### 3.103 $$\int \csc ^3(x) \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.006974, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, Rules used = {3768, 3770} $-\frac{1}{2} \tanh ^{-1}(\cos (x))-\frac{1}{2} \cot (x) \csc (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Csc[x]^3,x]

[Out]

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

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 \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*}

Mathematica [B]  time = 0.0045829, size = 47, normalized size = 2.94 $-\frac{1}{8} \csc ^2\left (\frac{x}{2}\right )+\frac{1}{8} \sec ^2\left (\frac{x}{2}\right )+\frac{1}{2} \log \left (\sin \left (\frac{x}{2}\right )\right )-\frac{1}{2} \log \left (\cos \left (\frac{x}{2}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Csc[x]^3,x]

[Out]

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

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Maple [A]  time = 0.031, size = 18, normalized size = 1.1 \begin{align*} -{\frac{\cot \left ( x \right ) \csc \left ( x \right ) }{2}}+{\frac{\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^3,x)

[Out]

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

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Maxima [B]  time = 0.932211, size = 36, normalized size = 2.25 \begin{align*} \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{align*}

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]  time = 2.04442, size = 150, normalized size = 9.38 \begin{align*} -\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{align*}

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 = 0.112021, size = 27, normalized size = 1.69 \begin{align*} \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{align*}

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]  time = 1.06475, size = 73, normalized size = 4.56 \begin{align*} -\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{align*}

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))