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

Optimal. Leaf size=45 \[ \frac{1}{2} x^2 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b \sqrt{1-c x}}{2 c^2 \sqrt{\frac{1}{c x+1}}} \]

[Out]

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

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Rubi [A]  time = 0.0136379, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6283, 74} \[ \frac{1}{2} x^2 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b \sqrt{1-c x}}{2 c^2 \sqrt{\frac{1}{c x+1}}} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcSech[c*x]),x]

[Out]

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

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 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps

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

Mathematica [A]  time = 0.0515405, size = 57, normalized size = 1.27 \[ \frac{a x^2}{2}+b \left (-\frac{1}{2 c^2}-\frac{x}{2 c}\right ) \sqrt{\frac{1-c x}{c x+1}}+\frac{1}{2} b x^2 \text{sech}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcSech[c*x]),x]

[Out]

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

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Maple [A]  time = 0.178, size = 63, normalized size = 1.4 \begin{align*}{\frac{1}{{c}^{2}} \left ({\frac{{c}^{2}{x}^{2}a}{2}}+b \left ({\frac{{\rm arcsech} \left (cx\right ){c}^{2}{x}^{2}}{2}}-{\frac{cx}{2}\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(x*(a+b*arcsech(c*x)),x)

[Out]

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

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Maxima [A]  time = 0.979394, size = 49, normalized size = 1.09 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{1}{2} \,{\left (x^{2} \operatorname{arsech}\left (c x\right ) - \frac{x \sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsech(c*x)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.90979, size = 157, normalized size = 3.49 \begin{align*} \frac{b c x^{2} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + a c x^{2} - b x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.31947, size = 46, normalized size = 1.02 \begin{align*} \begin{cases} \frac{a x^{2}}{2} + \frac{b x^{2} \operatorname{asech}{\left (c x \right )}}{2} - \frac{b \sqrt{- c^{2} x^{2} + 1}}{2 c^{2}} & \text{for}\: c \neq 0 \\\frac{x^{2} \left (a + \infty b\right )}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*x**2/2 + b*x**2*asech(c*x)/2 - b*sqrt(-c**2*x**2 + 1)/(2*c**2), Ne(c, 0)), (x**2*(a + oo*b)/2, Tr
ue))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsech(c*x)),x, algorithm="giac")

[Out]

integrate((b*arcsech(c*x) + a)*x, x)