3.1195 \(\int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=623 \[ \frac {3 a \left (a^2 (n+6)+3 b^2 (n+1)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(c+d x)\right )}{d (n+1) (n+2) (n+4) (n+6) \sqrt {\cos ^2(c+d x)}}+\frac {3 b \left (3 a^2 (n+7)+b^2 (n+2)\right ) \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {n+2}{2};\frac {n+4}{2};\sin ^2(c+d x)\right )}{d (n+2) (n+3) (n+5) (n+7) \sqrt {\cos ^2(c+d x)}}-\frac {3 a \left (a^2 \left (n^2+5 n+6\right )-b^2 \left (n^2+15 n+53\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (n+4) (n+5) (n+6) (n+7)}-\frac {\left (a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (n+5) (n+6) (n+7)}-\frac {3 a \left (2 a^4 \left (n^2+5 n+6\right )-2 a^2 b^2 \left (n^2+16 n+58\right )+3 b^4 \left (n^2+12 n+35\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x)}{b^2 d (n+2) (n+4) (n+5) (n+6) (n+7)}-\frac {3 \left (2 a^4 \left (n^2+5 n+6\right )-2 a^2 b^2 \left (n^2+16 n+57\right )+b^4 \left (n^2+10 n+24\right )\right ) \cos (c+d x) \sin ^{n+2}(c+d x)}{b d (n+3) (n+4) (n+5) (n+6) (n+7)}+\frac {a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)} \]

[Out]

-3*a*(2*a^4*(n^2+5*n+6)+3*b^4*(n^2+12*n+35)-2*a^2*b^2*(n^2+16*n+58))*cos(d*x+c)*sin(d*x+c)^(1+n)/b^2/d/(5+n)/(
6+n)/(7+n)/(n^2+6*n+8)-3*(2*a^4*(n^2+5*n+6)+b^4*(n^2+10*n+24)-2*a^2*b^2*(n^2+16*n+57))*cos(d*x+c)*sin(d*x+c)^(
2+n)/b/d/(3+n)/(4+n)/(5+n)/(6+n)/(7+n)-3*a*(a^2*(n^2+5*n+6)-b^2*(n^2+15*n+53))*cos(d*x+c)*sin(d*x+c)^(1+n)*(a+
b*sin(d*x+c))^2/b^2/d/(4+n)/(5+n)/(6+n)/(7+n)-(a^2*(2+n)*(3+n)-b^2*(6+n)*(8+n))*cos(d*x+c)*sin(d*x+c)^(1+n)*(a
+b*sin(d*x+c))^3/b^2/d/(5+n)/(6+n)/(7+n)+a*(3+n)*cos(d*x+c)*sin(d*x+c)^(1+n)*(a+b*sin(d*x+c))^4/b^2/d/(6+n)/(7
+n)-cos(d*x+c)*sin(d*x+c)^(2+n)*(a+b*sin(d*x+c))^4/b/d/(7+n)+3*a*(3*b^2*(1+n)+a^2*(6+n))*cos(d*x+c)*hypergeom(
[1/2, 1/2+1/2*n],[3/2+1/2*n],sin(d*x+c)^2)*sin(d*x+c)^(1+n)/d/(6+n)/(n^3+7*n^2+14*n+8)/(cos(d*x+c)^2)^(1/2)+3*
b*(b^2*(2+n)+3*a^2*(7+n))*cos(d*x+c)*hypergeom([1/2, 1+1/2*n],[1/2*n+2],sin(d*x+c)^2)*sin(d*x+c)^(2+n)/d/(5+n)
/(7+n)/(n^2+5*n+6)/(cos(d*x+c)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.76, antiderivative size = 623, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2895, 3049, 3033, 3023, 2748, 2643} \[ \frac {3 a \left (a^2 (n+6)+3 b^2 (n+1)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(c+d x)\right )}{d (n+1) (n+2) (n+4) (n+6) \sqrt {\cos ^2(c+d x)}}+\frac {3 b \left (3 a^2 (n+7)+b^2 (n+2)\right ) \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left (\frac {1}{2},\frac {n+2}{2};\frac {n+4}{2};\sin ^2(c+d x)\right )}{d (n+2) (n+3) (n+5) (n+7) \sqrt {\cos ^2(c+d x)}}-\frac {3 a \left (a^2 \left (n^2+5 n+6\right )-b^2 \left (n^2+15 n+53\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (n+4) (n+5) (n+6) (n+7)}-\frac {3 a \left (-2 a^2 b^2 \left (n^2+16 n+58\right )+2 a^4 \left (n^2+5 n+6\right )+3 b^4 \left (n^2+12 n+35\right )\right ) \cos (c+d x) \sin ^{n+1}(c+d x)}{b^2 d (n+2) (n+4) (n+5) (n+6) (n+7)}-\frac {3 \left (-2 a^2 b^2 \left (n^2+16 n+57\right )+2 a^4 \left (n^2+5 n+6\right )+b^4 \left (n^2+10 n+24\right )\right ) \cos (c+d x) \sin ^{n+2}(c+d x)}{b d (n+3) (n+4) (n+5) (n+6) (n+7)}-\frac {\left (a^2 (n+2) (n+3)-b^2 (n+6) (n+8)\right ) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (n+5) (n+6) (n+7)}+\frac {a (n+3) \cos (c+d x) \sin ^{n+1}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (n+6) (n+7)}-\frac {\cos (c+d x) \sin ^{n+2}(c+d x) (a+b \sin (c+d x))^4}{b d (n+7)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*(2*a^4*(6 + 5*n + n^2) + 3*b^4*(35 + 12*n + n^2) - 2*a^2*b^2*(58 + 16*n + n^2))*Cos[c + d*x]*Sin[c + d*x
