3.1116 \(\int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=354 \[ \frac {b \left (27 a^2+4 b^2\right ) \cos ^3(c+d x)}{315 d}-\frac {b \left (27 a^2+4 b^2\right ) \cos (c+d x)}{105 d}-\frac {5 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}-\frac {a \left (20 a^2-87 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac {a \left (8 a^2+9 b^2\right ) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {1}{128} a x \left (8 a^2+9 b^2\right )-\frac {\left (20 a^4-93 a^2 b^2+24 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{2520 b d}-\frac {a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4032 b^2 d}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d} \]

[Out]

1/128*a*(8*a^2+9*b^2)*x-1/105*b*(27*a^2+4*b^2)*cos(d*x+c)/d+1/315*b*(27*a^2+4*b^2)*cos(d*x+c)^3/d-1/128*a*(8*a
^2+9*b^2)*cos(d*x+c)*sin(d*x+c)/d-1/4032*a*(40*a^4-188*a^2*b^2+189*b^4)*cos(d*x+c)*sin(d*x+c)^3/b^2/d-1/2520*(
20*a^4-93*a^2*b^2+24*b^4)*cos(d*x+c)*sin(d*x+c)^4/b/d-1/1008*a*(20*a^2-87*b^2)*cos(d*x+c)*sin(d*x+c)^3*(a+b*si
n(d*x+c))^2/b^2/d-5/126*(a^2-4*b^2)*cos(d*x+c)*sin(d*x+c)^3*(a+b*sin(d*x+c))^3/b^2/d+5/72*a*cos(d*x+c)*sin(d*x
+c)^3*(a+b*sin(d*x+c))^4/b^2/d-1/9*cos(d*x+c)*sin(d*x+c)^4*(a+b*sin(d*x+c))^4/b/d

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Rubi [A]  time = 0.93, antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2895, 3049, 3033, 3023, 2748, 2635, 8, 2633} \[ \frac {b \left (27 a^2+4 b^2\right ) \cos ^3(c+d x)}{315 d}-\frac {b \left (27 a^2+4 b^2\right ) \cos (c+d x)}{105 d}-\frac {\left (-93 a^2 b^2+20 a^4+24 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{2520 b d}-\frac {5 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}-\frac {a \left (20 a^2-87 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac {a \left (-188 a^2 b^2+40 a^4+189 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4032 b^2 d}-\frac {a \left (8 a^2+9 b^2\right ) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {1}{128} a x \left (8 a^2+9 b^2\right )+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(8*a^2 + 9*b^2)*x)/128 - (b*(27*a^2 + 4*b^2)*Cos[c + d*x])/(105*d) + (b*(27*a^2 + 4*b^2)*Cos[c + d*x]^3)/(3
15*d) - (a*(8*a^2 + 9*b^2)*Cos[c + d*x]*Sin[c + d*x])/(128*d) - (a*(40*a^4 - 188*a^2*b^2 + 189*b^4)*Cos[c + d*
x]*Sin[c + d*x]^3)/(4032*b^2*d) - ((20*a^4 - 93*a^2*b^2 + 24*b^4)*Cos[c + d*x]*Sin[c + d*x]^4)/(2520*b*d) - (a
*(20*a^2 - 87*b^2)*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(1008*b^2*d) - (5*(a^2 - 4*b^2)*Cos[c +
d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(126*b^2*d) + (5*a*Cos[c + d*x]*Sin[c + d*x]^3*(a + b*Sin[c + d*x
])^4)/(72*b^2*d) - (Cos[c + d*x]*Sin[c + d*x]^4*(a + b*Sin[c + d*x])^4)/(9*b*d)

Rule 8

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

Rule 2633

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 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 2748

