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

Optimal. Leaf size=40 \[ a x+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{c}+b x \text{sech}^{-1}(c x) \]

[Out]

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

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Rubi [A]  time = 0.0151617, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6277, 216} \[ a x+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{c}+b x \text{sech}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Int[a + b*ArcSech[c*x],x]

[Out]

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

Rule 6277

Int[ArcSech[(c_.)*(x_)], x_Symbol] :> Simp[x*ArcSech[c*x], x] + Dist[Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)], Int[1/Sq
rt[1 - c^2*x^2], x], x] /; FreeQ[c, x]

Rule 216

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

Rubi steps

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

Mathematica [A]  time = 0.0874106, size = 60, normalized size = 1.5 \[ a x-\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c (c x-1)}+b x \text{sech}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcSech[c*x],x]

[Out]

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

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Maple [A]  time = 0.165, size = 42, normalized size = 1.1 \begin{align*} ax+bx{\rm arcsech} \left (cx\right )-{\frac{b}{c}\arctan \left ( \sqrt{-1+{\frac{1}{cx}}}\sqrt{1+{\frac{1}{cx}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*arcsech(c*x),x)

[Out]

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

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Maxima [A]  time = 0.970528, size = 42, normalized size = 1.05 \begin{align*} a x + \frac{{\left (c x \operatorname{arsech}\left (c x\right ) - \arctan \left (\sqrt{\frac{1}{c^{2} x^{2}} - 1}\right )\right )} b}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.69404, size = 262, normalized size = 6.55 \begin{align*} \frac{a c x - b c \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) - 2 \, b \arctan \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) +{\left (b c x - b c\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(a + b*asech(c*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(b*arcsech(c*x) + a, x)