\(\int \frac {\csc ^3(\sqrt {x})}{\sqrt {x}} \, dx\) [61]

Optimal result

Integrand size = 14, antiderivative size = 24 \[ \int \frac {\csc ^3\left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-\text {arctanh}\left (\cos \left (\sqrt {x}\right )\right )-\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4290, 3853, 3855} \[ \int \frac {\csc ^3\left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-\text {arctanh}\left (\cos \left (\sqrt {x}\right )\right )-\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right ) \]

[In]

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

[Out]

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

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]

Rule 4290

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]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \csc ^3(x) \, dx,x,\sqrt {x}\right ) \\ & = -\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right )+\text {Subst}\left (\int \csc (x) \, dx,x,\sqrt {x}\right ) \\ & = -\text {arctanh}\left (\cos \left (\sqrt {x}\right )\right )-\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(24)=48\).

Time = 0.06 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.38 \[ \int \frac {\csc ^3\left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-\frac {1}{4} \csc ^2\left (\frac {\sqrt {x}}{2}\right )-\log \left (\cos \left (\frac {\sqrt {x}}{2}\right )\right )+\log \left (\sin \left (\frac {\sqrt {x}}{2}\right )\right )+\frac {1}{4} \sec ^2\left (\frac {\sqrt {x}}{2}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(-\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right )+\ln \left (\csc \left (\sqrt {x}\right )-\cot \left (\sqrt {x}\right )\right )\)	\(24\)
default	\(-\cot \left (\sqrt {x}\right ) \csc \left (\sqrt {x}\right )+\ln \left (\csc \left (\sqrt {x}\right )-\cot \left (\sqrt {x}\right )\right )\)	\(24\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (18) = 36\).

Time = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.33 \[ \int \frac {\csc ^3\left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-\frac {{\left (\cos \left (\sqrt {x}\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (\sqrt {x}\right ) + \frac {1}{2}\right ) - {\left (\cos \left (\sqrt {x}\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (\sqrt {x}\right ) + \frac {1}{2}\right ) - 2 \, \cos \left (\sqrt {x}\right )}{2 \, {\left (\cos \left (\sqrt {x}\right )^{2} - 1\right )}} \]

[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)

Sympy [F]

\[ \int \frac {\csc ^3\left (\sqrt {x}\right )}{\sqrt {x}} \, dx=\int \frac {\csc ^{3}{\left (\sqrt {x} \right )}}{\sqrt {x}}\, dx \]

[In]

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

[Out]

Integral(csc(sqrt(x))**3/sqrt(x), x)

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {\csc ^3\left (\sqrt {x}\right )}{\sqrt {x}} \, dx=\frac {\cos \left (\sqrt {x}\right )}{\cos \left (\sqrt {x}\right )^{2} - 1} - \frac {1}{2} \, \log \left (\cos \left (\sqrt {x}\right ) + 1\right ) + \frac {1}{2} \, \log \left (\cos \left (\sqrt {x}\right ) - 1\right ) \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (18) = 36\).

Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.92 \[ \int \frac {\csc ^3\left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-\frac {{\left (\frac {2 \, {\left (\cos \left (\sqrt {x}\right ) - 1\right )}}{\cos \left (\sqrt {x}\right ) + 1} - 1\right )} {\left (\cos \left (\sqrt {x}\right ) + 1\right )}}{4 \, {\left (\cos \left (\sqrt {x}\right ) - 1\right )}} - \frac {\cos \left (\sqrt {x}\right ) - 1}{4 \, {\left (\cos \left (\sqrt {x}\right ) + 1\right )}} + \frac {1}{2} \, \log \left (-\frac {\cos \left (\sqrt {x}\right ) - 1}{\cos \left (\sqrt {x}\right ) + 1}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 19.55 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.92 \[ \int \frac {\csc ^3\left (\sqrt {x}\right )}{\sqrt {x}} \, dx=-\ln \left (-\frac {{\mathrm {e}}^{\sqrt {x}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{\sqrt {x}}-\frac {1{}\mathrm {i}}{\sqrt {x}}\right )+\ln \left (-\frac {{\mathrm {e}}^{\sqrt {x}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{\sqrt {x}}+\frac {1{}\mathrm {i}}{\sqrt {x}}\right )+\frac {4\,{\mathrm {e}}^{\sqrt {x}\,1{}\mathrm {i}}}{1+{\mathrm {e}}^{\sqrt {x}\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{\sqrt {x}\,2{}\mathrm {i}}}+\frac {2\,{\mathrm {e}}^{\sqrt {x}\,1{}\mathrm {i}}}{{\mathrm {e}}^{\sqrt {x}\,2{}\mathrm {i}}-1} \]

[In]

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

[Out]

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