\(\int (b \cot (e+f x))^n (a \csc (e+f x))^m \, dx\) [52]

Optimal result

Integrand size = 21, antiderivative size = 83 \[ \int (b \cot (e+f x))^n (a \csc (e+f x))^m \, dx=-\frac {(b \cot (e+f x))^{1+n} (a \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {1}{2} (1+m+n),\frac {3+n}{2},\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {1}{2} (1+m+n)}}{b f (1+n)} \]

[Out]

-(b*cot(f*x+e))^(1+n)*(a*csc(f*x+e))^m*hypergeom([1/2+1/2*n, 1/2+1/2*m+1/2*n],[3/2+1/2*n],cos(f*x+e)^2)*(sin(f
*x+e)^2)^(1/2+1/2*m+1/2*n)/b/f/(1+n)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2697} \[ \int (b \cot (e+f x))^n (a \csc (e+f x))^m \, dx=-\frac {(a \csc (e+f x))^m (b \cot (e+f x))^{n+1} \sin ^2(e+f x)^{\frac {1}{2} (m+n+1)} \operatorname {Hypergeometric2F1}\left (\frac {n+1}{2},\frac {1}{2} (m+n+1),\frac {n+3}{2},\cos ^2(e+f x)\right )}{b f (n+1)} \]

[In]

Int[(b*Cot[e + f*x])^n*(a*Csc[e + f*x])^m,x]

[Out]

-(((b*Cot[e + f*x])^(1 + n)*(a*Csc[e + f*x])^m*Hypergeometric2F1[(1 + n)/2, (1 + m + n)/2, (3 + n)/2, Cos[e +
f*x]^2]*(Sin[e + f*x]^2)^((1 + m + n)/2))/(b*f*(1 + n)))

Rule 2697

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*Sec[e + f
*x])^m*(b*Tan[e + f*x])^(n + 1)*((Cos[e + f*x]^2)^((m + n + 1)/2)/(b*f*(n + 1)))*Hypergeometric2F1[(n + 1)/2,
(m + n + 1)/2, (n + 3)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[(n - 1)/2] &&  !In
tegerQ[m/2]

Rubi steps \begin{align*} \text {integral}& = -\frac {(b \cot (e+f x))^{1+n} (a \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {1}{2} (1+m+n),\frac {3+n}{2},\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {1}{2} (1+m+n)}}{b f (1+n)} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 3.03 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.69 \[ \int (b \cot (e+f x))^n (a \csc (e+f x))^m \, dx=-\frac {a (-3+m+n) \operatorname {AppellF1}\left (\frac {1}{2} (1-m-n),-n,1-m,\frac {1}{2} (3-m-n),\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (b \cot (e+f x))^n (a \csc (e+f x))^{-1+m}}{f (-1+m+n) \left ((-3+m+n) \operatorname {AppellF1}\left (\frac {1}{2} (1-m-n),-n,1-m,\frac {1}{2} (3-m-n),\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+2 \left (n \operatorname {AppellF1}\left (\frac {1}{2} (3-m-n),1-n,1-m,\frac {1}{2} (5-m-n),\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-(-1+m) \operatorname {AppellF1}\left (\frac {1}{2} (3-m-n),-n,2-m,\frac {1}{2} (5-m-n),\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )} \]

[In]

Integrate[(b*Cot[e + f*x])^n*(a*Csc[e + f*x])^m,x]

[Out]

-((a*(-3 + m + n)*AppellF1[(1 - m - n)/2, -n, 1 - m, (3 - m - n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(
b*Cot[e + f*x])^n*(a*Csc[e + f*x])^(-1 + m))/(f*(-1 + m + n)*((-3 + m + n)*AppellF1[(1 - m - n)/2, -n, 1 - m,
(3 - m - n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*(n*AppellF1[(3 - m - n)/2, 1 - n, 1 - m, (5 - m -
n)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - (-1 + m)*AppellF1[(3 - m - n)/2, -n, 2 - m, (5 - m - n)/2, Ta
n[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))

Maple [F]

\[\int \left (b \cot \left (f x +e \right )\right )^{n} \left (a \csc \left (f x +e \right )\right )^{m}d x\]

[In]

int((b*cot(f*x+e))^n*(a*csc(f*x+e))^m,x)

[Out]

int((b*cot(f*x+e))^n*(a*csc(f*x+e))^m,x)

Fricas [F]

\[ \int (b \cot (e+f x))^n (a \csc (e+f x))^m \, dx=\int { \left (b \cot \left (f x + e\right )\right )^{n} \left (a \csc \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((b*cot(f*x+e))^n*(a*csc(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((b*cot(f*x + e))^n*(a*csc(f*x + e))^m, x)

Sympy [F]

\[ \int (b \cot (e+f x))^n (a \csc (e+f x))^m \, dx=\int \left (a \csc {\left (e + f x \right )}\right )^{m} \left (b \cot {\left (e + f x \right )}\right )^{n}\, dx \]

[In]

integrate((b*cot(f*x+e))**n*(a*csc(f*x+e))**m,x)

[Out]

Integral((a*csc(e + f*x))**m*(b*cot(e + f*x))**n, x)

Maxima [F]

\[ \int (b \cot (e+f x))^n (a \csc (e+f x))^m \, dx=\int { \left (b \cot \left (f x + e\right )\right )^{n} \left (a \csc \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((b*cot(f*x+e))^n*(a*csc(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((b*cot(f*x + e))^n*(a*csc(f*x + e))^m, x)

Giac [F]

\[ \int (b \cot (e+f x))^n (a \csc (e+f x))^m \, dx=\int { \left (b \cot \left (f x + e\right )\right )^{n} \left (a \csc \left (f x + e\right )\right )^{m} \,d x } \]

[In]

integrate((b*cot(f*x+e))^n*(a*csc(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((b*cot(f*x + e))^n*(a*csc(f*x + e))^m, x)

Mupad [F(-1)]

Timed out. \[ \int (b \cot (e+f x))^n (a \csc (e+f x))^m \, dx=\int {\left (b\,\mathrm {cot}\left (e+f\,x\right )\right )}^n\,{\left (\frac {a}{\sin \left (e+f\,x\right )}\right )}^m \,d x \]

[In]

int((b*cot(e + f*x))^n*(a/sin(e + f*x))^m,x)

[Out]

int((b*cot(e + f*x))^n*(a/sin(e + f*x))^m, x)