3.263 \(\int \frac {\cos (c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^2} \, dx\)

Optimal. Leaf size=38 \[ \frac {\sin ^{n+1}(c+d x) \, _2F_1(2,n+1;n+2;-\sin (c+d x))}{a^2 d (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2833, 64} \[ \frac {\sin ^{n+1}(c+d x) \, _2F_1(2,n+1;n+2;-\sin (c+d x))}{a^2 d (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^2,x]

[Out]

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

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-(d/(b*c)), 0])))

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

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Mathematica [A]  time = 0.04, size = 38, normalized size = 1.00 \[ \frac {\sin ^{n+1}(c+d x) \, _2F_1(2,n+1;n+2;-\sin (c+d x))}{a^2 d (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^2,x]

[Out]

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

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fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )}{a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^n/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

integral(-sin(d*x + c)^n*cos(d*x + c)/(a^2*cos(d*x + c)^2 - 2*a^2*sin(d*x + c) - 2*a^2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^n/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

integrate(sin(d*x + c)^n*cos(d*x + c)/(a*sin(d*x + c) + a)^2, x)

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maple [F]  time = 4.02, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x +c \right ) \left (\sin ^{n}\left (d x +c \right )\right )}{\left (a +a \sin \left (d x +c \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*sin(d*x+c)^n/(a+a*sin(d*x+c))^2,x)

[Out]

int(cos(d*x+c)*sin(d*x+c)^n/(a+a*sin(d*x+c))^2,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^n/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

integrate(sin(d*x + c)^n*cos(d*x + c)/(a*sin(d*x + c) + a)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^n}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)*sin(c + d*x)^n)/(a + a*sin(c + d*x))^2,x)

[Out]

int((cos(c + d*x)*sin(c + d*x)^n)/(a + a*sin(c + d*x))^2, 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(d*x+c)*sin(d*x+c)**n/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

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