3.11 \(\int \frac{a+b \text{csch}^{-1}(c x)}{x^4} \, dx\)

Optimal. Leaf size=58 \[ -\frac{a+b \text{csch}^{-1}(c x)}{3 x^3}+\frac{1}{9} b c^3 \left (\frac{1}{c^2 x^2}+1\right )^{3/2}-\frac{1}{3} b c^3 \sqrt{\frac{1}{c^2 x^2}+1} \]

[Out]

-(b*c^3*Sqrt[1 + 1/(c^2*x^2)])/3 + (b*c^3*(1 + 1/(c^2*x^2))^(3/2))/9 - (a + b*ArcCsch[c*x])/(3*x^3)

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Rubi [A]  time = 0.0428057, antiderivative size = 58, 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 = {6284, 266, 43} \[ -\frac{a+b \text{csch}^{-1}(c x)}{3 x^3}+\frac{1}{9} b c^3 \left (\frac{1}{c^2 x^2}+1\right )^{3/2}-\frac{1}{3} b c^3 \sqrt{\frac{1}{c^2 x^2}+1} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsch[c*x])/x^4,x]

[Out]

-(b*c^3*Sqrt[1 + 1/(c^2*x^2)])/3 + (b*c^3*(1 + 1/(c^2*x^2))^(3/2))/9 - (a + b*ArcCsch[c*x])/(3*x^3)

Rule 6284

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCsch[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 \text{csch}^{-1}(c x)}{x^4} \, dx &=-\frac{a+b \text{csch}^{-1}(c x)}{3 x^3}-\frac{b \int \frac{1}{\sqrt{1+\frac{1}{c^2 x^2}} x^5} \, dx}{3 c}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{3 x^3}+\frac{b \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{6 c}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{3 x^3}+\frac{b \operatorname{Subst}\left (\int \left (-\frac{c^2}{\sqrt{1+\frac{x}{c^2}}}+c^2 \sqrt{1+\frac{x}{c^2}}\right ) \, dx,x,\frac{1}{x^2}\right )}{6 c}\\ &=-\frac{1}{3} b c^3 \sqrt{1+\frac{1}{c^2 x^2}}+\frac{1}{9} b c^3 \left (1+\frac{1}{c^2 x^2}\right )^{3/2}-\frac{a+b \text{csch}^{-1}(c x)}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0455896, size = 59, normalized size = 1.02 \[ -\frac{a}{3 x^3}+b \left (\frac{c}{9 x^2}-\frac{2 c^3}{9}\right ) \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}-\frac{b \text{csch}^{-1}(c x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCsch[c*x])/x^4,x]

[Out]

-a/(3*x^3) + b*((-2*c^3)/9 + c/(9*x^2))*Sqrt[(1 + c^2*x^2)/(c^2*x^2)] - (b*ArcCsch[c*x])/(3*x^3)

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Maple [A]  time = 0.197, size = 75, normalized size = 1.3 \begin{align*}{c}^{3} \left ( -{\frac{a}{3\,{c}^{3}{x}^{3}}}+b \left ( -{\frac{{\rm arccsch} \left (cx\right )}{3\,{c}^{3}{x}^{3}}}-{\frac{ \left ({c}^{2}{x}^{2}+1 \right ) \left ( 2\,{c}^{2}{x}^{2}-1 \right ) }{9\,{c}^{4}{x}^{4}}{\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*arccsch(c*x))/x^4,x)

[Out]

c^3*(-1/3*a/c^3/x^3+b*(-1/3/c^3/x^3*arccsch(c*x)-1/9*(c^2*x^2+1)*(2*c^2*x^2-1)/((c^2*x^2+1)/c^2/x^2)^(1/2)/c^4
/x^4))

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Maxima [A]  time = 0.995903, size = 76, normalized size = 1.31 \begin{align*} \frac{1}{9} \, b{\left (\frac{c^{4}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 3 \, c^{4} \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c} - \frac{3 \, \operatorname{arcsch}\left (c x\right )}{x^{3}}\right )} - \frac{a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))/x^4,x, algorithm="maxima")

[Out]

1/9*b*((c^4*(1/(c^2*x^2) + 1)^(3/2) - 3*c^4*sqrt(1/(c^2*x^2) + 1))/c - 3*arccsch(c*x)/x^3) - 1/3*a/x^3

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Fricas [A]  time = 2.13714, size = 171, normalized size = 2.95 \begin{align*} -\frac{3 \, b \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) +{\left (2 \, b c^{3} x^{3} - b c x\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 3 \, a}{9 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))/x^4,x, algorithm="fricas")

[Out]

-1/9*(3*b*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + (2*b*c^3*x^3 - b*c*x)*sqrt((c^2*x^2 + 1)/(c^2*x
^2)) + 3*a)/x^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsch(c*x))/x**4,x)

[Out]

Integral((a + b*acsch(c*x))/x**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcsch}\left (c x\right ) + a}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))/x^4,x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)/x^4, x)