3.1.24 \(\int x (a+b \text {sech}^{-1}(c x)) \, dx\) [24]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6418, 75} \begin {gather*} \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}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcSech[c*x]),x]

[Out]

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

Rule 75

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]

Rule 6418

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]/(m + 1))*Sqrt[1/(1 + c*x)], Int[(d*x)^m/(Sqrt[1 - c*x]*Sqrt[1 + c
*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]

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*}

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Mathematica [A]
time = 0.04, size = 57, normalized size = 1.27 \begin {gather*} \frac {a x^2}{2}+b \left (-\frac {1}{2 c^2}-\frac {x}{2 c}\right ) \sqrt {\frac {1-c x}{1+c x}}+\frac {1}{2} b x^2 \text {sech}^{-1}(c x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcSech[c*x]),x]

[Out]

(a*x^2)/2 + b*(-1/2*1/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 = 0.16, size = 63, normalized size = 1.40

method	result	size
derivativedivides	\(\frac {\frac {a \,c^{2} x^{2}}{2}+b \left (\frac {c^{2} x^{2} \mathrm {arcsech}\left (c x \right )}{2}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}}{2}\right )}{c^{2}}\)	\(63\)
default	\(\frac {\frac {a \,c^{2} x^{2}}{2}+b \left (\frac {c^{2} x^{2} \mathrm {arcsech}\left (c x \right )}{2}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}}{2}\right )}{c^{2}}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arcsech(c*x)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 36, normalized size = 0.80 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (35) = 70\).
time = 0.35, size = 73, normalized size = 1.62 \begin {gather*} \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 {gather*}

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 = 0.19, size = 46, normalized size = 1.02 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 1.39, size = 50, normalized size = 1.11 \begin {gather*} \frac {a\,x^2}{2}+\frac {b\,x^2\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{2}-\frac {b\,x\,\sqrt {\frac {1}{c\,x}-1}\,\sqrt {\frac {1}{c\,x}+1}}{2\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*acosh(1/(c*x))),x)

[Out]

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

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