3.31 \(\int \frac{a+b \text{sech}^{-1}(c x)}{x^6} \, dx\)

Optimal. Leaf size=109 \[ -\frac{a+b \text{sech}^{-1}(c x)}{5 x^5}+\frac{4 b c^2 \sqrt{1-c x}}{75 x^3 \sqrt{\frac{1}{c x+1}}}+\frac{8 b c^4 \sqrt{1-c x}}{75 x \sqrt{\frac{1}{c x+1}}}+\frac{b \sqrt{1-c x}}{25 x^5 \sqrt{\frac{1}{c x+1}}} \]

[Out]

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

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Rubi [A]  time = 0.0503686, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6283, 103, 12, 95} \[ -\frac{a+b \text{sech}^{-1}(c x)}{5 x^5}+\frac{4 b c^2 \sqrt{1-c x}}{75 x^3 \sqrt{\frac{1}{c x+1}}}+\frac{8 b c^4 \sqrt{1-c x}}{75 x \sqrt{\frac{1}{c x+1}}}+\frac{b \sqrt{1-c x}}{25 x^5 \sqrt{\frac{1}{c x+1}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSech[c*x])/x^6,x]

[Out]

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

Rule 6283

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

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] /; FreeQ[{a, b, c, d,
e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0
] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{a+b \text{sech}^{-1}(c x)}{x^6} \, dx &=-\frac{a+b \text{sech}^{-1}(c x)}{5 x^5}-\frac{1}{5} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^6 \sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=\frac{b \sqrt{1-c x}}{25 x^5 \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{5 x^5}+\frac{1}{25} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int -\frac{4 c^2}{x^4 \sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=\frac{b \sqrt{1-c x}}{25 x^5 \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{5 x^5}-\frac{1}{25} \left (4 b c^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^4 \sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=\frac{b \sqrt{1-c x}}{25 x^5 \sqrt{\frac{1}{1+c x}}}+\frac{4 b c^2 \sqrt{1-c x}}{75 x^3 \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{5 x^5}+\frac{1}{75} \left (4 b c^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int -\frac{2 c^2}{x^2 \sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=\frac{b \sqrt{1-c x}}{25 x^5 \sqrt{\frac{1}{1+c x}}}+\frac{4 b c^2 \sqrt{1-c x}}{75 x^3 \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{5 x^5}-\frac{1}{75} \left (8 b c^4 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x^2 \sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=\frac{b \sqrt{1-c x}}{25 x^5 \sqrt{\frac{1}{1+c x}}}+\frac{4 b c^2 \sqrt{1-c x}}{75 x^3 \sqrt{\frac{1}{1+c x}}}+\frac{8 b c^4 \sqrt{1-c x}}{75 x \sqrt{\frac{1}{1+c x}}}-\frac{a+b \text{sech}^{-1}(c x)}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0860176, size = 94, normalized size = 0.86 \[ -\frac{a}{5 x^5}+b \left (\frac{4 c^3}{75 x^2}+\frac{4 c^2}{75 x^3}+\frac{8 c^4}{75 x}+\frac{8 c^5}{75}+\frac{c}{25 x^4}+\frac{1}{25 x^5}\right ) \sqrt{\frac{1-c x}{c x+1}}-\frac{b \text{sech}^{-1}(c x)}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSech[c*x])/x^6,x]

[Out]

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

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Maple [A]  time = 0.18, size = 85, normalized size = 0.8 \begin{align*}{c}^{5} \left ( -{\frac{a}{5\,{c}^{5}{x}^{5}}}+b \left ( -{\frac{{\rm arcsech} \left (cx\right )}{5\,{c}^{5}{x}^{5}}}+{\frac{8\,{c}^{4}{x}^{4}+4\,{c}^{2}{x}^{2}+3}{75\,{c}^{4}{x}^{4}}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.99591, size = 99, normalized size = 0.91 \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{arsech}\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*arcsech(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
*arcsech(c*x)/x^5) - 1/5*a/x^5

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Fricas [A]  time = 1.87304, size = 200, normalized size = 1.83 \begin{align*} -\frac{15 \, b \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) -{\left (8 \, b c^{5} x^{5} + 4 \, b c^{3} x^{3} + 3 \, b c x\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 15 \, a}{75 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*asech(c*x))/x**6, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arsech}\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*arcsech(c*x))/x^6,x, algorithm="giac")

[Out]

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