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

Optimal. Leaf size=30 \[ a x+\frac{b \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{c}+b x \text{csch}^{-1}(c x) \]

[Out]

a*x + b*x*ArcCsch[c*x] + (b*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/c

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Rubi [A]  time = 0.0220054, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6278, 266, 63, 208} \[ a x+\frac{b \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{c}+b x \text{csch}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Int[a + b*ArcCsch[c*x],x]

[Out]

a*x + b*x*ArcCsch[c*x] + (b*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/c

Rule 6278

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 0.0481137, size = 44, normalized size = 1.47 \[ a x+\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+b x \text{csch}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*ArcCsch[c*x],x]

[Out]

a*x + b*x*ArcCsch[c*x] + (b*Sqrt[1 + 1/(c^2*x^2)]*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2]

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Maple [A]  time = 0.169, size = 36, normalized size = 1.2 \begin{align*} ax+bx{\rm arccsch} \left (cx\right )+{\frac{b}{c}\ln \left ( cx+cx\sqrt{1+{\frac{1}{{c}^{2}{x}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x+b*x*arccsch(c*x)+b/c*ln(c*x+c*x*(1+1/c^2/x^2)^(1/2))

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Maxima [A]  time = 0.987604, size = 66, normalized size = 2.2 \begin{align*} a x + \frac{{\left (2 \, c x \operatorname{arcsch}\left (c x\right ) + \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )\right )} b}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.38492, size = 320, normalized size = 10.67 \begin{align*} \frac{a c x + b c \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - b c \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) - b \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - 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*arccsch(c*x),x, algorithm="fricas")

[Out]

(a*c*x + b*c*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x + 1) - b*c*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*
x - 1) - b*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - 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{acsch}{\left (c x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(a + b*acsch(c*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(b*arccsch(c*x) + a, x)