]^(1 + n))/(b^2*d*(2 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)) + (3*a*(3*b^2*(1 + n) + a^2*(6 + n))*Cos[c + d*x]*H
ypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 + n)*(2 + n)*(4 + n)*(
6 + n)*Sqrt[Cos[c + d*x]^2]) - (3*(2*a^4*(6 + 5*n + n^2) + b^4*(24 + 10*n + n^2) - 2*a^2*b^2*(57 + 16*n + n^2)
)*Cos[c + d*x]*Sin[c + d*x]^(2 + n))/(b*d*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)) + (3*b*(b^2*(2 + n) + 3*a^2
*(7 + n))*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + n))/(d*(
2 + n)*(3 + n)*(5 + n)*(7 + n)*Sqrt[Cos[c + d*x]^2]) - (3*a*(a^2*(6 + 5*n + n^2) - b^2*(53 + 15*n + n^2))*Cos[
c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^2)/(b^2*d*(4 + n)*(5 + n)*(6 + n)*(7 + n)) - ((a^2*(2 + n)*
(3 + n) - b^2*(6 + n)*(8 + n))*Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^3)/(b^2*d*(5 + n)*(6 + n
)*(7 + n)) + (a*(3 + n)*Cos[c + d*x]*Sin[c + d*x]^(1 + n)*(a + b*Sin[c + d*x])^4)/(b^2*d*(6 + n)*(7 + n)) - (C
os[c + d*x]*Sin[c + d*x]^(2 + n)*(a + b*Sin[c + d*x])^4)/(b*d*(7 + n))

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[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 ^n(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac {a (3+n) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (6+n) (7+n)}-\frac {\cos (c+d x) \sin ^{2+n}(c+d x) (a+b \sin (c+d x))^4}{b d (7+n)}-\frac {\int \sin ^n(c+d x) (a+b \sin (c+d x))^3 \left (a^2 (1+n) (3+n)-b^2 (6+n) (7+n)+3 a b \sin (c+d x)-\left (a^2 (2+n) (3+n)-b^2 (6+n) (8+n)\right ) \sin ^2(c+d x)\right ) \, dx}{b^2 (6+n) (7+n)}\\ &=-\frac {\left (a^2 (2+n) (3+n)-b^2 (6+n) (8+n)\right ) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (5+n) (6+n) (7+n)}+\frac {a (3+n) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (6+n) (7+n)}-\frac {\cos (c+d x) \sin ^{2+n}(c+d x) (a+b \sin (c+d x))^4}{b d (7+n)}-\frac {\int \sin ^n(c+d x) (a+b \sin (c+d x))^2 \left (3 a \left (a^2 \left (3+4 n+n^2\right )-b^2 \left (54+15 n+n^2\right )\right )+3 b \left (2 a^2-b^2 (6+n)\right ) \sin (c+d x)-3 a \left (a^2 \left (6+5 n+n^2\right )-b^2 \left (53+15 n+n^2\right )\right ) \sin ^2(c+d x)\right ) \, dx}{b^2 (5+n) (6+n) (7+n)}\\ &=-\frac {3 a \left (a^2 \left (6+5 n+n^2\right )-b^2 \left (53+15 n+n^2\right )\right ) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (4+n) (5+n) (6+n) (7+n)}-\frac {\left (a^2 (2+n) (3+n)-b^2 (6+n) (8+n)\right ) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (5+n) (6+n) (7+n)}+\frac {a (3+n) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (6+n) (7+n)}-\frac {\cos (c+d x) \sin ^{2+n}(c+d x) (a+b \sin (c+d x))^4}{b d (7+n)}-\frac {\int \sin ^n(c+d x) (a+b \sin (c+d x)) \left (3 a^2 \left (2 a^2 \left (3+4 n+n^2\right )-b^2 \left (163+46 n+3 n^2\right )\right )+3 a b \left (2 a^2-b^2 \left (81+26 n+2 n^2\right )\right ) \sin (c+d x)-3 \left (2 a^4 \left (6+5 n+n^2\right )+b^4 \left (24+10 n+n^2\right )-2 a^2 