3.378 \(\int \cot ^5(e+f x) (b \csc (e+f x))^m \, dx\)

Optimal. Leaf size=69 \[ \frac{2 (b \csc (e+f x))^{m+2}}{b^2 f (m+2)}-\frac{(b \csc (e+f x))^{m+4}}{b^4 f (m+4)}-\frac{(b \csc (e+f x))^m}{f m} \]

[Out]

-((b*Csc[e + f*x])^m/(f*m)) + (2*(b*Csc[e + f*x])^(2 + m))/(b^2*f*(2 + m)) - (b*Csc[e + f*x])^(4 + m)/(b^4*f*(
4 + m))

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Rubi [A]  time = 0.0604031, antiderivative size = 69, 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, 270} \[ \frac{2 (b \csc (e+f x))^{m+2}}{b^2 f (m+2)}-\frac{(b \csc (e+f x))^{m+4}}{b^4 f (m+4)}-\frac{(b \csc (e+f x))^m}{f m} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^5*(b*Csc[e + f*x])^m,x]

[Out]

-((b*Csc[e + f*x])^m/(f*m)) + (2*(b*Csc[e + f*x])^(2 + m))/(b^2*f*(2 + m)) - (b*Csc[e + f*x])^(4 + m)/(b^4*f*(
4 + 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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \cot ^5(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 )^2 \, dx,x,\csc (e+f x)\right )}{f}\\ &=-\frac{b \operatorname{Subst}\left (\int \left ((b x)^{-1+m}-\frac{2 (b x)^{1+m}}{b^2}+\frac{(b x)^{3+m}}{b^4}\right ) \, dx,x,\csc (e+f x)\right )}{f}\\ &=-\frac{(b \csc (e+f x))^m}{f m}+\frac{2 (b \csc (e+f x))^{2+m}}{b^2 f (2+m)}-\frac{(b \csc (e+f x))^{4+m}}{b^4 f (4+m)}\\ \end{align*}

Mathematica [A]  time = 0.293394, size = 63, normalized size = 0.91 \[ -\frac{\left (m (m+2) \csc ^4(e+f x)-2 m (m+4) \csc ^2(e+f x)+m^2+6 m+8\right ) (b \csc (e+f x))^m}{f m (m+2) (m+4)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^5*(b*Csc[e + f*x])^m,x]

[Out]

-(((b*Csc[e + f*x])^m*(8 + 6*m + m^2 - 2*m*(4 + m)*Csc[e + f*x]^2 + m*(2 + m)*Csc[e + f*x]^4))/(f*m*(2 + m)*(4
+ m)))

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Maple [C]  time = 0.553, size = 16599, normalized size = 240.6 \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)^5*(b*csc(f*x+e))^m,x)

[Out]

result too large to display

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Maxima [A]  time = 0.995602, size = 105, normalized size = 1.52 \begin{align*} -\frac{\frac{b^{m} \sin \left (f x + e\right )^{-m}}{m} - \frac{2 \, b^{m} \sin \left (f x + e\right )^{-m}}{{\left (m + 2\right )} \sin \left (f x + e\right )^{2}} + \frac{b^{m} \sin \left (f x + e\right )^{-m}}{{\left (m + 4\right )} \sin \left (f x + e\right )^{4}}}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^5*(b*csc(f*x+e))^m,x, algorithm="maxima")

[Out]

-(b^m*sin(f*x + e)^(-m)/m - 2*b^m*sin(f*x + e)^(-m)/((m + 2)*sin(f*x + e)^2) + b^m*sin(f*x + e)^(-m)/((m + 4)*
sin(f*x + e)^4))/f

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Fricas [A]  time = 1.76921, size = 269, normalized size = 3.9 \begin{align*} -\frac{{\left ({\left (m^{2} + 6 \, m + 8\right )} \cos \left (f x + e\right )^{4} - 4 \,{\left (m + 4\right )} \cos \left (f x + e\right )^{2} + 8\right )} \left (\frac{b}{\sin \left (f x + e\right )}\right )^{m}}{{\left (f m^{3} + 6 \, f m^{2} + 8 \, f m\right )} \cos \left (f x + e\right )^{4} + f m^{3} + 6 \, f m^{2} - 2 \,{\left (f m^{3} + 6 \, f m^{2} + 8 \, f m\right )} \cos \left (f x + e\right )^{2} + 8 \, f m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^5*(b*csc(f*x+e))^m,x, algorithm="fricas")

[Out]

-((m^2 + 6*m + 8)*cos(f*x + e)^4 - 4*(m + 4)*cos(f*x + e)^2 + 8)*(b/sin(f*x + e))^m/((f*m^3 + 6*f*m^2 + 8*f*m)
*cos(f*x + e)^4 + f*m^3 + 6*f*m^2 - 2*(f*m^3 + 6*f*m^2 + 8*f*m)*cos(f*x + e)^2 + 8*f*m)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**5*(b*csc(f*x+e))**m,x)

[Out]

Timed out

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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 )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^5*(b*csc(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((b*csc(f*x + e))^m*cot(f*x + e)^5, x)