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

Optimal. Leaf size=83 \[ -\frac{(a \csc (e+f x))^m (b \cot (e+f x))^{n+1} \sin ^2(e+f x)^{\frac{1}{2} (m+n+1)} \text{Hypergeometric2F1}\left (\frac{n+1}{2},\frac{1}{2} (m+n+1),\frac{n+3}{2},\cos ^2(e+f x)\right )}{b f (n+1)} \]

[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)))

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Rubi [A]  time = 0.0457312, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2617} \[ -\frac{(a \csc (e+f x))^m (b \cot (e+f x))^{n+1} \sin ^2(e+f x)^{\frac{1}{2} (m+n+1)} \, _2F_1\left (\frac{n+1}{2},\frac{1}{2} (m+n+1);\frac{n+3}{2};\cos ^2(e+f x)\right )}{b f (n+1)} \]

Antiderivative was successfully verified.

[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 2617

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)*Hypergeometric2F1[(n + 1)/2, (m + n + 1)/2,
(n + 3)/2, Sin[e + f*x]^2])/(b*f*(n + 1)), 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 \csc (e+f x))^m \, dx &=-\frac{(b \cot (e+f x))^{1+n} (a \csc (e+f x))^m \, _2F_1\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]  time = 1.79764, size = 306, normalized size = 3.69 \[ -\frac{a (m+n-3) (a \csc (e+f x))^{m-1} (b \cot (e+f x))^n F_1\left (\frac{1}{2} (-m-n+1);-n,1-m;\frac{1}{2} (-m-n+3);\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{f (m+n-1) \left (2 \tan ^2\left (\frac{1}{2} (e+f x)\right ) \left (n F_1\left (\frac{1}{2} (-m-n+3);1-n,1-m;\frac{1}{2} (-m-n+5);\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-(m-1) F_1\left (\frac{1}{2} (-m-n+3);-n,2-m;\frac{1}{2} (-m-n+5);\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )+(m+n-3) F_1\left (\frac{1}{2} (-m-n+1);-n,1-m;\frac{1}{2} (-m-n+3);\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[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)))

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Maple [F]  time = 1.115, size = 0, normalized size = 0. \begin{align*} \int \left ( b\cot \left ( fx+e \right ) \right ) ^{n} \left ( a\csc \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \cot \left (f x + e\right )\right )^{n} \left (a \csc \left (f x + e\right )\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b \cot \left (f x + e\right )\right )^{n} \left (a \csc \left (f x + e\right )\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \cot \left (f x + e\right )\right )^{n} \left (a \csc \left (f x + e\right )\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)