3.357 \(\int \cot ^2(x) \csc ^3(x) \, dx\)

Optimal. Leaf size=26 \[ \frac{1}{8} \tanh ^{-1}(\cos (x))-\frac{1}{4} \cot (x) \csc ^3(x)+\frac{1}{8} \cot (x) \csc (x) \]

[Out]

ArcTanh[Cos[x]]/8 + (Cot[x]*Csc[x])/8 - (Cot[x]*Csc[x]^3)/4

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Rubi [A]  time = 0.0313486, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2611, 3768, 3770} \[ \frac{1}{8} \tanh ^{-1}(\cos (x))-\frac{1}{4} \cot (x) \csc ^3(x)+\frac{1}{8} \cot (x) \csc (x) \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^2*Csc[x]^3,x]

[Out]

ArcTanh[Cos[x]]/8 + (Cot[x]*Csc[x])/8 - (Cot[x]*Csc[x]^3)/4

Rule 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \cot ^2(x) \csc ^3(x) \, dx &=-\frac{1}{4} \cot (x) \csc ^3(x)-\frac{1}{4} \int \csc ^3(x) \, dx\\ &=\frac{1}{8} \cot (x) \csc (x)-\frac{1}{4} \cot (x) \csc ^3(x)-\frac{1}{8} \int \csc (x) \, dx\\ &=\frac{1}{8} \tanh ^{-1}(\cos (x))+\frac{1}{8} \cot (x) \csc (x)-\frac{1}{4} \cot (x) \csc ^3(x)\\ \end{align*}

Mathematica [B]  time = 0.018436, size = 71, normalized size = 2.73 \[ -\frac{1}{64} \csc ^4\left (\frac{x}{2}\right )+\frac{1}{32} \csc ^2\left (\frac{x}{2}\right )+\frac{1}{64} \sec ^4\left (\frac{x}{2}\right )-\frac{1}{32} \sec ^2\left (\frac{x}{2}\right )-\frac{1}{8} \log \left (\sin \left (\frac{x}{2}\right )\right )+\frac{1}{8} \log \left (\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^2*Csc[x]^3,x]

[Out]

Csc[x/2]^2/32 - Csc[x/2]^4/64 + Log[Cos[x/2]]/8 - Log[Sin[x/2]]/8 - Sec[x/2]^2/32 + Sec[x/2]^4/64

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Maple [A]  time = 0.012, size = 36, normalized size = 1.4 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3}}{4\, \left ( \sin \left ( x \right ) \right ) ^{4}}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3}}{8\, \left ( \sin \left ( x \right ) \right ) ^{2}}}-{\frac{\cos \left ( x \right ) }{8}}-{\frac{\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^2*csc(x)^3,x)

[Out]

-1/4*cos(x)^3/sin(x)^4-1/8/sin(x)^2*cos(x)^3-1/8*cos(x)-1/8*ln(csc(x)-cot(x))

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Maxima [A]  time = 0.933742, size = 51, normalized size = 1.96 \begin{align*} -\frac{\cos \left (x\right )^{3} + \cos \left (x\right )}{8 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} + \frac{1}{16} \, \log \left (\cos \left (x\right ) + 1\right ) - \frac{1}{16} \, \log \left (\cos \left (x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^2*csc(x)^3,x, algorithm="maxima")

[Out]

-1/8*(cos(x)^3 + cos(x))/(cos(x)^4 - 2*cos(x)^2 + 1) + 1/16*log(cos(x) + 1) - 1/16*log(cos(x) - 1)

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Fricas [B]  time = 1.87957, size = 221, normalized size = 8.5 \begin{align*} -\frac{2 \, \cos \left (x\right )^{3} -{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) +{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 2 \, \cos \left (x\right )}{16 \,{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^2*csc(x)^3,x, algorithm="fricas")

[Out]

-1/16*(2*cos(x)^3 - (cos(x)^4 - 2*cos(x)^2 + 1)*log(1/2*cos(x) + 1/2) + (cos(x)^4 - 2*cos(x)^2 + 1)*log(-1/2*c
os(x) + 1/2) + 2*cos(x))/(cos(x)^4 - 2*cos(x)^2 + 1)

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Sympy [A]  time = 0.132958, size = 39, normalized size = 1.5 \begin{align*} - \frac{\cos ^{3}{\left (x \right )} + \cos{\left (x \right )}}{8 \cos ^{4}{\left (x \right )} - 16 \cos ^{2}{\left (x \right )} + 8} - \frac{\log{\left (\cos{\left (x \right )} - 1 \right )}}{16} + \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)**2*csc(x)**3,x)

[Out]

-(cos(x)**3 + cos(x))/(8*cos(x)**4 - 16*cos(x)**2 + 8) - log(cos(x) - 1)/16 + log(cos(x) + 1)/16

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Giac [B]  time = 1.08961, size = 63, normalized size = 2.42 \begin{align*} -\frac{\frac{1}{\cos \left (x\right )} + \cos \left (x\right )}{8 \,{\left ({\left (\frac{1}{\cos \left (x\right )} + \cos \left (x\right )\right )}^{2} - 4\right )}} + \frac{1}{32} \, \log \left ({\left | \frac{1}{\cos \left (x\right )} + \cos \left (x\right ) + 2 \right |}\right ) - \frac{1}{32} \, \log \left ({\left | \frac{1}{\cos \left (x\right )} + \cos \left (x\right ) - 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^2*csc(x)^3,x, algorithm="giac")

[Out]

-1/8*(1/cos(x) + cos(x))/((1/cos(x) + cos(x))^2 - 4) + 1/32*log(abs(1/cos(x) + cos(x) + 2)) - 1/32*log(abs(1/c
os(x) + cos(x) - 2))