 \, b^{2}\right )} d x + 35 \, {\left (48 \, b^{2} \cos \left (d x + c\right )^{7} - 8 \, {\left (8 \, a^{2} + 9 \, b^{2}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (8 \, a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{13440 \, d} \]

[In]

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

[Out]

1/13440*(3840*a*b*cos(d*x + c)^7 - 5376*a*b*cos(d*x + c)^5 + 105*(8*a^2 + 3*b^2)*d*x + 35*(48*b^2*cos(d*x + c)
^7 - 8*(8*a^2 + 9*b^2)*cos(d*x + c)^5 + 2*(8*a^2 + 3*b^2)*cos(d*x + c)^3 + 3*(8*a^2 + 3*b^2)*cos(d*x + c))*sin
(d*x + c))/d

Sympy [A] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.51 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\begin {cases} \frac {a^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {a^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {a^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {2 a b \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {4 a b \cos ^{7}{\left (c + d x \right )}}{35 d} + \frac {3 b^{2} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {3 b^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {9 b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {3 b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {3 b^{2} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 b^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {11 b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {11 b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {3 b^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{2} \sin ^{2}{\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a**2*x*sin(c + d*x)**6/16 + 3*a**2*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 3*a**2*x*sin(c + d*x)**2*
cos(c + d*x)**4/16 + a**2*x*cos(c + d*x)**6/16 + a**2*sin(c + d*x)**5*cos(c + d*x)/(16*d) + a**2*sin(c + d*x)*
*3*cos(c + d*x)**3/(6*d) - a**2*sin(c + d*x)*cos(c + d*x)**5/(16*d) - 2*a*b*sin(c + d*x)**2*cos(c + d*x)**5/(5
*d) - 4*a*b*cos(c + d*x)**7/(35*d) + 3*b**2*x*sin(c + d*x)**8/128 + 3*b**2*x*sin(c + d*x)**6*cos(c + d*x)**2/3
2 + 9*b**2*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 3*b**2*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 3*b**2*x*cos(c
+ d*x)**8/128 + 3*b**2*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 11*b**2*sin(c + d*x)**5*cos(c + d*x)**3/(128*d)
- 11*b**2*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) - 3*b**2*sin(c + d*x)*cos(c + d*x)**7/(128*d), Ne(d, 0)), (
x*(a + b*sin(c))**2*sin(c)**2*cos(c)**4, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.36 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {560 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} + 6144 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a b + 105 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{2}}{107520 \, d} \]

[In]

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

[Out]

1/107520*(560*(4*sin(2*d*x + 2*c)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c))*a^2 + 6144*(5*cos(d*x + c)^7 - 7*cos
(d*x + c)^5)*a*b + 105*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*sin(4*d*x + 4*c))*b^2)/d

Giac [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.54 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {1}{128} \, {\left (8 \, a^{2} + 3 \, b^{2}\right )} x + \frac {a b \cos \left (7 \, d x + 7 \, c\right )}{224 \, d} + \frac {a b \cos \left (5 \, d x + 5 \, c\right )}{160 \, d} - \frac {a b \cos \left (3 \, d x + 3 \, c\right )}{32 \, d} - \frac {3 \, a b \cos \left (d x + c\right )}{32 \, d} + \frac {b^{2} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a^{2} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} - \frac {{\left (2 \, a^{2} + b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} \]

[In]

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

[Out]

1/128*(8*a^2 + 3*b^2)*x + 1/224*a*b*cos(7*d*x + 7*c)/d + 1/160*a*b*cos(5*d*x + 5*c)/d - 1/32*a*b*cos(3*d*x + 3
*c)/d - 3/32*a*b*cos(d*x + c)/d + 1/1024*b^2*sin(8*d*x + 8*c)/d - 1/192*a^2*sin(6*d*x + 6*c)/d + 1/64*a^2*sin(
2*d*x + 2*c)/d - 1/128*(2*a^2 + b^2)*sin(4*d*x + 4*c)/d

Mupad [B] (verification not implemented)

Time = 11.99 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.50 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {210\,a^2\,\sin \left (2\,c+2\,d\,x\right )-210\,a^2\,\sin \left (4\,c+4\,d\,x\right )-70\,a^2\,\sin \left (6\,c+6\,d\,x\right )-105\,b^2\,\sin \left (4\,c+4\,d\,x\right )+\frac {105\,b^2\,\sin \left (8\,c+8\,d\,x\right )}{8}-1260\,a\,b\,\cos \left (c+d\,x\right )-420\,a\,b\,\cos \left (3\,c+3\,d\,x\right )+84\,a\,b\,\cos \left (5\,c+5\,d\,x\right )+60\,a\,b\,\cos \left (7\,c+7\,d\,x\right )+840\,a^2\,d\,x+315\,b^2\,d\,x}{13440\,d} \]

[In]

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

[Out]

(210*a^2*sin(2*c + 2*d*x) - 210*a^2*sin(4*c + 4*d*x) - 70*a^2*sin(6*c + 6*d*x) - 105*b^2*sin(4*c + 4*d*x) + (1
05*b^2*sin(8*c + 8*d*x))/8 - 1260*a*b*cos(c + d*x) - 420*a*b*cos(3*c + 3*d*x) + 84*a*b*cos(5*c + 5*d*x) + 60*a
*b*cos(7*c + 7*d*x) + 840*a^2*d*x + 315*b^2*d*x)/(13440*d)