\(\int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx\) [653]

Optimal result

Integrand size = 29, antiderivative size = 267 \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(c+d x)\right ) \sin ^{1+n}(c+d x)}{d (1+n) \sqrt {\cos ^2(c+d x)}}+\frac {3 a^3 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(c+d x)\right ) \sin ^{2+n}(c+d x)}{d (2+n) \sqrt {\cos ^2(c+d x)}}+\frac {3 a^3 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3+n}{2},\frac {5+n}{2},\sin ^2(c+d x)\right ) \sin ^{3+n}(c+d x)}{d (3+n) \sqrt {\cos ^2(c+d x)}}+\frac {a^3 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {4+n}{2},\frac {6+n}{2},\sin ^2(c+d x)\right ) \sin ^{4+n}(c+d x)}{d (4+n) \sqrt {\cos ^2(c+d x)}} \]

[Out]

a^3*cos(d*x+c)*hypergeom([-5/2, 1/2+1/2*n],[3/2+1/2*n],sin(d*x+c)^2)*sin(d*x+c)^(1+n)/d/(1+n)/(cos(d*x+c)^2)^(
1/2)+3*a^3*cos(d*x+c)*hypergeom([-5/2, 1+1/2*n],[1/2*n+2],sin(d*x+c)^2)*sin(d*x+c)^(2+n)/d/(2+n)/(cos(d*x+c)^2
)^(1/2)+3*a^3*cos(d*x+c)*hypergeom([-5/2, 3/2+1/2*n],[5/2+1/2*n],sin(d*x+c)^2)*sin(d*x+c)^(3+n)/d/(3+n)/(cos(d
*x+c)^2)^(1/2)+a^3*cos(d*x+c)*hypergeom([-5/2, 1/2*n+2],[1/2*n+3],sin(d*x+c)^2)*sin(d*x+c)^(4+n)/d/(4+n)/(cos(
d*x+c)^2)^(1/2)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2952, 2657} \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \cos (c+d x) \sin ^{n+1}(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(c+d x)\right )}{d (n+1) \sqrt {\cos ^2(c+d x)}}+\frac {3 a^3 \cos (c+d x) \sin ^{n+2}(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {n+2}{2},\frac {n+4}{2},\sin ^2(c+d x)\right )}{d (n+2) \sqrt {\cos ^2(c+d x)}}+\frac {3 a^3 \cos (c+d x) \sin ^{n+3}(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {n+3}{2},\frac {n+5}{2},\sin ^2(c+d x)\right )}{d (n+3) \sqrt {\cos ^2(c+d x)}}+\frac {a^3 \cos (c+d x) \sin ^{n+4}(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {n+4}{2},\frac {n+6}{2},\sin ^2(c+d x)\right )}{d (n+4) \sqrt {\cos ^2(c+d x)}} \]

[In]

Int[Cos[c + d*x]^6*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*Cos[c + d*x]*Hypergeometric2F1[-5/2, (1 + n)/2, (3 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(1 + n))/(d*(1 +
n)*Sqrt[Cos[c + d*x]^2]) + (3*a^3*Cos[c + d*x]*Hypergeometric2F1[-5/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2]*S
in[c + d*x]^(2 + n))/(d*(2 + n)*Sqrt[Cos[c + d*x]^2]) + (3*a^3*Cos[c + d*x]*Hypergeometric2F1[-5/2, (3 + n)/2,
(5 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(3 + n))/(d*(3 + n)*Sqrt[Cos[c + d*x]^2]) + (a^3*Cos[c + d*x]*Hyperge
ometric2F1[-5/2, (4 + n)/2, (6 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(4 + n))/(d*(4 + n)*Sqrt[Cos[c + d*x]^2])

Rule 2657

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b^(2*IntPart[
(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2*FracPart[(n - 1)/2])*((a*Sin[e + f*x])^(m + 1)/(a*f*(m + 1)*(Cos[e + f*x]^
2)^FracPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b
, e, f, m, n}, x]

