\(\int \cot (c+d x) (a+a \sin (c+d x))^m \, dx\) [932]

Optimal result

Integrand size = 19, antiderivative size = 43 \[ \int \cot (c+d x) (a+a \sin (c+d x))^m \, dx=-\frac {\operatorname {Hypergeometric2F1}(1,1+m,2+m,1+\sin (c+d x)) (a+a \sin (c+d x))^{1+m}}{a d (1+m)} \]

[Out]

-hypergeom([1, 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 = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2786, 67} \[ \int \cot (c+d x) (a+a \sin (c+d x))^m \, dx=-\frac {(a \sin (c+d x)+a)^{m+1} \operatorname {Hypergeometric2F1}(1,m+1,m+2,\sin (c+d x)+1)}{a d (m+1)} \]

[In]

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

[Out]

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

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 2786

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[x^p*((a + x)^(m - (p + 1)/2)/(a - x)^((p + 1)/2)), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+x)^m}{x} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = -\frac {\operatorname {Hypergeometric2F1}(1,1+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 = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \cot (c+d x) (a+a \sin (c+d x))^m \, dx=-\frac {\operatorname {Hypergeometric2F1}(1,1+m,2+m,1+\sin (c+d x)) (a+a \sin (c+d x))^{1+m}}{a d (1+m)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

integrate(cos(d*x+c)*csc(d*x+c)*(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), x)

Sympy [F]

\[ \int \cot (c+d x) (a+a \sin (c+d x))^m \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{m} \cos {\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx \]

[In]

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

[Out]

Integral((a*(sin(c + d*x) + 1))**m*cos(c + d*x)*csc(c + d*x), x)

Maxima [F]

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

[In]

integrate(cos(d*x+c)*csc(d*x+c)*(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), x)

Giac [F]

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

[In]

integrate(cos(d*x+c)*csc(d*x+c)*(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), x)

Mupad [F(-1)]

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

[In]

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

[Out]

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