\(\int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx\) [934]

Optimal result

Integrand size = 27, antiderivative size = 43 \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx=-\frac {\operatorname {Hypergeometric2F1}(3,1+m,2+m,1+\sin (c+d x)) (a+a \sin (c+d x))^{1+m}}{a d (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2912, 12, 67} \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx=-\frac {(a \sin (c+d x)+a)^{m+1} \operatorname {Hypergeometric2F1}(3,m+1,m+2,\sin (c+d x)+1)}{a d (m+1)} \]

[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 67

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

Rule 2912

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/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a^3 (a+x)^m}{x^3} \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {a^2 \text {Subst}\left (\int \frac {(a+x)^m}{x^3} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = -\frac {\operatorname {Hypergeometric2F1}(3,1+m,2+m,1+\sin (c+d x)) (a+a \sin (c+d x))^{1+m}}{a d (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx=-\frac {\operatorname {Hypergeometric2F1}(3,1+m,2+m,1+\sin (c+d x)) (a+a \sin (c+d x))^{1+m}}{a d (1+m)} \]

[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)))

Maple [F]

\[\int \cos \left (d x +c \right ) \left (\csc ^{3}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{m}d x\]

[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)

Fricas [F]

\[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right ) \csc \left (d x + c\right )^{3} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right ) \csc \left (d x + c\right )^{3} \,d x } \]

[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)

Giac [F]

\[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{m} \cos \left (d x + c\right ) \csc \left (d x + c\right )^{3} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^m \, dx=\int \frac {\cos \left (c+d\,x\right )\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^m}{{\sin \left (c+d\,x\right )}^3} \,d x \]

[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)