b^2 \left (57+16 n+n^2\right )\right ) \sin ^2(c+d x)\right ) \, dx}{b^2 (4+n) (5+n) (6+n) (7+n)}\\ &=-\frac {3 \left (2 a^4 \left (6+5 n+n^2\right )+b^4 \left (24+10 n+n^2\right )-2 a^2 b^2 \left (57+16 n+n^2\right )\right ) \cos (c+d x) \sin ^{2+n}(c+d x)}{b d (3+n) (4+n) (5+n) (6+n) (7+n)}-\frac {3 a \left (a^2 \left (6+5 n+n^2\right )-b^2 \left (53+15 n+n^2\right )\right ) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (4+n) (5+n) (6+n) (7+n)}-\frac {\left (a^2 (2+n) (3+n)-b^2 (6+n) (8+n)\right ) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (5+n) (6+n) (7+n)}+\frac {a (3+n) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (6+n) (7+n)}-\frac {\cos (c+d x) \sin ^{2+n}(c+d x) (a+b \sin (c+d x))^4}{b d (7+n)}-\frac {\int \sin ^n(c+d x) \left (3 a^3 (3+n) \left (2 a^2 \left (3+4 n+n^2\right )-b^2 \left (163+46 n+3 n^2\right )\right )-3 b^3 \left (24+10 n+n^2\right ) \left (b^2 (2+n)+3 a^2 (7+n)\right ) \sin (c+d x)-3 a (3+n) \left (2 a^4 \left (6+5 n+n^2\right )+3 b^4 \left (35+12 n+n^2\right )-2 a^2 b^2 \left (58+16 n+n^2\right )\right ) \sin ^2(c+d x)\right ) \, dx}{b^2 (3+n) (4+n) (5+n) (6+n) (7+n)}\\ &=-\frac {3 a \left (2 a^4 \left (6+5 n+n^2\right )+3 b^4 \left (35+12 n+n^2\right )-2 a^2 b^2 \left (58+16 n+n^2\right )\right ) \cos (c+d x) \sin ^{1+n}(c+d x)}{b^2 d (2+n) (4+n) (5+n) (6+n) (7+n)}-\frac {3 \left (2 a^4 \left (6+5 n+n^2\right )+b^4 \left (24+10 n+n^2\right )-2 a^2 b^2 \left (57+16 n+n^2\right )\right ) \cos (c+d x) \sin ^{2+n}(c+d x)}{b d (3+n) (4+n) (5+n) (6+n) (7+n)}-\frac {3 a \left (a^2 \left (6+5 n+n^2\right )-b^2 \left (53+15 n+n^2\right )\right ) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (4+n) (5+n) (6+n) (7+n)}-\frac {\left (a^2 (2+n) (3+n)-b^2 (6+n) (8+n)\right ) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (5+n) (6+n) (7+n)}+\frac {a (3+n) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (6+n) (7+n)}-\frac {\cos (c+d x) \sin ^{2+n}(c+d x) (a+b \sin (c+d x))^4}{b d (7+n)}-\frac {\int \sin ^n(c+d x) \left (-3 a b^2 (3+n) \left (35+12 n+n^2\right ) \left (3 b^2 (1+n)+a^2 (6+n)\right )-3 b^3 (2+n) \left (24+10 n+n^2\right ) \left (b^2 (2+n)+3 a^2 (7+n)\right ) \sin (c+d x)\right ) \, dx}{b^2 (2+n) (3+n) (4+n) (5+n) (6+n) (7+n)}\\ &=-\frac {3 a \left (2 a^4 \left (6+5 n+n^2\right )+3 b^4 \left (35+12 n+n^2\right )-2 a^2 b^2 \left (58+16 n+n^2\right )\right ) \cos (c+d x) \sin ^{1+n}(c+d x)}{b^2 d (2+n) (4+n) (5+n) (6+n) (7+n)}-\frac {3 \left (2 a^4 \left (6+5 n+n^2\right )+b^4 \left (24+10 n+n^2\right )-2 a^2 b^2 \left (57+16 n+n^2\right )\right ) \cos (c+d x) \sin ^{2+n}(c+d x)}{b d (3+n) (4+n) (5+n) (6+n) (7+n)}-\frac {3 a \left (a^2 \left (6+5 n+n^2\right )-b^2 \left (53+15 n+n^2\right )\right ) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (4+n) (5+n) (6+n) (7+n)}-\frac {\left (a^2 (2+n) (3+n)-b^2 (6+n) (8+n)\right ) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (5+n) (6+n) (7+n)}+\frac {a (3+n) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (6+n) (7+n)}-\frac {\cos (c+d x) \sin ^{2+n}(c+d x) (a+b \sin (c+d x))^4}{b d (7+n)}+\frac {\left (3 a \left (3 b^2 (1+n)+a^2 (6+n)\right )\right ) \int \sin ^n(c+d x) \, dx}{(2+n) (4+n) (6+n)}+\frac {\left (3 b \left (b^2 (2+n)+3 a^2 (7+n)\right )\right ) \int \sin ^{1+n}(c+d