3.1.39 \(\int (b \cot (e+f x))^n (a \sin (e+f x))^m \, dx\) [39]

Optimal. Leaf size=87 \[ -\frac {(b \cot (e+f x))^{1+n} \, _2F_1\left (\frac {1+n}{2},\frac {1}{2} (1-m+n);\frac {3+n}{2};\cos ^2(e+f x)\right ) (a \sin (e+f x))^m \sin ^2(e+f x)^{\frac {1}{2} (1-m+n)}}{b f (1+n)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2683, 2697} \begin {gather*} -\frac {(a \sin (e+f x))^m (b \cot (e+f x))^{n+1} \sin ^2(e+f x)^{\frac {1}{2} (-m+n+1)} \, _2F_1\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)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2683

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(a*Cos[e + f*
x])^FracPart[m]*(Sec[e + f*x]/a)^FracPart[m], Int[(b*Tan[e + f*x])^n/(Sec[e + f*x]/a)^m, x], x] /; FreeQ[{a, b
, e, f, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[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*} \int (b \cot (e+f x))^n (a \sin (e+f x))^m \, dx &=\left (\left (\frac {\csc (e+f x)}{a}\right )^m (a \sin (e+f x))^m\right ) \int (b \cot (e+f x))^n \left (\frac {\csc (e+f x)}{a}\right )^{-m} \, dx\\ &=-\frac {(b \cot (e+f x))^{1+n} \, _2F_1\left (\frac {1+n}{2},\frac {1}{2} (1-m+n);\frac {3+n}{2};\cos ^2(e+f x)\right ) (a \sin (e+f x))^m \sin ^2(e+f x)^{\frac {1}{2} (1-m+n)}}{b f (1+n)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 1.23, size = 289, normalized size = 3.32 \begin {gather*} \frac {(3+m-n) F_1\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 \sin (e+f x) (a \sin (e+f x))^m}{f (1+m-n) \left ((3+m-n) F_1\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 F_1\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) F_1\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 )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((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*Sin[e + f*x]*(a*Sin[e + f*x])^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, Tan[
(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))

________________________________________________________________________________________

Maple [F]
time = 0.38, size = 0, normalized size = 0.00 \[\int \left (b \cot \left (f x +e \right )\right )^{n} \left (a \sin \left (f x +e \right )\right )^{m}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \sin {\left (e + f x \right )}\right )^{m} \left (b \cot {\left (e + f x \right )}\right )^{n}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (b\,\mathrm {cot}\left (e+f\,x\right )\right )}^n\,{\left (a\,\sin \left (e+f\,x\right )\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________