3.47 \(\int \frac{(a+b \text{sech}^{-1}(c x))^3}{x^2} \, dx\)

Optimal. Leaf size=102 \[ -\frac{6 b^2 \left (a+b \text{sech}^{-1}(c x)\right )}{x}+\frac{3 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )^2}{x}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^3}{x}+\frac{6 b^3 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{x} \]

[Out]

(6*b^3*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/x - (6*b^2*(a + b*ArcSech[c*x]))/x + (3*b*Sqrt[(1 - c*x)/(1 + c*x)
]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/x - (a + b*ArcSech[c*x])^3/x

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Rubi [A]  time = 0.102711, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {6285, 3296, 2637} \[ -\frac{6 b^2 \left (a+b \text{sech}^{-1}(c x)\right )}{x}+\frac{3 b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )^2}{x}-\frac{\left (a+b \text{sech}^{-1}(c x)\right )^3}{x}+\frac{6 b^3 \sqrt{\frac{1-c x}{c x+1}} (c x+1)}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSech[c*x])^3/x^2,x]

[Out]

(6*b^3*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/x - (6*b^2*(a + b*ArcSech[c*x]))/x + (3*b*Sqrt[(1 - c*x)/(1 + c*x)
]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/x - (a + b*ArcSech[c*x])^3/x

Rule 6285

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b
*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, ArcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] &
& (GtQ[n, 0] || LtQ[m, -1])

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.321964, size = 165, normalized size = 1.62 \[ -\frac{3 b \text{sech}^{-1}(c x) \left (a^2-2 a b \sqrt{\frac{1-c x}{c x+1}} (c x+1)+2 b^2\right )-3 a^2 b \sqrt{\frac{1-c x}{c x+1}} (c x+1)+a^3-3 b^2 \text{sech}^{-1}(c x)^2 \left (b \sqrt{\frac{1-c x}{c x+1}} (c x+1)-a\right )+6 a b^2-6 b^3 \sqrt{\frac{1-c x}{c x+1}} (c x+1)+b^3 \text{sech}^{-1}(c x)^3}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSech[c*x])^3/x^2,x]

[Out]

-((a^3 + 6*a*b^2 - 3*a^2*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) - 6*b^3*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x) + 3
*b*(a^2 + 2*b^2 - 2*a*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))*ArcSech[c*x] - 3*b^2*(-a + b*Sqrt[(1 - c*x)/(1 +
c*x)]*(1 + c*x))*ArcSech[c*x]^2 + b^3*ArcSech[c*x]^3)/x)

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Maple [B]  time = 0.252, size = 227, normalized size = 2.2 \begin{align*} c \left ( -{\frac{{a}^{3}}{cx}}+{b}^{3} \left ( -{\frac{ \left ({\rm arcsech} \left (cx\right ) \right ) ^{3}}{cx}}+3\, \left ({\rm arcsech} \left (cx\right ) \right ) ^{2}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}-6\,{\frac{{\rm arcsech} \left (cx\right )}{cx}}+6\,\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}} \right ) +3\,a{b}^{2} \left ( -{\frac{ \left ({\rm arcsech} \left (cx\right ) \right ) ^{2}}{cx}}+2\,{\rm arcsech} \left (cx\right )\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}-2\,{\frac{1}{cx}} \right ) +3\,{a}^{2}b \left ( -{\frac{{\rm arcsech} \left (cx\right )}{cx}}+\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))^3/x^2,x)

[Out]

c*(-a^3/c/x+b^3*(-arcsech(c*x)^3/c/x+3*arcsech(c*x)^2*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)-6/c/x*arcsech(c
*x)+6*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2))+3*a*b^2*(-arcsech(c*x)^2/c/x+2*arcsech(c*x)*(-(c*x-1)/c/x)^(1/
2)*((c*x+1)/c/x)^(1/2)-2/c/x)+3*a^2*b*(-1/c/x*arcsech(c*x)+(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)))

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Maxima [A]  time = 1.03547, size = 194, normalized size = 1.9 \begin{align*} -\frac{b^{3} \operatorname{arsech}\left (c x\right )^{3}}{x} + 3 \,{\left (c \sqrt{\frac{1}{c^{2} x^{2}} - 1} - \frac{\operatorname{arsech}\left (c x\right )}{x}\right )} a^{2} b + 6 \,{\left (c \sqrt{\frac{1}{c^{2} x^{2}} - 1} \operatorname{arsech}\left (c x\right ) - \frac{1}{x}\right )} a b^{2} + 3 \,{\left (c \sqrt{\frac{1}{c^{2} x^{2}} - 1} \operatorname{arsech}\left (c x\right )^{2} + 2 \, c \sqrt{\frac{1}{c^{2} x^{2}} - 1} - \frac{2 \, \operatorname{arsech}\left (c x\right )}{x}\right )} b^{3} - \frac{3 \, a b^{2} \operatorname{arsech}\left (c x\right )^{2}}{x} - \frac{a^{3}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(c*x))^3/x^2,x, algorithm="maxima")

[Out]

-b^3*arcsech(c*x)^3/x + 3*(c*sqrt(1/(c^2*x^2) - 1) - arcsech(c*x)/x)*a^2*b + 6*(c*sqrt(1/(c^2*x^2) - 1)*arcsec
h(c*x) - 1/x)*a*b^2 + 3*(c*sqrt(1/(c^2*x^2) - 1)*arcsech(c*x)^2 + 2*c*sqrt(1/(c^2*x^2) - 1) - 2*arcsech(c*x)/x
)*b^3 - 3*a*b^2*arcsech(c*x)^2/x - a^3/x

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Fricas [B]  time = 1.68919, size = 485, normalized size = 4.75 \begin{align*} -\frac{b^{3} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} - 3 \,{\left (a^{2} b + 2 \, b^{3}\right )} c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + a^{3} + 6 \, a b^{2} - 3 \,{\left (b^{3} c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - a b^{2}\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 3 \,{\left (2 \, a b^{2} c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - a^{2} b - 2 \, b^{3}\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(c*x))^3/x^2,x, algorithm="fricas")

[Out]

-(b^3*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x))^3 - 3*(a^2*b + 2*b^3)*c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x
^2)) + a^3 + 6*a*b^2 - 3*(b^3*c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - a*b^2)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^
2)) + 1)/(c*x))^2 - 3*(2*a*b^2*c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - a^2*b - 2*b^3)*log((c*x*sqrt(-(c^2*x^2 - 1
)/(c^2*x^2)) + 1)/(c*x)))/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{asech}{\left (c x \right )}\right )^{3}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asech(c*x))**3/x**2,x)

[Out]

Integral((a + b*asech(c*x))**3/x**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}^{3}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(c*x))^3/x^2,x, algorithm="giac")

[Out]

integrate((b*arcsech(c*x) + a)^3/x^2, x)