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 2895

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 3023

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 3033

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 3049

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*} \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}-\frac {\int \sin ^2(c+d x) (a+b \sin (c+d x))^3 \left (3 \left (5 a^2-24 b^2\right )+3 a b \sin (c+d x)-20 \left (a^2-4 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{72 b^2}\\ &=-\frac {5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}-\frac {\int \sin ^2(c+d x) (a+b \sin (c+d x))^2 \left (3 a \left (15 a^2-88 b^2\right )+6 b \left (a^2-4 b^2\right ) \sin (c+d x)-3 a \left (20 a^2-87 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{504 b^2}\\ &=-\frac {a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac {5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}-\frac {\int \sin ^2(c+d x) (a+b \sin (c+d x)) \left (9 a^2 \left (10 a^2-89 b^2\right )+3 a b \left (2 a^2-141 b^2\right ) \sin (c+d x)-6 \left (20 a^4-93 a^2 b^2+24 b^4\right ) \sin ^2(c+d x)\right ) \, dx}{3024 b^2}\\ &=-\frac {\left (20 a^4-93 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{2520 b d}-\frac {a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac {5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}-\frac {\int \sin ^2(c+d x) \left (45 a^3 \left (10 a^2-89 b^2\right )-144 b^3 \left (27 a^2+4 b^2\right ) \sin (c+d x)-15 a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \sin ^2(c+d x)\right ) \, dx}{15120 b^2}\\ &=-\frac {a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{4032 b^2 d}-\frac {\left (20 a^4-93 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{2520 b d}-\frac {a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac {5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}-\frac {\int \sin ^2(c+d x) \left (-945 a b^2 \left (8 a^2+9 b^2\right )-576 b^3 \left (27 a^2+4 b^2\right ) \sin (c+d x)\right ) \, dx}{60480 b^2}\\ &=-\frac {a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{4032 b^2 d}-\frac {\left (20 a^4-93 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{2520 b d}-\frac {a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac {5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}+\frac {1}{105} \left (b \left (27 a^2+4 b^2\right )\right ) \int \sin ^3(c+d x) \, dx+\frac {1}{64} \left (a \left (8 a^2+9 b^2\right )\right ) \int \sin ^2(c+d x) \, dx\\ &=-\frac {a \left (8 a^2+9 b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}-\frac {a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{4032 b^2 d}-\frac {\left (20 a^4-93 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{2520 b d}-\frac {a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac {5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}+\frac {1}{128} \left (a \left (8 a^2+9 b^2\right )\right ) \int 1 \, dx-\frac {\left (b \left (27 a^2+4 b^2\right )\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{105 d}\\ &=\frac {1}{128} a \left (8 a^2+9 b^2\right ) x-\frac {b \left (27 a^2+4 b^2\right ) \cos (c+d x)}{105 d}+\frac {b \left (27 a^2+4 b^2\right ) \cos ^3(c+d x)}{315 d}-\frac {a \left (8 a^2+9 b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}-\frac {a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{4032 b^2 d}-\frac {\left (20 a^4-93 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{2520 b d}-\frac {a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac {5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}\\ \end {align*}

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Mathematica [A]  time = 1.28, size = 204, normalized size = 0.58 \[ \frac {2520 a^3 \sin (2 (c+d x))-2520 a^3 \sin (4 (c+d x))-840 a^3 \sin (6 (c+d x))+10080 a^3 d x-840 \left (9 a^2 b+b^3\right ) \cos (3 (c+d x))-3780 b \left (6 a^2+b^2\right ) \cos (c+d x)+1512 a^2 b \cos (5 (c+d x))+1080 a^2 b \cos (7 (c+d x))-3780 a b^2 \sin (4 (c+d x))+\frac {945}{2} a b^2 \sin (8 (c+d x))+15120 a b^2 c+11340 a b^2 d x+504 b^3 \cos (5 (c+d x))+90 b^3 \cos (7 (c+d x))-70 b^3 \cos (9 (c+d x))}{161280 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(15120*a*b^2*c + 10080*a^3*d*x + 11340*a*b^2*d*x - 3780*b*(6*a^2 + b^2)*Cos[c + d*x] - 840*(9*a^2*b + b^3)*Cos
[3*(c + d*x)] + 1512*a^2*b*Cos[5*(c + d*x)] + 504*b^3*Cos[5*(c + d*x)] + 1080*a^2*b*Cos[7*(c + d*x)] + 90*b^3*
Cos[7*(c + d*x)] - 70*b^3*Cos[9*(c + d*x)] + 2520*a^3*Sin[2*(c + d*x)] - 2520*a^3*Sin[4*(c + d*x)] - 3780*a*b^
2*Sin[4*(c + d*x)] - 840*a^3*Sin[6*(c + d*x)] + (945*a*b^2*Sin[8*(c + d*x)])/2)/(161280*d)

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fricas [A]  time = 0.98, size = 164, normalized size = 0.46 \[ -\frac {4480 \, b^{3} \cos \left (d x + c\right )^{9} - 5760 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{7} + 8064 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{5} - 315 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )} d x - 105 \, {\left (144 \, a b^{2} \cos \left (d x + c\right )^{7} - 8 \, {\left (8 \, a^{3} + 27 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40320 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40320*(4480*b^3*cos(d*x + c)^9 - 5760*(3*a^2*b + 2*b^3)*cos(d*x + c)^7 + 8064*(3*a^2*b + b^3)*cos(d*x + c)^
5 - 315*(8*a^3 + 9*a*b^2)*d*x - 105*(144*a*b^2*cos(d*x + c)^7 - 8*(8*a^3 + 27*a*b^2)*cos(d*x + c)^5 + 2*(8*a^3
+ 9*a*b^2)*cos(d*x + c)^3 + 3*(8*a^3 + 9*a*b^2)*cos(d*x + c))*sin(d*x + c))/d