Rule 2952

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 \cos ^6(c+d x) \sin ^n(c+d x)+3 a^3 \cos ^6(c+d x) \sin ^{1+n}(c+d x)+3 a^3 \cos ^6(c+d x) \sin ^{2+n}(c+d x)+a^3 \cos ^6(c+d x) \sin ^{3+n}(c+d x)\right ) \, dx \\ & = a^3 \int \cos ^6(c+d x) \sin ^n(c+d x) \, dx+a^3 \int \cos ^6(c+d x) \sin ^{3+n}(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^{1+n}(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^{2+n}(c+d x) \, dx \\ & = \frac {a^3 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(c+d x)\right ) \sin ^{1+n}(c+d x)}{d (1+n) \sqrt {\cos ^2(c+d x)}}+\frac {3 a^3 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(c+d x)\right ) \sin ^{2+n}(c+d x)}{d (2+n) \sqrt {\cos ^2(c+d x)}}+\frac {3 a^3 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3+n}{2},\frac {5+n}{2},\sin ^2(c+d x)\right ) \sin ^{3+n}(c+d x)}{d (3+n) \sqrt {\cos ^2(c+d x)}}+\frac {a^3 \cos (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {4+n}{2},\frac {6+n}{2},\sin ^2(c+d x)\right ) \sin ^{4+n}(c+d x)}{d (4+n) \sqrt {\cos ^2(c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.70 \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \sqrt {\cos ^2(c+d x)} \sec (c+d x) \sin ^{1+n}(c+d x) \left (\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(c+d x)\right )}{1+n}+\sin (c+d x) \left (\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2+n}{2},\frac {4+n}{2},\sin ^2(c+d x)\right )}{2+n}+\sin (c+d x) \left (\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3+n}{2},\frac {5+n}{2},\sin ^2(c+d x)\right )}{3+n}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {4+n}{2},\frac {6+n}{2},\sin ^2(c+d x)\right ) \sin (c+d x)}{4+n}\right )\right )\right )}{d} \]

[In]

Integrate[Cos[c + d*x]^6*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*Sqrt[Cos[c + d*x]^2]*Sec[c + d*x]*Sin[c + d*x]^(1 + n)*(Hypergeometric2F1[-5/2, (1 + n)/2, (3 + n)/2, Sin
[c + d*x]^2]/(1 + n) + Sin[c + d*x]*((3*Hypergeometric2F1[-5/2, (2 + n)/2, (4 + n)/2, Sin[c + d*x]^2])/(2 + n)
+ Sin[c + d*x]*((3*Hypergeometric2F1[-5/2, (3 + n)/2, (5 + n)/2, Sin[c + d*x]^2])/(3 + n) + (Hypergeometric2F
1[-5/2, (4 + n)/2, (6 + n)/2, Sin[c + d*x]^2]*Sin[c + d*x])/(4 + n)))))/d

Maple [F]

\[\int \left (\cos ^{6}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{3}d x\]

[In]

int(cos(d*x+c)^6*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x)

[Out]

int(cos(d*x+c)^6*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x)

Fricas [F]

\[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{3} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{6} \,d x } \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

integral(-(3*a^3*cos(d*x + c)^8 - 4*a^3*cos(d*x + c)^6 + (a^3*cos(d*x + c)^8 - 4*a^3*cos(d*x + c)^6)*sin(d*x +
c))*sin(d*x + c)^n, x)

Sympy [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*sin(d*x+c)**n*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{3} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{6} \,d x } \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

integrate((a*sin(d*x + c) + a)^3*sin(d*x + c)^n*cos(d*x + c)^6, x)

Giac [F]

\[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{3} \sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{6} \,d x } \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((a*sin(d*x + c) + a)^3*sin(d*x + c)^n*cos(d*x + c)^6, x)

Mupad [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx=\int {\cos \left (c+d\,x\right )}^6\,{\sin \left (c+d\,x\right )}^n\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3 \,d x \]

[In]

int(cos(c + d*x)^6*sin(c + d*x)^n*(a + a*sin(c + d*x))^3,x)

[Out]

int(cos(c + d*x)^6*sin(c + d*x)^n*(a + a*sin(c + d*x))^3, x)