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

Optimal. Leaf size=267 \[ \frac{a^3 \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left (-\frac{5}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right )}{d (n+1) \sqrt{\cos ^2(c+d x)}}+\frac{3 a^3 \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left (-\frac{5}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right )}{d (n+2) \sqrt{\cos ^2(c+d x)}}+\frac{3 a^3 \cos (c+d x) \sin ^{n+3}(c+d x) \, _2F_1\left (-\frac{5}{2},\frac{n+3}{2};\frac{n+5}{2};\sin ^2(c+d x)\right )}{d (n+3) \sqrt{\cos ^2(c+d x)}}+\frac{a^3 \cos (c+d x) \sin ^{n+4}(c+d x) \, _2F_1\left (-\frac{5}{2},\frac{n+4}{2};\frac{n+6}{2};\sin ^2(c+d x)\right )}{d (n+4) \sqrt{\cos ^2(c+d x)}} \]

[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])

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Rubi [A]  time = 0.284884, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2873, 2577} \[ \frac{a^3 \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left (-\frac{5}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right )}{d (n+1) \sqrt{\cos ^2(c+d x)}}+\frac{3 a^3 \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left (-\frac{5}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right )}{d (n+2) \sqrt{\cos ^2(c+d x)}}+\frac{3 a^3 \cos (c+d x) \sin ^{n+3}(c+d x) \, _2F_1\left (-\frac{5}{2},\frac{n+3}{2};\frac{n+5}{2};\sin ^2(c+d x)\right )}{d (n+3) \sqrt{\cos ^2(c+d x)}}+\frac{a^3 \cos (c+d x) \sin ^{n+4}(c+d x) \, _2F_1\left (-\frac{5}{2},\frac{n+4}{2};\frac{n+6}{2};\sin ^2(c+d x)\right )}{d (n+4) \sqrt{\cos ^2(c+d x)}} \]

Antiderivative was successfully verified.

[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 2873

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]

Rule 2577

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)*Hypergeometric2F1[(1 + m)/2
, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2])/(a*f*(m + 1)*(Cos[e + f*x]^2)^FracPart[(n - 1)/2]), x] /; FreeQ[{a, b
, e, f, m, n}, x]

Rubi steps

\begin{align*} \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx &=\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) \, _2F_1\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) \, _2F_1\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) \, _2F_1\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) \, _2F_1\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]  time = 0.599743, size = 188, normalized size = 0.7 \[ \frac{a^3 \sqrt{\cos ^2(c+d x)} \sec (c+d x) \sin ^{n+1}(c+d x) \left (\frac{\, _2F_1\left (-\frac{5}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right )}{n+1}+\sin (c+d x) \left (\frac{3 \, _2F_1\left (-\frac{5}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right )}{n+2}+\sin (c+d x) \left (\frac{3 \, _2F_1\left (-\frac{5}{2},\frac{n+3}{2};\frac{n+5}{2};\sin ^2(c+d x)\right )}{n+3}+\frac{\sin (c+d x) \, _2F_1\left (-\frac{5}{2},\frac{n+4}{2};\frac{n+6}{2};\sin ^2(c+d x)\right )}{n+4}\right )\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^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

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Maple [F]  time = 14.569, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{6} \left ( \sin \left ( dx+c \right ) \right ) ^{n} \left ( a+a\sin \left ( dx+c \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (3 \, a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} +{\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out