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

Optimal. Leaf size=129 \[ \frac{a \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left (-\frac{5}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right )}{d (n+1) \sqrt{\cos ^2(c+d x)}}+\frac{a \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left (-\frac{5}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right )}{d (n+2) \sqrt{\cos ^2(c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.136853, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2838, 2577} \[ \frac{a \cos (c+d x) \sin ^{n+1}(c+d x) \, _2F_1\left (-\frac{5}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(c+d x)\right )}{d (n+1) \sqrt{\cos ^2(c+d x)}}+\frac{a \cos (c+d x) \sin ^{n+2}(c+d x) \, _2F_1\left (-\frac{5}{2},\frac{n+2}{2};\frac{n+4}{2};\sin ^2(c+d x)\right )}{d (n+2) \sqrt{\cos ^2(c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2838

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)]), x_Symbol] :> Dist[a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[(g*Cos[e + f*x
])^p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]

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)) \, dx &=a \int \cos ^6(c+d x) \sin ^n(c+d x) \, dx+a \int \cos ^6(c+d x) \sin ^{1+n}(c+d x) \, dx\\ &=\frac{a \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{a \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)}}\\ \end{align*}

Mathematica [F]  time = 0.297752, size = 0, normalized size = 0. \[ \int \cos ^6(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 5.675, 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 ) \, 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)),x)

[Out]

int(cos(d*x+c)^6*sin(d*x+c)^n*(a+a*sin(d*x+c)),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 )} \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)),x, algorithm="maxima")

[Out]

integrate((a*sin(d*x + c) + a)*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 (a \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a \cos \left (d x + c\right )^{6}\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)),x, algorithm="fricas")

[Out]

integral((a*cos(d*x + c)^6*sin(d*x + c) + a*cos(d*x + c)^6)*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)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )} \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)),x, algorithm="giac")

[Out]

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