3.377 \(\int \cot ^3(e+f x) (b \csc (e+f x))^m \, dx\)

Optimal. Leaf size=43 \[ \frac{(b \csc (e+f x))^m}{f m}-\frac{(b \csc (e+f x))^{m+2}}{b^2 f (m+2)} \]

[Out]

(b*Csc[e + f*x])^m/(f*m) - (b*Csc[e + f*x])^(2 + m)/(b^2*f*(2 + m))

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Rubi [A]  time = 0.0472561, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2606, 14} \[ \frac{(b \csc (e+f x))^m}{f m}-\frac{(b \csc (e+f x))^{m+2}}{b^2 f (m+2)} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^3*(b*Csc[e + f*x])^m,x]

[Out]

(b*Csc[e + f*x])^m/(f*m) - (b*Csc[e + f*x])^(2 + m)/(b^2*f*(2 + m))

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \cot ^3(e+f x) (b \csc (e+f x))^m \, dx &=-\frac{b \operatorname{Subst}\left (\int (b x)^{-1+m} \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{f}\\ &=-\frac{b \operatorname{Subst}\left (\int \left (-(b x)^{-1+m}+\frac{(b x)^{1+m}}{b^2}\right ) \, dx,x,\csc (e+f x)\right )}{f}\\ &=\frac{(b \csc (e+f x))^m}{f m}-\frac{(b \csc (e+f x))^{2+m}}{b^2 f (2+m)}\\ \end{align*}

Mathematica [A]  time = 0.0819227, size = 36, normalized size = 0.84 \[ \frac{\left (-m \csc ^2(e+f x)+m+2\right ) (b \csc (e+f x))^m}{f m (m+2)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^3*(b*Csc[e + f*x])^m,x]

[Out]

((b*Csc[e + f*x])^m*(2 + m - m*Csc[e + f*x]^2))/(f*m*(2 + m))

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Maple [C]  time = 0.601, size = 6612, normalized size = 153.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^3*(b*csc(f*x+e))^m,x)

[Out]

result too large to display

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Maxima [A]  time = 0.968742, size = 68, normalized size = 1.58 \begin{align*} \frac{\frac{b^{m} \sin \left (f x + e\right )^{-m}}{m} - \frac{b^{m} \sin \left (f x + e\right )^{-m}}{{\left (m + 2\right )} \sin \left (f x + e\right )^{2}}}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^3*(b*csc(f*x+e))^m,x, algorithm="maxima")

[Out]

(b^m*sin(f*x + e)^(-m)/m - b^m*sin(f*x + e)^(-m)/((m + 2)*sin(f*x + e)^2))/f

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Fricas [A]  time = 1.66319, size = 134, normalized size = 3.12 \begin{align*} -\frac{{\left ({\left (m + 2\right )} \cos \left (f x + e\right )^{2} - 2\right )} \left (\frac{b}{\sin \left (f x + e\right )}\right )^{m}}{f m^{2} -{\left (f m^{2} + 2 \, f m\right )} \cos \left (f x + e\right )^{2} + 2 \, f m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^3*(b*csc(f*x+e))^m,x, algorithm="fricas")

[Out]

-((m + 2)*cos(f*x + e)^2 - 2)*(b/sin(f*x + e))^m/(f*m^2 - (f*m^2 + 2*f*m)*cos(f*x + e)^2 + 2*f*m)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**3*(b*csc(f*x+e))**m,x)

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \csc \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^3*(b*csc(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((b*csc(f*x + e))^m*cot(f*x + e)^3, x)