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

Optimal. Leaf size=38 \[ \frac{1}{2} x^2 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b x \sqrt{\frac{1}{c^2 x^2}+1}}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.0128323, antiderivative size = 38, 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 = {6284, 191} \[ \frac{1}{2} x^2 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b x \sqrt{\frac{1}{c^2 x^2}+1}}{2 c} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcCsch[c*x]),x]

[Out]

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

Rule 6284

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCsch[c*
x]))/(d*(m + 1)), x] + Dist[(b*d)/(c*(m + 1)), Int[(d*x)^(m - 1)/Sqrt[1 + 1/(c^2*x^2)], x], x] /; FreeQ[{a, b,
c, d, m}, x] && NeQ[m, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.0229745, size = 50, normalized size = 1.32 \[ \frac{a x^2}{2}+\frac{b x \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}}{2 c}+\frac{1}{2} b x^2 \text{csch}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcCsch[c*x]),x]

[Out]

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

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Maple [A]  time = 0.197, size = 65, normalized size = 1.7 \begin{align*}{\frac{1}{{c}^{2}} \left ({\frac{{c}^{2}{x}^{2}a}{2}}+b \left ({\frac{{c}^{2}{x}^{2}{\rm arccsch} \left (cx\right )}{2}}+{\frac{{c}^{2}{x}^{2}+1}{2\,cx}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00651, size = 47, normalized size = 1.24 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{1}{2} \,{\left (x^{2} \operatorname{arcsch}\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*arccsch(c*x)),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.20757, size = 154, normalized size = 4.05 \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*arccsch(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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \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(x*(a+b*acsch(c*x)),x)

[Out]

Integral(x*(a + b*acsch(c*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arcsch}\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*arccsch(c*x)),x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)*x, x)