3.61 \(\int \frac{\csc ^3(\sqrt{x})}{\sqrt{x}} \, dx\)

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

[Out]

-ArcTanh[Cos[Sqrt[x]]] - Cot[Sqrt[x]]*Csc[Sqrt[x]]

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Rubi [A]  time = 0.0213424, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4205, 3768, 3770} \[ -\tanh ^{-1}\left (\cos \left (\sqrt{x}\right )\right )-\cot \left (\sqrt{x}\right ) \csc \left (\sqrt{x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTanh[Cos[Sqrt[x]]] - Cot[Sqrt[x]]*Csc[Sqrt[x]]

Rule 4205

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Csc[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

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

Mathematica [B]  time = 0.0386761, size = 57, normalized size = 2.38 \[ -\frac{1}{4} \csc ^2\left (\frac{\sqrt{x}}{2}\right )+\frac{1}{4} \sec ^2\left (\frac{\sqrt{x}}{2}\right )+\log \left (\sin \left (\frac{\sqrt{x}}{2}\right )\right )-\log \left (\cos \left (\frac{\sqrt{x}}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.023, size = 24, normalized size = 1. \begin{align*} -\cot \left ( \sqrt{x} \right ) \csc \left ( \sqrt{x} \right ) +\ln \left ( \csc \left ( \sqrt{x} \right ) -\cot \left ( \sqrt{x} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.988923, size = 46, normalized size = 1.92 \begin{align*} \frac{\cos \left (\sqrt{x}\right )}{\cos \left (\sqrt{x}\right )^{2} - 1} - \frac{1}{2} \, \log \left (\cos \left (\sqrt{x}\right ) + 1\right ) + \frac{1}{2} \, \log \left (\cos \left (\sqrt{x}\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 0.488625, size = 198, normalized size = 8.25 \begin{align*} -\frac{{\left (\cos \left (\sqrt{x}\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (\sqrt{x}\right ) + \frac{1}{2}\right ) -{\left (\cos \left (\sqrt{x}\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (\sqrt{x}\right ) + \frac{1}{2}\right ) - 2 \, \cos \left (\sqrt{x}\right )}{2 \,{\left (\cos \left (\sqrt{x}\right )^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{3}{\left (\sqrt{x} \right )}}{\sqrt{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csc(sqrt(x))**3/sqrt(x), x)

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Giac [B]  time = 1.31325, size = 95, normalized size = 3.96 \begin{align*} -\frac{{\left (\frac{2 \,{\left (\cos \left (\sqrt{x}\right ) - 1\right )}}{\cos \left (\sqrt{x}\right ) + 1} - 1\right )}{\left (\cos \left (\sqrt{x}\right ) + 1\right )}}{4 \,{\left (\cos \left (\sqrt{x}\right ) - 1\right )}} - \frac{\cos \left (\sqrt{x}\right ) - 1}{4 \,{\left (\cos \left (\sqrt{x}\right ) + 1\right )}} + \frac{1}{2} \, \log \left (-\frac{\cos \left (\sqrt{x}\right ) - 1}{\cos \left (\sqrt{x}\right ) + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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