3.4.44 \(\int \csc ^3(\frac {x}{2}) \, dx\) [344]

Optimal. Leaf size=24 \[ -\tanh ^{-1}\left (\cos \left (\frac {x}{2}\right )\right )-\cot \left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right ) \]

[Out]

-arctanh(cos(1/2*x))-cot(1/2*x)*csc(1/2*x)

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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3853, 3855} \begin {gather*} -\tanh ^{-1}\left (\cos \left (\frac {x}{2}\right )\right )-\cot \left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[x/2]^3,x]

[Out]

-ArcTanh[Cos[x/2]] - Cot[x/2]*Csc[x/2]

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

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Mathematica [A]
time = 0.01, size = 41, normalized size = 1.71 \begin {gather*} -\frac {1}{4} \csc ^2\left (\frac {x}{4}\right )-\log \left (\cos \left (\frac {x}{4}\right )\right )+\log \left (\sin \left (\frac {x}{4}\right )\right )+\frac {1}{4} \sec ^2\left (\frac {x}{4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[x/2]^3,x]

[Out]

-1/4*Csc[x/4]^2 - Log[Cos[x/4]] + Log[Sin[x/4]] + Sec[x/4]^2/4

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Mathics [A]
time = 2.18, size = 37, normalized size = 1.54 \begin {gather*} \frac {4 \text {Cos}\left [\frac {x}{2}\right ]+\left (-1+\text {Cos}\left [x\right ]\right ) \left (\text {Log}\left [-1+\text {Cos}\left [\frac {x}{2}\right ]\right ]-\text {Log}\left [1+\text {Cos}\left [\frac {x}{2}\right ]\right ]\right )}{-2+2 \text {Cos}\left [x\right ]} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Csc[x/2]^3,x]')

[Out]

(4 Cos[x / 2] + (-1 + Cos[x]) (Log[-1 + Cos[x / 2]] - Log[1 + Cos[x / 2]])) / (2 (-1 + Cos[x]))

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Maple [A]
time = 0.04, size = 24, normalized size = 1.00

method	result	size
derivativedivides	\(-\cot \left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right )+\ln \left (\csc \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )\)	\(24\)
default	\(-\cot \left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right )+\ln \left (\csc \left (\frac {x}{2}\right )-\cot \left (\frac {x}{2}\right )\right )\)	\(24\)
norman	\(\frac {-\frac {1}{4}+\frac {\left (\tan ^{4}\left (\frac {x}{4}\right )\right )}{4}}{\tan \left (\frac {x}{4}\right )^{2}}+\ln \left (\tan \left (\frac {x}{4}\right )\right )\)	\(24\)
risch	\(\frac {2 \,{\mathrm e}^{\frac {3 i x}{2}}+2 \,{\mathrm e}^{\frac {i x}{2}}}{\left ({\mathrm e}^{i x}-1\right )^{2}}-\ln \left ({\mathrm e}^{\frac {i x}{2}}+1\right )+\ln \left (-1+{\mathrm e}^{\frac {i x}{2}}\right )\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(1/2*x)^3,x,method=_RETURNVERBOSE)

[Out]

-cot(1/2*x)*csc(1/2*x)+ln(csc(1/2*x)-cot(1/2*x))

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Maxima [A]
time = 0.27, size = 34, normalized size = 1.42 \begin {gather*} \frac {\cos \left (\frac {1}{2} \, x\right )}{\cos \left (\frac {1}{2} \, x\right )^{2} - 1} - \frac {1}{2} \, \log \left (\cos \left (\frac {1}{2} \, x\right ) + 1\right ) + \frac {1}{2} \, \log \left (\cos \left (\frac {1}{2} \, x\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(1/2*x)^3,x, algorithm="maxima")

[Out]

cos(1/2*x)/(cos(1/2*x)^2 - 1) - 1/2*log(cos(1/2*x) + 1) + 1/2*log(cos(1/2*x) - 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (18) = 36\).
time = 0.35, size = 56, normalized size = 2.33 \begin {gather*} -\frac {{\left (\cos \left (\frac {1}{2} \, x\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (\frac {1}{2} \, x\right ) + \frac {1}{2}\right ) - {\left (\cos \left (\frac {1}{2} \, x\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (\frac {1}{2} \, x\right ) + \frac {1}{2}\right ) - 2 \, \cos \left (\frac {1}{2} \, x\right )}{2 \, {\left (\cos \left (\frac {1}{2} \, x\right )^{2} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(1/2*x)^3,x, algorithm="fricas")

[Out]

-1/2*((cos(1/2*x)^2 - 1)*log(1/2*cos(1/2*x) + 1/2) - (cos(1/2*x)^2 - 1)*log(-1/2*cos(1/2*x) + 1/2) - 2*cos(1/2
*x))/(cos(1/2*x)^2 - 1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\)
time = 0.07, size = 36, normalized size = 1.50 \begin {gather*} \frac {\log {\left (\cos {\left (\frac {x}{2} \right )} - 1 \right )}}{2} - \frac {\log {\left (\cos {\left (\frac {x}{2} \right )} + 1 \right )}}{2} + \frac {2 \cos {\left (\frac {x}{2} \right )}}{2 \cos ^{2}{\left (\frac {x}{2} \right )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(1/2*x)**3,x)

[Out]

log(cos(x/2) - 1)/2 - log(cos(x/2) + 1)/2 + 2*cos(x/2)/(2*cos(x/2)**2 - 2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (18) = 36\).
time = 0.00, size = 84, normalized size = 3.50 \begin {gather*} 2\cdot 2 \left (\frac {1-\cos \left (\frac {x}{2}\right )}{\left (1+\cos \left (\frac {x}{2}\right )\right )\cdot 16}+\frac {\left (-\frac {2 \left (1-\cos \left (\frac {x}{2}\right )\right )}{1+\cos \left (\frac {x}{2}\right )}-1\right ) \left (1+\cos \left (\frac {x}{2}\right )\right )}{16 \left (1-\cos \left (\frac {x}{2}\right )\right )}+\frac {\ln \left (\frac {1-\cos \left (\frac {x}{2}\right )}{1+\cos \left (\frac {x}{2}\right )}\right )}{8}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(1/2*x)^3,x)

[Out]

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

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Mupad [B]
time = 0.07, size = 18, normalized size = 0.75 \begin {gather*} \ln \left (\mathrm {tan}\left (\frac {x}{4}\right )\right )-\frac {\cos \left (\frac {x}{2}\right )}{{\sin \left (\frac {x}{2}\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(x/2)^3,x)

[Out]

log(tan(x/4)) - cos(x/2)/sin(x/2)^2

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