3.917 \(\int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx\)

Optimal. Leaf size=60 \[ -\frac {(c+d \sin (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {c+d \sin (e+f x)}{c-d}\right )}{a f (n+1) (c-d)} \]

[Out]

-hypergeom([1, 1+n],[2+n],(c+d*sin(f*x+e))/(c-d))*(c+d*sin(f*x+e))^(1+n)/a/(c-d)/f/(1+n)

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Rubi [A]  time = 0.12, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2833, 68} \[ -\frac {(c+d \sin (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {c+d \sin (e+f x)}{c-d}\right )}{a f (n+1) (c-d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Hypergeometric2F1[1, 1 + n, 2 + n, (c + d*Sin[e + f*x])/(c - d)]*(c + d*Sin[e + f*x])^(1 + n))/(a*(c - d)*f
*(1 + n)))

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rubi steps

\begin {align*} \int \frac {\cos (e+f x) (c+d \sin (e+f x))^n}{a+a \sin (e+f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (c+\frac {d x}{a}\right )^n}{a+x} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=-\frac {\, _2F_1\left (1,1+n;2+n;\frac {c+d \sin (e+f x)}{c-d}\right ) (c+d \sin (e+f x))^{1+n}}{a (c-d) f (1+n)}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 60, normalized size = 1.00 \[ -\frac {(c+d \sin (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {c+d \sin (e+f x)}{c-d}\right )}{a f (n+1) (c-d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Hypergeometric2F1[1, 1 + n, 2 + n, (c + d*Sin[e + f*x])/(c - d)]*(c + d*Sin[e + f*x])^(1 + n))/(a*(c - d)*f
*(1 + n)))

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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )}{a \sin \left (f x + e\right ) + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e)),x, algorithm="fricas")

[Out]

integral((d*sin(f*x + e) + c)^n*cos(f*x + e)/(a*sin(f*x + e) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e)),x, algorithm="giac")

[Out]

integrate((d*sin(f*x + e) + c)^n*cos(f*x + e)/(a*sin(f*x + e) + a), x)

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maple [F]  time = 2.42, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (f x +e \right ) \left (c +d \sin \left (f x +e \right )\right )^{n}}{a +a \sin \left (f x +e \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e)),x)

[Out]

int(cos(f*x+e)*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e)),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)*(c+d*sin(f*x+e))^n/(a+a*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate((d*sin(f*x + e) + c)^n*cos(f*x + e)/(a*sin(f*x + e) + a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\cos \left (e+f\,x\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n}{a+a\,\sin \left (e+f\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(e + f*x)*(c + d*sin(e + f*x))^n)/(a + a*sin(e + f*x)),x)

[Out]

int((cos(e + f*x)*(c + d*sin(e + f*x))^n)/(a + a*sin(e + f*x)), x)

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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(f*x+e)*(c+d*sin(f*x+e))**n/(a+a*sin(f*x+e)),x)

[Out]

Timed out

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