Integrand size = 21, antiderivative size = 55 \[ \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \sin ^n(c+d x)}{a}\right ) \left (a+b \sin ^n(c+d x)\right )^{1+p}}{a d n (1+p)} \] Output:
-hypergeom([1, p+1],[2+p],1+b*sin(d*x+c)^n/a)*(a+b*sin(d*x+c)^n)^(p+1)/a/d /n/(p+1)
Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b \sin ^n(c+d x)}{a}\right ) \left (a+b \sin ^n(c+d x)\right )^{1+p}}{a d n (1+p)} \] Input:
Integrate[Cot[c + d*x]*(a + b*Sin[c + d*x]^n)^p,x]
Output:
-((Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^n)/a]*(a + b*Sin [c + d*x]^n)^(1 + p))/(a*d*n*(1 + p)))
Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3709, 798, 75}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \sin (c+d x)^n\right )^p}{\tan (c+d x)}dx\) |
\(\Big \downarrow \) 3709 |
\(\displaystyle \frac {\int \csc (c+d x) \left (b \sin ^n(c+d x)+a\right )^pd\sin (c+d x)}{d}\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int \csc (c+d x) \left (b \sin ^n(c+d x)+a\right )^pd\sin ^n(c+d x)}{d n}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle -\frac {\left (a+b \sin ^n(c+d x)\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b \sin ^n(c+d x)}{a}+1\right )}{a d n (p+1)}\) |
Input:
Int[Cot[c + d*x]*(a + b*Sin[c + d*x]^n)^p,x]
Output:
-((Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Sin[c + d*x]^n)/a]*(a + b*Sin [c + d*x]^n)^(1 + p))/(a*d*n*(1 + p)))
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] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + ( f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Si mp[ff^(m + 1)/f Subst[Int[x^m*((a + b*(c*ff*x)^n)^p/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && ILtQ[(m - 1)/2, 0]
\[\int \cot \left (d x +c \right ) \left (a +b \sin \left (d x +c \right )^{n}\right )^{p}d x\]
Input:
int(cot(d*x+c)*(a+b*sin(d*x+c)^n)^p,x)
Output:
int(cot(d*x+c)*(a+b*sin(d*x+c)^n)^p,x)
\[ \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \cot \left (d x + c\right ) \,d x } \] Input:
integrate(cot(d*x+c)*(a+b*sin(d*x+c)^n)^p,x, algorithm="fricas")
Output:
integral((b*sin(d*x + c)^n + a)^p*cot(d*x + c), x)
\[ \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int \left (a + b \sin ^{n}{\left (c + d x \right )}\right )^{p} \cot {\left (c + d x \right )}\, dx \] Input:
integrate(cot(d*x+c)*(a+b*sin(d*x+c)**n)**p,x)
Output:
Integral((a + b*sin(c + d*x)**n)**p*cot(c + d*x), x)
\[ \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \cot \left (d x + c\right ) \,d x } \] Input:
integrate(cot(d*x+c)*(a+b*sin(d*x+c)^n)^p,x, algorithm="maxima")
Output:
integrate((b*sin(d*x + c)^n + a)^p*cot(d*x + c), x)
\[ \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \cot \left (d x + c\right ) \,d x } \] Input:
integrate(cot(d*x+c)*(a+b*sin(d*x+c)^n)^p,x, algorithm="giac")
Output:
integrate((b*sin(d*x + c)^n + a)^p*cot(d*x + c), x)
Timed out. \[ \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int \mathrm {cot}\left (c+d\,x\right )\,{\left (a+b\,{\sin \left (c+d\,x\right )}^n\right )}^p \,d x \] Input:
int(cot(c + d*x)*(a + b*sin(c + d*x)^n)^p,x)
Output:
int(cot(c + d*x)*(a + b*sin(c + d*x)^n)^p, x)
\[ \int \cot (c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int \left (\sin \left (d x +c \right )^{n} b +a \right )^{p} \cot \left (d x +c \right )d x \] Input:
int(cot(d*x+c)*(a+b*sin(d*x+c)^n)^p,x)
Output:
int((sin(c + d*x)**n*b + a)**p*cot(c + d*x),x)