3.934 \(\int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx\)

Optimal. Leaf size=43 \[ -\frac {(a \sin (c+d x)+a)^{m+1} \, _2F_1(3,m+1;m+2;\sin (c+d x)+1)}{a d (m+1)} \]

[Out]

-hypergeom([3, 1+m],[2+m],1+sin(d*x+c))*(a+a*sin(d*x+c))^(1+m)/a/d/(1+m)

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Rubi [A]  time = 0.08, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2833, 12, 65} \[ -\frac {(a \sin (c+d x)+a)^{m+1} \, _2F_1(3,m+1;m+2;\sin (c+d x)+1)}{a d (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^m,x]

[Out]

-((Hypergeometric2F1[3, 1 + m, 2 + m, 1 + Sin[c + d*x]]*(a + a*Sin[c + d*x])^(1 + m))/(a*d*(1 + m)))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || 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 \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^3 (a+x)^m}{x^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \frac {(a+x)^m}{x^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {\, _2F_1(3,1+m;2+m;1+\sin (c+d x)) (a+a \sin (c+d x))^{1+m}}{a d (1+m)}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 43, normalized size = 1.00 \[ -\frac {(a \sin (c+d x)+a)^{m+1} \, _2F_1(3,m+1;m+2;\sin (c+d x)+1)}{a d (m+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^m,x]

[Out]

-((Hypergeometric2F1[3, 1 + m, 2 + m, 1 + Sin[c + d*x]]*(a + a*Sin[c + d*x])^(1 + m))/(a*d*(1 + m)))

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fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right ) \csc \left (d x + c\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a*sin(d*x + c) + a)^m*cos(d*x + c)*csc(d*x + c)^3, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right ) \csc \left (d x + c\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sin(d*x + c) + a)^m*cos(d*x + c)*csc(d*x + c)^3, x)

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maple [F]  time = 1.20, size = 0, normalized size = 0.00 \[ \int \cos \left (d x +c \right ) \left (\csc ^{3}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*csc(d*x+c)^3*(a+a*sin(d*x+c))^m,x)

[Out]

int(cos(d*x+c)*csc(d*x+c)^3*(a+a*sin(d*x+c))^m,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right ) \csc \left (d x + c\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sin(d*x + c) + a)^m*cos(d*x + c)*csc(d*x + c)^3, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\cos \left (c+d\,x\right )\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{{\sin \left (c+d\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)*(a + a*sin(c + d*x))^m)/sin(c + d*x)^3,x)

[Out]

int((cos(c + d*x)*(a + a*sin(c + d*x))^m)/sin(c + d*x)^3, 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)*csc(d*x+c)**3*(a+a*sin(d*x+c))**m,x)

[Out]

Timed out

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