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giac [A]  time = 0.46, size = 204, normalized size = 0.58 \[ -\frac {b^{3} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {3 \, a b^{2} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {a^{3} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {1}{128} \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )} x + \frac {{\left (12 \, a^{2} b + b^{3}\right )} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {{\left (3 \, a^{2} b + b^{3}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {{\left (9 \, a^{2} b + b^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {3 \, {\left (6 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )}{128 \, d} - \frac {{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2304*b^3*cos(9*d*x + 9*c)/d + 3/1024*a*b^2*sin(8*d*x + 8*c)/d - 1/192*a^3*sin(6*d*x + 6*c)/d + 1/64*a^3*sin
(2*d*x + 2*c)/d + 1/128*(8*a^3 + 9*a*b^2)*x + 1/1792*(12*a^2*b + b^3)*cos(7*d*x + 7*c)/d + 1/320*(3*a^2*b + b^
3)*cos(5*d*x + 5*c)/d - 1/192*(9*a^2*b + b^3)*cos(3*d*x + 3*c)/d - 3/128*(6*a^2*b + b^3)*cos(d*x + c)/d - 1/12
8*(2*a^3 + 3*a*b^2)*sin(4*d*x + 4*c)/d

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maple [A]  time = 0.34, size = 218, normalized size = 0.62 \[ \frac {a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+3 a^{2} b \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+3 a \,b^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+b^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^3*(-1/6*sin(d*x+c)*cos(d*x+c)^5+1/24*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+1/16*d*x+1/16*c)+3*a^2*b*
(-1/7*sin(d*x+c)^2*cos(d*x+c)^5-2/35*cos(d*x+c)^5)+3*a*b^2*(-1/8*sin(d*x+c)^3*cos(d*x+c)^5-1/16*sin(d*x+c)*cos
(d*x+c)^5+1/64*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/128*d*x+3/128*c)+b^3*(-1/9*sin(d*x+c)^4*cos(d*x+c)^5
-4/63*sin(d*x+c)^2*cos(d*x+c)^5-8/315*cos(d*x+c)^5))

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maxima [A]  time = 0.45, size = 140, normalized size = 0.40 \[ \frac {1680 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} + 27648 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{2} b + 945 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{2} - 1024 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} b^{3}}{322560 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/322560*(1680*(4*sin(2*d*x + 2*c)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c))*a^3 + 27648*(5*cos(d*x + c)^7 - 7*c
os(d*x + c)^5)*a^2*b + 945*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*sin(4*d*x + 4*c))*a*b^2 - 1024*(35*cos(d*x +
c)^9 - 90*cos(d*x + c)^7 + 63*cos(d*x + c)^5)*b^3)/d

