3.13 \(\int \frac{a+b \csc ^{-1}(c x)}{x^6} \, dx\)

Optimal. Leaf size=82 \[ -\frac{a+b \csc ^{-1}(c x)}{5 x^5}-\frac{1}{25} b c^5 \left (1-\frac{1}{c^2 x^2}\right )^{5/2}+\frac{2}{15} b c^5 \left (1-\frac{1}{c^2 x^2}\right )^{3/2}-\frac{1}{5} b c^5 \sqrt{1-\frac{1}{c^2 x^2}} \]

[Out]

-(b*c^5*Sqrt[1 - 1/(c^2*x^2)])/5 + (2*b*c^5*(1 - 1/(c^2*x^2))^(3/2))/15 - (b*c^5*(1 - 1/(c^2*x^2))^(5/2))/25 -
(a + b*ArcCsc[c*x])/(5*x^5)

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Rubi [A]  time = 0.0491578, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5221, 266, 43} \[ -\frac{a+b \csc ^{-1}(c x)}{5 x^5}-\frac{1}{25} b c^5 \left (1-\frac{1}{c^2 x^2}\right )^{5/2}+\frac{2}{15} b c^5 \left (1-\frac{1}{c^2 x^2}\right )^{3/2}-\frac{1}{5} b c^5 \sqrt{1-\frac{1}{c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsc[c*x])/x^6,x]

[Out]

-(b*c^5*Sqrt[1 - 1/(c^2*x^2)])/5 + (2*b*c^5*(1 - 1/(c^2*x^2))^(3/2))/15 - (b*c^5*(1 - 1/(c^2*x^2))^(5/2))/25 -
(a + b*ArcCsc[c*x])/(5*x^5)

Rule 5221

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCsc[c*x]
))/(d*(m + 1)), x] + Dist[(b*d)/(c*(m + 1)), Int[(d*x)^(m - 1)/Sqrt[1 - 1/(c^2*x^2)], x], x] /; FreeQ[{a, b, c
, d, m}, x] && 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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{a+b \csc ^{-1}(c x)}{x^6} \, dx &=-\frac{a+b \csc ^{-1}(c x)}{5 x^5}-\frac{b \int \frac{1}{\sqrt{1-\frac{1}{c^2 x^2}} x^7} \, dx}{5 c}\\ &=-\frac{a+b \csc ^{-1}(c x)}{5 x^5}+\frac{b \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{10 c}\\ &=-\frac{a+b \csc ^{-1}(c x)}{5 x^5}+\frac{b \operatorname{Subst}\left (\int \left (\frac{c^4}{\sqrt{1-\frac{x}{c^2}}}-2 c^4 \sqrt{1-\frac{x}{c^2}}+c^4 \left (1-\frac{x}{c^2}\right )^{3/2}\right ) \, dx,x,\frac{1}{x^2}\right )}{10 c}\\ &=-\frac{1}{5} b c^5 \sqrt{1-\frac{1}{c^2 x^2}}+\frac{2}{15} b c^5 \left (1-\frac{1}{c^2 x^2}\right )^{3/2}-\frac{1}{25} b c^5 \left (1-\frac{1}{c^2 x^2}\right )^{5/2}-\frac{a+b \csc ^{-1}(c x)}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0850999, size = 69, normalized size = 0.84 \[ -\frac{a}{5 x^5}+b \left (-\frac{4 c^3}{75 x^2}-\frac{8 c^5}{75}-\frac{c}{25 x^4}\right ) \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}-\frac{b \csc ^{-1}(c x)}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCsc[c*x])/x^6,x]

[Out]

-a/(5*x^5) + b*((-8*c^5)/75 - c/(25*x^4) - (4*c^3)/(75*x^2))*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)] - (b*ArcCsc[c*x])/
(5*x^5)

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Maple [A]  time = 0.173, size = 83, normalized size = 1. \begin{align*}{c}^{5} \left ( -{\frac{a}{5\,{c}^{5}{x}^{5}}}+b \left ( -{\frac{{\rm arccsc} \left (cx\right )}{5\,{c}^{5}{x}^{5}}}-{\frac{ \left ({c}^{2}{x}^{2}-1 \right ) \left ( 8\,{c}^{4}{x}^{4}+4\,{c}^{2}{x}^{2}+3 \right ) }{75\,{c}^{6}{x}^{6}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsc(c*x))/x^6,x)

[Out]

c^5*(-1/5*a/c^5/x^5+b*(-1/5/c^5/x^5*arccsc(c*x)-1/75*(c^2*x^2-1)*(8*c^4*x^4+4*c^2*x^2+3)/((c^2*x^2-1)/c^2/x^2)
^(1/2)/c^6/x^6))

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Maxima [A]  time = 0.978024, size = 103, normalized size = 1.26 \begin{align*} -\frac{1}{75} \, b{\left (\frac{3 \, c^{6}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{5}{2}} - 10 \, c^{6}{\left (-\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + 15 \, c^{6} \sqrt{-\frac{1}{c^{2} x^{2}} + 1}}{c} + \frac{15 \, \operatorname{arccsc}\left (c x\right )}{x^{5}}\right )} - \frac{a}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))/x^6,x, algorithm="maxima")

[Out]

-1/75*b*((3*c^6*(-1/(c^2*x^2) + 1)^(5/2) - 10*c^6*(-1/(c^2*x^2) + 1)^(3/2) + 15*c^6*sqrt(-1/(c^2*x^2) + 1))/c
+ 15*arccsc(c*x)/x^5) - 1/5*a/x^5

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Fricas [A]  time = 2.26885, size = 123, normalized size = 1.5 \begin{align*} -\frac{15 \, b \operatorname{arccsc}\left (c x\right ) +{\left (8 \, b c^{4} x^{4} + 4 \, b c^{2} x^{2} + 3 \, b\right )} \sqrt{c^{2} x^{2} - 1} + 15 \, a}{75 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))/x^6,x, algorithm="fricas")

[Out]

-1/75*(15*b*arccsc(c*x) + (8*b*c^4*x^4 + 4*b*c^2*x^2 + 3*b)*sqrt(c^2*x^2 - 1) + 15*a)/x^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{acsc}{\left (c x \right )}}{x^{6}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsc(c*x))/x**6,x)

[Out]

Integral((a + b*acsc(c*x))/x**6, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arccsc}\left (c x\right ) + a}{x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))/x^6,x, algorithm="giac")

[Out]

integrate((b*arccsc(c*x) + a)/x^6, x)