x) \, dx}{(3+n) (5+n) (7+n)}\\ &=-\frac {3 a \left (2 a^4 \left (6+5 n+n^2\right )+3 b^4 \left (35+12 n+n^2\right )-2 a^2 b^2 \left (58+16 n+n^2\right )\right ) \cos (c+d x) \sin ^{1+n}(c+d x)}{b^2 d (2+n) (4+n) (5+n) (6+n) (7+n)}+\frac {3 a \left (3 b^2 (1+n)+a^2 (6+n)\right ) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};\sin ^2(c+d x)\right ) \sin ^{1+n}(c+d x)}{d (1+n) (2+n) (4+n) (6+n) \sqrt {\cos ^2(c+d x)}}-\frac {3 \left (2 a^4 \left (6+5 n+n^2\right )+b^4 \left (24+10 n+n^2\right )-2 a^2 b^2 \left (57+16 n+n^2\right )\right ) \cos (c+d x) \sin ^{2+n}(c+d x)}{b d (3+n) (4+n) (5+n) (6+n) (7+n)}+\frac {3 b \left (b^2 (2+n)+3 a^2 (7+n)\right ) \cos (c+d x) \, _2F_1\left (\frac {1}{2},\frac {2+n}{2};\frac {4+n}{2};\sin ^2(c+d x)\right ) \sin ^{2+n}(c+d x)}{d (2+n) (3+n) (5+n) (7+n) \sqrt {\cos ^2(c+d x)}}-\frac {3 a \left (a^2 \left (6+5 n+n^2\right )-b^2 \left (53+15 n+n^2\right )\right ) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^2}{b^2 d (4+n) (5+n) (6+n) (7+n)}-\frac {\left (a^2 (2+n) (3+n)-b^2 (6+n) (8+n)\right ) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^3}{b^2 d (5+n) (6+n) (7+n)}+\frac {a (3+n) \cos (c+d x) \sin ^{1+n}(c+d x) (a+b \sin (c+d x))^4}{b^2 d (6+n) (7+n)}-\frac {\cos (c+d x) \sin ^{2+n}(c+d x) (a+b \sin (c+d x))^4}{b d (7+n)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.85, size = 195, normalized size = 0.31 \[ \frac {\sqrt {\cos ^2(c+d x)} \sec (c+d x) \sin ^{n+1}(c+d x) \left (\frac {a^3 \, _2F_1\left (-\frac {3}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(c+d x)\right )}{n+1}+b \sin (c+d x) \left (\frac {3 a^2 \, _2F_1\left (-\frac {3}{2},\frac {n+2}{2};\frac {n+4}{2};\sin ^2(c+d x)\right )}{n+2}+b \sin (c+d x) \left (\frac {3 a \, _2F_1\left (-\frac {3}{2},\frac {n+3}{2};\frac {n+5}{2};\sin ^2(c+d x)\right )}{n+3}+\frac {b \sin (c+d x) \, _2F_1\left (-\frac {3}{2},\frac {n+4}{2};\frac {n+6}{2};\sin ^2(c+d x)\right )}{n+4}\right )\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Cos[c + d*x]^2]*Sec[c + d*x]*Sin[c + d*x]^(1 + n)*((a^3*Hypergeometric2F1[-3/2, (1 + n)/2, (3 + n)/2, Si
n[c + d*x]^2])/(1 + n) + b*Sin[c + d*x]*((3*a^2*Hypergeometric2F1[-3/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2])
/(2 + n) + b*Sin[c + d*x]*((3*a*Hypergeometric2F1[-3/2, (3 + n)/2, (5 + n)/2, Sin[c + d*x]^2])/(3 + n) + (b*Hy
pergeometric2F1[-3/2, (4 + n)/2, (6 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x])/(4 + n)))))/d

________________________________________________________________________________________

fricas [F]  time = 1.21, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (3 \, a b^{2} \cos \left (d x + c\right )^{6} - {\left (a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (b^{3} \cos \left (d x + c\right )^{6} - {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(3*a*b^2*cos(d*x + c)^6 - (a^3 + 3*a*b^2)*cos(d*x + c)^4 + (b^3*cos(d*x + c)^6 - (3*a^2*b + b^3)*cos
(d*x + c)^4)*sin(d*x + c))*sin(d*x + c)^n, x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{3} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 28.10, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{3} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^n\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________