3.2.3 \(\int \csc ^3(x) \, dx\) [103]

Optimal. Leaf size=16 \[ -\frac {1}{2} \tanh ^{-1}(\cos (x))-\frac {1}{2} \cot (x) \csc (x) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3853, 3855} \begin {gather*} -\frac {1}{2} \tanh ^{-1}(\cos (x))-\frac {1}{2} \cot (x) \csc (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[x]^3,x]

[Out]

-1/2*ArcTanh[Cos[x]] - (Cot[x]*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(x) \, dx &=-\frac {1}{2} \cot (x) \csc (x)+\frac {1}{2} \int \csc (x) \, dx\\ &=-\frac {1}{2} \tanh ^{-1}(\cos (x))-\frac {1}{2} \cot (x) \csc (x)\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(16)=32\).
time = 0.00, size = 47, normalized size = 2.94 \begin {gather*} -\frac {1}{8} \csc ^2\left (\frac {x}{2}\right )-\frac {1}{2} \log \left (\cos \left (\frac {x}{2}\right )\right )+\frac {1}{2} \log \left (\sin \left (\frac {x}{2}\right )\right )+\frac {1}{8} \sec ^2\left (\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^3,x]

[Out]

-1/8*Csc[x/2]^2 - Log[Cos[x/2]]/2 + Log[Sin[x/2]]/2 + Sec[x/2]^2/8

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Maple [A]
time = 0.06, size = 18, normalized size = 1.12

method	result	size
default	\(-\frac {\cot \left (x \right ) \csc \left (x \right )}{2}+\frac {\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )}{2}\)	\(18\)
norman	\(\frac {-\frac {1}{8}+\frac {\left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8}}{\tan \left (\frac {x}{2}\right )^{2}}+\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{2}\)	\(26\)
risch	\(\frac {{\mathrm e}^{3 i x}+{\mathrm e}^{i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}-\frac {\ln \left (1+{\mathrm e}^{i x}\right )}{2}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{2}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (12) = 24\).
time = 0.88, size = 27, normalized size = 1.69 \begin {gather*} \frac {\cos \left (x\right )}{2 \, {\left (\cos \left (x\right )^{2} - 1\right )}} - \frac {1}{4} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac {1}{4} \, \log \left (\cos \left (x\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (12) = 24\).
time = 0.83, size = 44, normalized size = 2.75 \begin {gather*} -\frac {{\left (\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 )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 2 \, \cos \left (x\right )}{4 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.05, size = 27, normalized size = 1.69 \begin {gather*} \frac {\log {\left (\cos {\left (x \right )} - 1 \right )}}{4} - \frac {\log {\left (\cos {\left (x \right )} + 1 \right )}}{4} + \frac {\cos {\left (x \right )}}{2 \cos ^{2}{\left (x \right )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (12) = 24\).
time = 0.90, size = 54, normalized size = 3.38 \begin {gather*} -\frac {{\left (\frac {2 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 1\right )} {\left (\cos \left (x\right ) + 1\right )}}{8 \, {\left (\cos \left (x\right ) - 1\right )}} - \frac {\cos \left (x\right ) - 1}{8 \, {\left (\cos \left (x\right ) + 1\right )}} + \frac {1}{4} \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.15, size = 16, normalized size = 1.00 \begin {gather*} \frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{2}-\frac {\cos \left (x\right )}{2\,{\sin \left (x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(tan(x/2))/2 - cos(x)/(2*sin(x)^2)

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