3.4.57 \(\int \cot ^2(x) \csc ^3(x) \, dx\) [357]

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

[Out]

1/8*arctanh(cos(x))+1/8*cot(x)*csc(x)-1/4*cot(x)*csc(x)^3

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Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, 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 = {2691, 3853, 3855} \begin {gather*} \frac {1}{8} \tanh ^{-1}(\cos (x))-\frac {1}{4} \cot (x) \csc ^3(x)+\frac {1}{8} \cot (x) \csc (x) \end {gather*}

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 2691

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 3853

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 3855

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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(71\) vs. \(2(26)=52\).
time = 0.02, size = 71, normalized size = 2.73 \begin {gather*} \frac {1}{32} \csc ^2\left (\frac {x}{2}\right )-\frac {1}{64} \csc ^4\left (\frac {x}{2}\right )+\frac {1}{8} \log \left (\cos \left (\frac {x}{2}\right )\right )-\frac {1}{8} \log \left (\sin \left (\frac {x}{2}\right )\right )-\frac {1}{32} \sec ^2\left (\frac {x}{2}\right )+\frac {1}{64} \sec ^4\left (\frac {x}{2}\right ) \end {gather*}

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 = 0.05, size = 36, normalized size = 1.38

method	result	size
default	\(-\frac {\cos ^{3}\left (x \right )}{4 \sin \left (x \right )^{4}}-\frac {\cos ^{3}\left (x \right )}{8 \sin \left (x \right )^{2}}-\frac {\cos \left (x \right )}{8}-\frac {\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )}{8}\)	\(36\)
risch	\(-\frac {{\mathrm e}^{7 i x}+7 \,{\mathrm e}^{5 i x}+7 \,{\mathrm e}^{3 i x}+{\mathrm e}^{i x}}{4 \left ({\mathrm e}^{2 i x}-1\right )^{4}}+\frac {\ln \left (1+{\mathrm e}^{i x}\right )}{8}-\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{8}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^2*csc(x)^3,x,method=_RETURNVERBOSE)

[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 = 2.33, size = 38, normalized size = 1.46 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (20) = 40\).
time = 1.77, size = 68, normalized size = 2.62 \begin {gather*} -\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 {gather*}

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 = 0.05, size = 41, normalized size = 1.58 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (20) = 40\).
time = 0.98, size = 47, normalized size = 1.81 \begin {gather*} -\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 {gather*}

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))

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Mupad [B]
time = 0.32, size = 24, normalized size = 0.92 \begin {gather*} \frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{64}-\frac {1}{64\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}-\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^2/sin(x)^3,x)

[Out]

tan(x/2)^4/64 - 1/(64*tan(x/2)^4) - log(tan(x/2))/8

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