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

Optimal. Leaf size=89 \[ \frac{\cos ^2(e+f x)^{\frac{n+1}{2}} (a \csc (e+f x))^m (b \tan (e+f x))^{n+1} \, _2F_1\left (\frac{n+1}{2},\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);\sin ^2(e+f x)\right )}{b f (-m+n+1)} \]

[Out]

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

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Rubi [A]  time = 0.149748, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2618, 2602, 2577} \[ \frac{\cos ^2(e+f x)^{\frac{n+1}{2}} (a \csc (e+f x))^m (b \tan (e+f x))^{n+1} \, _2F_1\left (\frac{n+1}{2},\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);\sin ^2(e+f x)\right )}{b f (-m+n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2618

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(a*Csc[e + f*
x])^FracPart[m]*(Sin[e + f*x]/a)^FracPart[m], Int[(b*Tan[e + f*x])^n/(Sin[e + f*x]/a)^m, x], x] /; FreeQ[{a, b
, e, f, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rule 2602

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(a*Cos[e + f
*x]^(n + 1)*(b*Tan[e + f*x])^(n + 1))/(b*(a*Sin[e + f*x])^(n + 1)), Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^
n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[n]

Rule 2577

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b^(2*IntPart
[(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2*FracPart[(n - 1)/2])*(a*Sin[e + f*x])^(m + 1)*Hypergeometric2F1[(1 + m)/2
, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2])/(a*f*(m + 1)*(Cos[e + f*x]^2)^FracPart[(n - 1)/2]), x] /; FreeQ[{a, b
, e, f, m, n}, x]

Rubi steps

\begin{align*} \int (a \csc (e+f x))^m (b \tan (e+f x))^n \, dx &=\left ((a \csc (e+f x))^m \left (\frac{\sin (e+f x)}{a}\right )^m\right ) \int \left (\frac{\sin (e+f x)}{a}\right )^{-m} (b \tan (e+f x))^n \, dx\\ &=\frac{\left (\cos ^{1+n}(e+f x) (a \csc (e+f x))^{1+m} \left (\frac{\sin (e+f x)}{a}\right )^{m-n} (b \tan (e+f x))^{1+n}\right ) \int \cos ^{-n}(e+f x) \left (\frac{\sin (e+f x)}{a}\right )^{-m+n} \, dx}{a b}\\ &=\frac{\cos ^2(e+f x)^{\frac{1+n}{2}} (a \csc (e+f x))^{1+m} \, _2F_1\left (\frac{1+n}{2},\frac{1}{2} (1-m+n);\frac{1}{2} (3-m+n);\sin ^2(e+f x)\right ) \sin (e+f x) (b \tan (e+f x))^{1+n}}{a b f (1-m+n)}\\ \end{align*}

Mathematica [C]  time = 1.96385, size = 287, normalized size = 3.22 \[ -\frac{a (m-n-3) (a \csc (e+f x))^{m-1} (b \tan (e+f x))^n F_1\left (\frac{1}{2} (-m+n+1);n,1-m;\frac{1}{2} (-m+n+3);\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{f (m-n-1) \left ((m-n-3) F_1\left (\frac{1}{2} (-m+n+1);n,1-m;\frac{1}{2} (-m+n+3);\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-2 \tan ^2\left (\frac{1}{2} (e+f x)\right ) \left ((m-1) F_1\left (\frac{1}{2} (-m+n+3);n,2-m;\frac{1}{2} (-m+n+5);\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )+n F_1\left (\frac{1}{2} (-m+n+3);n+1,1-m;\frac{1}{2} (-m+n+5);\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a*Csc[e + f*x])^m*(b*Tan[e + f*x])^n,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]*(a
*Csc[e + f*x])^(-1 + m)*(b*Tan[e + f*x])^n)/(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*((-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] + 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])*Tan[(e + f*x)/2]^2)))

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Maple [F]  time = 0.789, size = 0, normalized size = 0. \begin{align*} \int \left ( a\csc \left ( fx+e \right ) \right ) ^{m} \left ( b\tan \left ( fx+e \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a*csc(f*x + e))^m*(b*tan(f*x + e))^n, 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 \tan{\left (e + f x \right )}\right )^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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