3.15 \(\int \frac{\csc ^{-1}(\frac{a}{x})}{x^4} \, dx\)

Optimal. Leaf size=60 \[ -\frac{\sqrt{1-\frac{x^2}{a^2}}}{6 a x^2}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{x^2}{a^2}}\right )}{6 a^3}-\frac{\sin ^{-1}\left (\frac{x}{a}\right )}{3 x^3} \]

[Out]

-Sqrt[1 - x^2/a^2]/(6*a*x^2) - ArcSin[x/a]/(3*x^3) - ArcTanh[Sqrt[1 - x^2/a^2]]/(6*a^3)

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Rubi [A]  time = 0.0418601, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5265, 4627, 266, 51, 63, 208} \[ -\frac{\sqrt{1-\frac{x^2}{a^2}}}{6 a x^2}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{x^2}{a^2}}\right )}{6 a^3}-\frac{\sin ^{-1}\left (\frac{x}{a}\right )}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcCsc[a/x]/x^4,x]

[Out]

-Sqrt[1 - x^2/a^2]/(6*a*x^2) - ArcSin[x/a]/(3*x^3) - ArcTanh[Sqrt[1 - x^2/a^2]]/(6*a^3)

Rule 5265

Int[ArcCsc[(c_.)/((a_.) + (b_.)*(x_)^(n_.))]^(m_.)*(u_.), x_Symbol] :> Int[u*ArcSin[a/c + (b*x^n)/c]^m, x] /;
FreeQ[{a, b, c, n, m}, x]

Rule 4627

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1
- c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 266

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\csc ^{-1}\left (\frac{a}{x}\right )}{x^4} \, dx &=\int \frac{\sin ^{-1}\left (\frac{x}{a}\right )}{x^4} \, dx\\ &=-\frac{\sin ^{-1}\left (\frac{x}{a}\right )}{3 x^3}+\frac{\int \frac{1}{x^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx}{3 a}\\ &=-\frac{\sin ^{-1}\left (\frac{x}{a}\right )}{3 x^3}+\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x}{a^2}}} \, dx,x,x^2\right )}{6 a}\\ &=-\frac{\sqrt{1-\frac{x^2}{a^2}}}{6 a x^2}-\frac{\sin ^{-1}\left (\frac{x}{a}\right )}{3 x^3}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,x^2\right )}{12 a^3}\\ &=-\frac{\sqrt{1-\frac{x^2}{a^2}}}{6 a x^2}-\frac{\sin ^{-1}\left (\frac{x}{a}\right )}{3 x^3}-\frac{\operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{x^2}{a^2}}\right )}{6 a}\\ &=-\frac{\sqrt{1-\frac{x^2}{a^2}}}{6 a x^2}-\frac{\sin ^{-1}\left (\frac{x}{a}\right )}{3 x^3}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{x^2}{a^2}}\right )}{6 a^3}\\ \end{align*}

Mathematica [A]  time = 0.0519034, size = 69, normalized size = 1.15 \[ -\frac{a^2 x \sqrt{1-\frac{x^2}{a^2}}+x^3 \log \left (\sqrt{1-\frac{x^2}{a^2}}+1\right )+2 a^3 \csc ^{-1}\left (\frac{a}{x}\right )-x^3 \log (x)}{6 a^3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCsc[a/x]/x^4,x]

[Out]

-(a^2*x*Sqrt[1 - x^2/a^2] + 2*a^3*ArcCsc[a/x] - x^3*Log[x] + x^3*Log[1 + Sqrt[1 - x^2/a^2]])/(6*a^3*x^3)

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Maple [A]  time = 0.181, size = 98, normalized size = 1.6 \begin{align*} -{\frac{1}{3\,{x}^{3}}{\rm arccsc} \left ({\frac{a}{x}}\right )}-{\frac{1}{6\,{a}^{3}} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ){\frac{1}{\sqrt{{\frac{{x}^{2}}{{a}^{2}} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ) }}}}}-{\frac{x}{6\,{a}^{4}}\sqrt{-1+{\frac{{a}^{2}}{{x}^{2}}}}\ln \left ({\frac{a}{x}}+\sqrt{-1+{\frac{{a}^{2}}{{x}^{2}}}} \right ){\frac{1}{\sqrt{{\frac{{x}^{2}}{{a}^{2}} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ) }}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccsc(a/x)/x^4,x)

[Out]

-1/3*arccsc(a/x)/x^3-1/6/a^3*(-1+a^2/x^2)/((-1+a^2/x^2)/a^2*x^2)^(1/2)-1/6/a^4*(-1+a^2/x^2)^(1/2)/((-1+a^2/x^2
)/a^2*x^2)^(1/2)*x*ln(a/x+(-1+a^2/x^2)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(a/x)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.68391, size = 198, normalized size = 3.3 \begin{align*} -\frac{4 \, a^{3} \operatorname{arccsc}\left (\frac{a}{x}\right ) + x^{3} \log \left (x \sqrt{\frac{a^{2} - x^{2}}{x^{2}}} + a\right ) - x^{3} \log \left (x \sqrt{\frac{a^{2} - x^{2}}{x^{2}}} - a\right ) + 2 \, a x^{2} \sqrt{\frac{a^{2} - x^{2}}{x^{2}}}}{12 \, a^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(a/x)/x^4,x, algorithm="fricas")

[Out]

-1/12*(4*a^3*arccsc(a/x) + x^3*log(x*sqrt((a^2 - x^2)/x^2) + a) - x^3*log(x*sqrt((a^2 - x^2)/x^2) - a) + 2*a*x
^2*sqrt((a^2 - x^2)/x^2))/(a^3*x^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acsc}{\left (\frac{a}{x} \right )}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acsc(a/x)/x**4,x)

[Out]

Integral(acsc(a/x)/x**4, x)

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Giac [A]  time = 1.1126, size = 108, normalized size = 1.8 \begin{align*} -\frac{a{\left (\frac{\log \left ({\left | a + \sqrt{a^{2} - x^{2}} \right |}\right )}{a^{3}} - \frac{\log \left ({\left | -a + \sqrt{a^{2} - x^{2}} \right |}\right )}{a^{3}} + \frac{2 \, \sqrt{a^{2} - x^{2}}}{a^{2} x^{2}}\right )}}{12 \,{\left | a \right |}} - \frac{\arcsin \left (\frac{x}{a}\right )}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccsc(a/x)/x^4,x, algorithm="giac")

[Out]

-1/12*a*(log(abs(a + sqrt(a^2 - x^2)))/a^3 - log(abs(-a + sqrt(a^2 - x^2)))/a^3 + 2*sqrt(a^2 - x^2)/(a^2*x^2))
/abs(a) - 1/3*arcsin(x/a)/x^3