3.1 \(\int \frac{\csc ^{-1}(a x^5)}{x} \, dx\)

Optimal. Leaf size=62 \[ \frac{1}{10} i \text{PolyLog}\left (2,e^{2 i \csc ^{-1}\left (a x^5\right )}\right )+\frac{1}{10} i \csc ^{-1}\left (a x^5\right )^2-\frac{1}{5} \csc ^{-1}\left (a x^5\right ) \log \left (1-e^{2 i \csc ^{-1}\left (a x^5\right )}\right ) \]

[Out]

(I/10)*ArcCsc[a*x^5]^2 - (ArcCsc[a*x^5]*Log[1 - E^((2*I)*ArcCsc[a*x^5])])/5 + (I/10)*PolyLog[2, E^((2*I)*ArcCs
c[a*x^5])]

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Rubi [A]  time = 0.0904843, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5219, 4625, 3717, 2190, 2279, 2391} \[ \frac{1}{10} i \text{PolyLog}\left (2,e^{2 i \csc ^{-1}\left (a x^5\right )}\right )+\frac{1}{10} i \csc ^{-1}\left (a x^5\right )^2-\frac{1}{5} \csc ^{-1}\left (a x^5\right ) \log \left (1-e^{2 i \csc ^{-1}\left (a x^5\right )}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCsc[a*x^5]/x,x]

[Out]

(I/10)*ArcCsc[a*x^5]^2 - (ArcCsc[a*x^5]*Log[1 - E^((2*I)*ArcCsc[a*x^5])])/5 + (I/10)*PolyLog[2, E^((2*I)*ArcCs
c[a*x^5])]

Rule 5219

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> -Subst[Int[(a + b*ArcSin[x/c])/x, x], x, 1/x] /; Fre
eQ[{a, b, c}, x]

Rule 4625

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Tan[x], x], x, ArcSin[c*
x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*
(m + 1)), x] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{\csc ^{-1}\left (a x^5\right )}{x} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{\csc ^{-1}(a x)}{x} \, dx,x,x^5\right )\\ &=-\left (\frac{1}{5} \operatorname{Subst}\left (\int \frac{\sin ^{-1}\left (\frac{x}{a}\right )}{x} \, dx,x,\frac{1}{x^5}\right )\right )\\ &=-\left (\frac{1}{5} \operatorname{Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}\left (a x^5\right )\right )\right )\\ &=\frac{1}{10} i \csc ^{-1}\left (a x^5\right )^2+\frac{2}{5} i \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}\left (a x^5\right )\right )\\ &=\frac{1}{10} i \csc ^{-1}\left (a x^5\right )^2-\frac{1}{5} \csc ^{-1}\left (a x^5\right ) \log \left (1-e^{2 i \csc ^{-1}\left (a x^5\right )}\right )+\frac{1}{5} \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}\left (a x^5\right )\right )\\ &=\frac{1}{10} i \csc ^{-1}\left (a x^5\right )^2-\frac{1}{5} \csc ^{-1}\left (a x^5\right ) \log \left (1-e^{2 i \csc ^{-1}\left (a x^5\right )}\right )-\frac{1}{10} i \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}\left (a x^5\right )}\right )\\ &=\frac{1}{10} i \csc ^{-1}\left (a x^5\right )^2-\frac{1}{5} \csc ^{-1}\left (a x^5\right ) \log \left (1-e^{2 i \csc ^{-1}\left (a x^5\right )}\right )+\frac{1}{10} i \text{Li}_2\left (e^{2 i \csc ^{-1}\left (a x^5\right )}\right )\\ \end{align*}

Mathematica [A]  time = 0.0386731, size = 56, normalized size = 0.9 \[ \frac{1}{10} i \left (\text{PolyLog}\left (2,e^{2 i \csc ^{-1}\left (a x^5\right )}\right )+\csc ^{-1}\left (a x^5\right ) \left (\csc ^{-1}\left (a x^5\right )+2 i \log \left (1-e^{2 i \csc ^{-1}\left (a x^5\right )}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCsc[a*x^5]/x,x]

[Out]

(I/10)*(ArcCsc[a*x^5]*(ArcCsc[a*x^5] + (2*I)*Log[1 - E^((2*I)*ArcCsc[a*x^5])]) + PolyLog[2, E^((2*I)*ArcCsc[a*
x^5])])

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Maple [F]  time = 0.214, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm arccsc} \left (a{x}^{5}\right )}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccsc(a*x^5)/x,x)

[Out]

int(arccsc(a*x^5)/x,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 5 \, a^{2} \int \frac{\sqrt{a x^{5} + 1} \sqrt{a x^{5} - 1} \log \left (x\right )}{a^{4} x^{11} - a^{2} x}\,{d x} - 5 i \, a^{2} \int \frac{\log \left (x\right )}{a^{4} x^{11} - a^{2} x}\,{d x} +{\left (\arctan \left (1, \sqrt{a x^{5} + 1} \sqrt{a x^{5} - 1}\right ) + i \, \log \left (a\right )\right )} \log \left (x\right ) - \frac{1}{2} i \, \log \left (a^{2} x^{10}\right ) \log \left (x\right ) + \frac{1}{2} i \, \log \left (a x^{5} + 1\right ) \log \left (x\right ) + \frac{1}{2} i \, \log \left (-a x^{5} + 1\right ) \log \left (x\right ) + \frac{5}{2} i \, \log \left (x\right )^{2} + \frac{1}{10} i \,{\rm Li}_2\left (a x^{5}\right ) + \frac{1}{10} i \,{\rm Li}_2\left (-a x^{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(a*x^5)/x,x, algorithm="maxima")

[Out]

5*a^2*integrate(sqrt(a*x^5 + 1)*sqrt(a*x^5 - 1)*log(x)/(a^4*x^11 - a^2*x), x) - 5*I*a^2*integrate(log(x)/(a^4*
x^11 - a^2*x), x) + (arctan2(1, sqrt(a*x^5 + 1)*sqrt(a*x^5 - 1)) + I*log(a))*log(x) - 1/2*I*log(a^2*x^10)*log(
x) + 1/2*I*log(a*x^5 + 1)*log(x) + 1/2*I*log(-a*x^5 + 1)*log(x) + 5/2*I*log(x)^2 + 1/10*I*dilog(a*x^5) + 1/10*
I*dilog(-a*x^5)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arccsc}\left (a x^{5}\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(a*x^5)/x,x, algorithm="fricas")

[Out]

integral(arccsc(a*x^5)/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acsc}{\left (a x^{5} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acsc(a*x**5)/x,x)

[Out]

Integral(acsc(a*x**5)/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccsc}\left (a x^{5}\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(a*x^5)/x,x, algorithm="giac")

[Out]

integrate(arccsc(a*x^5)/x, x)