3.381 \(\int \cot ^2(e+f x) (b \csc (e+f x))^m \, dx\)

Optimal. Leaf size=63 \[ -\frac {\cot ^3(e+f x) \sin ^2(e+f x)^{\frac {m+3}{2}} (b \csc (e+f x))^m \, _2F_1\left (\frac {3}{2},\frac {m+3}{2};\frac {5}{2};\cos ^2(e+f x)\right )}{3 f} \]

[Out]

-1/3*cot(f*x+e)^3*(b*csc(f*x+e))^m*hypergeom([3/2, 3/2+1/2*m],[5/2],cos(f*x+e)^2)*(sin(f*x+e)^2)^(3/2+1/2*m)/f

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Rubi [A]  time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2617} \[ -\frac {\cot ^3(e+f x) \sin ^2(e+f x)^{\frac {m+3}{2}} (b \csc (e+f x))^m \, _2F_1\left (\frac {3}{2},\frac {m+3}{2};\frac {5}{2};\cos ^2(e+f x)\right )}{3 f} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^2*(b*Csc[e + f*x])^m,x]

[Out]

-(Cot[e + f*x]^3*(b*Csc[e + f*x])^m*Hypergeometric2F1[3/2, (3 + m)/2, 5/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^((
3 + m)/2))/(3*f)

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 \cot ^2(e+f x) (b \csc (e+f x))^m \, dx &=-\frac {\cot ^3(e+f x) (b \csc (e+f x))^m \, _2F_1\left (\frac {3}{2},\frac {3+m}{2};\frac {5}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {3+m}{2}}}{3 f}\\ \end {align*}

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Mathematica [B]  time = 1.19, size = 186, normalized size = 2.95 \[ -\frac {\tan \left (\frac {1}{2} (e+f x)\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right )^{-m} (b \csc (e+f x))^m \left (-4 (m+1) \, _2F_1\left (1-m,\frac {1}{2}-\frac {m}{2};\frac {3}{2}-\frac {m}{2};-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(m+1) \, _2F_1\left (\frac {1}{2}-\frac {m}{2},-m;\frac {3}{2}-\frac {m}{2};-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(m-1) \cot ^2\left (\frac {1}{2} (e+f x)\right ) \, _2F_1\left (-\frac {m}{2}-\frac {1}{2},-m;\frac {1}{2}-\frac {m}{2};-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )}{2 f \left (m^2-1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^2*(b*Csc[e + f*x])^m,x]

[Out]

-1/2*((b*Csc[e + f*x])^m*(-4*(1 + m)*Hypergeometric2F1[1 - m, 1/2 - m/2, 3/2 - m/2, -Tan[(e + f*x)/2]^2] + (-1
+ m)*Cot[(e + f*x)/2]^2*Hypergeometric2F1[-1/2 - m/2, -m, 1/2 - m/2, -Tan[(e + f*x)/2]^2] + (1 + m)*Hypergeom
etric2F1[1/2 - m/2, -m, 3/2 - m/2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2])/(f*(-1 + m^2)*(Sec[(e + f*x)/2]^2)^
m)

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fricas [F]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \csc \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2*(b*csc(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((b*csc(f*x + e))^m*cot(f*x + e)^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \csc \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2*(b*csc(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((b*csc(f*x + e))^m*cot(f*x + e)^2, x)

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maple [F]  time = 0.42, size = 0, normalized size = 0.00 \[ \int \left (\cot ^{2}\left (f x +e \right )\right ) \left (b \csc \left (f x +e \right )\right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^2*(b*csc(f*x+e))^m,x)

[Out]

int(cot(f*x+e)^2*(b*csc(f*x+e))^m,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \csc \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2*(b*csc(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((b*csc(f*x + e))^m*cot(f*x + e)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {cot}\left (e+f\,x\right )}^2\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(e + f*x)^2*(b/sin(e + f*x))^m,x)

[Out]

int(cot(e + f*x)^2*(b/sin(e + f*x))^m, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \csc {\left (e + f x \right )}\right )^{m} \cot ^{2}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**2*(b*csc(f*x+e))**m,x)

[Out]

Integral((b*csc(e + f*x))**m*cot(e + f*x)**2, x)

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