3.12 \(\int \frac{a+b \text{csch}^{-1}(c x)}{x^5} \, dx\)

Optimal. Leaf size=74 \[ -\frac{a+b \text{csch}^{-1}(c x)}{4 x^4}-\frac{3 b c^3 \sqrt{\frac{1}{c^2 x^2}+1}}{32 x}+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1}}{16 x^3}+\frac{3}{32} b c^4 \text{csch}^{-1}(c x) \]

[Out]

(b*c*Sqrt[1 + 1/(c^2*x^2)])/(16*x^3) - (3*b*c^3*Sqrt[1 + 1/(c^2*x^2)])/(32*x) + (3*b*c^4*ArcCsch[c*x])/32 - (a
+ b*ArcCsch[c*x])/(4*x^4)

________________________________________________________________________________________

Rubi [A]  time = 0.0480371, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6284, 335, 321, 215} \[ -\frac{a+b \text{csch}^{-1}(c x)}{4 x^4}-\frac{3 b c^3 \sqrt{\frac{1}{c^2 x^2}+1}}{32 x}+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1}}{16 x^3}+\frac{3}{32} b c^4 \text{csch}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsch[c*x])/x^5,x]

[Out]

(b*c*Sqrt[1 + 1/(c^2*x^2)])/(16*x^3) - (3*b*c^3*Sqrt[1 + 1/(c^2*x^2)])/(32*x) + (3*b*c^4*ArcCsch[c*x])/32 - (a
+ b*ArcCsch[c*x])/(4*x^4)

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 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{a+b \text{csch}^{-1}(c x)}{x^5} \, dx &=-\frac{a+b \text{csch}^{-1}(c x)}{4 x^4}-\frac{b \int \frac{1}{\sqrt{1+\frac{1}{c^2 x^2}} x^6} \, dx}{4 c}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{4 x^4}+\frac{b \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )}{4 c}\\ &=\frac{b c \sqrt{1+\frac{1}{c^2 x^2}}}{16 x^3}-\frac{a+b \text{csch}^{-1}(c x)}{4 x^4}-\frac{1}{16} (3 b c) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{b c \sqrt{1+\frac{1}{c^2 x^2}}}{16 x^3}-\frac{3 b c^3 \sqrt{1+\frac{1}{c^2 x^2}}}{32 x}-\frac{a+b \text{csch}^{-1}(c x)}{4 x^4}+\frac{1}{32} \left (3 b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{c^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{b c \sqrt{1+\frac{1}{c^2 x^2}}}{16 x^3}-\frac{3 b c^3 \sqrt{1+\frac{1}{c^2 x^2}}}{32 x}+\frac{3}{32} b c^4 \text{csch}^{-1}(c x)-\frac{a+b \text{csch}^{-1}(c x)}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.041736, size = 78, normalized size = 1.05 \[ -\frac{a}{4 x^4}+b \left (\frac{c}{16 x^3}-\frac{3 c^3}{32 x}\right ) \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}+\frac{3}{32} b c^4 \sinh ^{-1}\left (\frac{1}{c x}\right )-\frac{b \text{csch}^{-1}(c x)}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCsch[c*x])/x^5,x]

[Out]

-a/(4*x^4) + b*(c/(16*x^3) - (3*c^3)/(32*x))*Sqrt[(1 + c^2*x^2)/(c^2*x^2)] - (b*ArcCsch[c*x])/(4*x^4) + (3*b*c
^4*ArcSinh[1/(c*x)])/32

________________________________________________________________________________________

Maple [A]  time = 0.195, size = 120, normalized size = 1.6 \begin{align*}{c}^{4} \left ( -{\frac{a}{4\,{c}^{4}{x}^{4}}}+b \left ( -{\frac{{\rm arccsch} \left (cx\right )}{4\,{c}^{4}{x}^{4}}}+{\frac{1}{32\,{c}^{5}{x}^{5}}\sqrt{{c}^{2}{x}^{2}+1} \left ( 3\,{\it Artanh} \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}} \right ){c}^{4}{x}^{4}-3\,{c}^{2}{x}^{2}\sqrt{{c}^{2}{x}^{2}+1}+2\,\sqrt{{c}^{2}{x}^{2}+1} \right ){\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^5,x)

[Out]

c^4*(-1/4*a/c^4/x^4+b*(-1/4/c^4/x^4*arccsch(c*x)+1/32*(c^2*x^2+1)^(1/2)*(3*arctanh(1/(c^2*x^2+1)^(1/2))*c^4*x^
4-3*c^2*x^2*(c^2*x^2+1)^(1/2)+2*(c^2*x^2+1)^(1/2))/((c^2*x^2+1)/c^2/x^2)^(1/2)/c^5/x^5))

________________________________________________________________________________________

Maxima [B]  time = 1.01471, size = 198, normalized size = 2.68 \begin{align*} \frac{1}{64} \, b{\left (\frac{3 \, c^{5} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right ) - 3 \, c^{5} \log \left (c x \sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right ) - \frac{2 \,{\left (3 \, c^{8} x^{3}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - 5 \, c^{6} x \sqrt{\frac{1}{c^{2} x^{2}} + 1}\right )}}{c^{4} x^{4}{\left (\frac{1}{c^{2} x^{2}} + 1\right )}^{2} - 2 \, c^{2} x^{2}{\left (\frac{1}{c^{2} x^{2}} + 1\right )} + 1}}{c} - \frac{16 \, \operatorname{arcsch}\left (c x\right )}{x^{4}}\right )} - \frac{a}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))/x^5,x, algorithm="maxima")

[Out]

1/64*b*((3*c^5*log(c*x*sqrt(1/(c^2*x^2) + 1) + 1) - 3*c^5*log(c*x*sqrt(1/(c^2*x^2) + 1) - 1) - 2*(3*c^8*x^3*(1
/(c^2*x^2) + 1)^(3/2) - 5*c^6*x*sqrt(1/(c^2*x^2) + 1))/(c^4*x^4*(1/(c^2*x^2) + 1)^2 - 2*c^2*x^2*(1/(c^2*x^2) +
1) + 1))/c - 16*arccsch(c*x)/x^4) - 1/4*a/x^4

________________________________________________________________________________________

Fricas [A]  time = 2.12589, size = 196, normalized size = 2.65 \begin{align*} \frac{{\left (3 \, b c^{4} x^{4} - 8 \, b\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) -{\left (3 \, b c^{3} x^{3} - 2 \, b c x\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - 8 \, a}{32 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))/x^5,x, algorithm="fricas")

[Out]

1/32*((3*b*c^4*x^4 - 8*b)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) - (3*b*c^3*x^3 - 2*b*c*x)*sqrt((c
^2*x^2 + 1)/(c^2*x^2)) - 8*a)/x^4

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsch(c*x))/x**5,x)

[Out]

Integral((a + b*acsch(c*x))/x**5, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))/x^5,x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)/x^5, x)