3.3.87 \(\int (a \cos (e+f x))^m (b \csc (e+f x))^n \, dx\) [287]

Optimal. Leaf size=91 \[ \frac {a b (a \cos (e+f x))^{-1+m} \cos ^2(e+f x)^{\frac {1-m}{2}} (b \csc (e+f x))^{-1+n} \, _2F_1\left (\frac {1-m}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 91, 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 = {2667, 2657} \begin {gather*} \frac {a b \cos ^2(e+f x)^{\frac {1-m}{2}} (a \cos (e+f x))^{m-1} (b \csc (e+f x))^{n-1} \, _2F_1\left (\frac {1-m}{2},\frac {1-n}{2};\frac {3-n}{2};\sin ^2(e+f x)\right )}{f (1-n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2657

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)/(a*f*(m + 1)*(Cos[e + f*x]^
2)^FracPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b
, e, f, m, n}, x]

Rule 2667

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

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 0.17, size = 316, normalized size = 3.47 \begin {gather*} -\frac {2 (-3+n) F_1\left (\frac {1}{2}-\frac {n}{2};-m,1+m-n;\frac {3}{2}-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) (a \cos (e+f x))^m (b \csc (e+f x))^n \sin \left (\frac {1}{2} (e+f x)\right )}{f (-1+n) \left ((-3+n) F_1\left (\frac {1}{2}-\frac {n}{2};-m,1+m-n;\frac {3}{2}-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )+2 \left (m F_1\left (\frac {3}{2}-\frac {n}{2};1-m,1+m-n;\frac {5}{2}-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(1+m-n) F_1\left (\frac {3}{2}-\frac {n}{2};-m,2+m-n;\frac {5}{2}-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \sin ^2\left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \left (a \cos \left (f x +e \right )\right )^{m} \left (b \csc \left (f x +e \right )\right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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