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mupad [B]  time = 10.77, size = 578, normalized size = 1.63 \[ \frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,a^2+9\,b^2\right )}{64\,\left (\frac {a^3}{8}+\frac {9\,a\,b^2}{64}\right )}\right )\,\left (8\,a^2+9\,b^2\right )}{64\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {a^3}{8}+\frac {9\,a\,b^2}{64}\right )+\frac {12\,a^2\,b}{35}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,\left (\frac {a^3}{8}+\frac {9\,a\,b^2}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {39\,a\,b^2}{32}-\frac {19\,a^3}{12}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (\frac {39\,a\,b^2}{32}-\frac {19\,a^3}{12}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {9\,a^3}{4}+\frac {465\,a\,b^2}{32}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {9\,a^3}{4}+\frac {465\,a\,b^2}{32}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {3\,a^3}{4}+\frac {507\,a\,b^2}{32}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {3\,a^3}{4}+\frac {507\,a\,b^2}{32}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (12\,a^2\,b-16\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (12\,a^2\,b+\frac {32\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {84\,a^2\,b}{5}-\frac {32\,b^3}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {12\,a^2\,b}{35}+\frac {64\,b^3}{35}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {108\,a^2\,b}{35}+\frac {16\,b^3}{35}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {156\,a^2\,b}{5}+\frac {112\,b^3}{5}\right )+\frac {16\,b^3}{315}+12\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+84\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+126\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+126\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+84\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a\,\left (8\,a^2+9\,b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{64\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*atan((a*tan(c/2 + (d*x)/2)*(8*a^2 + 9*b^2))/(64*((9*a*b^2)/64 + a^3/8)))*(8*a^2 + 9*b^2))/(64*d) - (tan(c/2
+ (d*x)/2)*((9*a*b^2)/64 + a^3/8) + (12*a^2*b)/35 - tan(c/2 + (d*x)/2)^17*((9*a*b^2)/64 + a^3/8) + tan(c/2 +
(d*x)/2)^3*((39*a*b^2)/32 - (19*a^3)/12) - tan(c/2 + (d*x)/2)^15*((39*a*b^2)/32 - (19*a^3)/12) - tan(c/2 + (d*
x)/2)^5*((465*a*b^2)/32 + (9*a^3)/4) + tan(c/2 + (d*x)/2)^13*((465*a*b^2)/32 + (9*a^3)/4) + tan(c/2 + (d*x)/2)
^7*((507*a*b^2)/32 + (3*a^3)/4) - tan(c/2 + (d*x)/2)^11*((507*a*b^2)/32 + (3*a^3)/4) + tan(c/2 + (d*x)/2)^10*(
12*a^2*b - 16*b^3) + tan(c/2 + (d*x)/2)^12*(12*a^2*b + (32*b^3)/3) + tan(c/2 + (d*x)/2)^6*((84*a^2*b)/5 - (32*
b^3)/5) + tan(c/2 + (d*x)/2)^4*((12*a^2*b)/35 + (64*b^3)/35) + tan(c/2 + (d*x)/2)^2*((108*a^2*b)/35 + (16*b^3)
/35) + tan(c/2 + (d*x)/2)^8*((156*a^2*b)/5 + (112*b^3)/5) + (16*b^3)/315 + 12*a^2*b*tan(c/2 + (d*x)/2)^14)/(d*
(9*tan(c/2 + (d*x)/2)^2 + 36*tan(c/2 + (d*x)/2)^4 + 84*tan(c/2 + (d*x)/2)^6 + 126*tan(c/2 + (d*x)/2)^8 + 126*t
an(c/2 + (d*x)/2)^10 + 84*tan(c/2 + (d*x)/2)^12 + 36*tan(c/2 + (d*x)/2)^14 + 9*tan(c/2 + (d*x)/2)^16 + tan(c/2
+ (d*x)/2)^18 + 1)) - (a*(8*a^2 + 9*b^2)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(64*d)

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sympy [A]  time = 16.18, size = 505, normalized size = 1.43 \[ \begin {cases} \frac {a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {3 a^{2} b \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {6 a^{2} b \cos ^{7}{\left (c + d x \right )}}{35 d} + \frac {9 a b^{2} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {9 a b^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {27 a b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {9 a b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {9 a b^{2} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {9 a b^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {33 a b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {33 a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {9 a b^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {b^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {4 b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {8 b^{3} \cos ^{9}{\left (c + d x \right )}}{315 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\relax (c )}\right )^{3} \sin ^{2}{\relax (c )} \cos ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*x*sin(c + d*x)**6/16 + 3*a**3*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 3*a**3*x*sin(c + d*x)**2*
cos(c + d*x)**4/16 + a**3*x*cos(c + d*x)**6/16 + a**3*sin(c + d*x)**5*cos(c + d*x)/(16*d) + a**3*sin(c + d*x)*
*3*cos(c + d*x)**3/(6*d) - a**3*sin(c + d*x)*cos(c + d*x)**5/(16*d) - 3*a**2*b*sin(c + d*x)**2*cos(c + d*x)**5
/(5*d) - 6*a**2*b*cos(c + d*x)**7/(35*d) + 9*a*b**2*x*sin(c + d*x)**8/128 + 9*a*b**2*x*sin(c + d*x)**6*cos(c +
d*x)**2/32 + 27*a*b**2*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 9*a*b**2*x*sin(c + d*x)**2*cos(c + d*x)**6/32 +
9*a*b**2*x*cos(c + d*x)**8/128 + 9*a*b**2*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 33*a*b**2*sin(c + d*x)**5*co
s(c + d*x)**3/(128*d) - 33*a*b**2*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) - 9*a*b**2*sin(c + d*x)*cos(c + d*x)
**7/(128*d) - b**3*sin(c + d*x)**4*cos(c + d*x)**5/(5*d) - 4*b**3*sin(c + d*x)**2*cos(c + d*x)**7/(35*d) - 8*b
**3*cos(c + d*x)**9/(315*d), Ne(d, 0)), (x*(a + b*sin(c))**3*sin(c)**2*cos(c)**4, True))

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