3.17 \(\int x (a+b \text{csch}^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=54 \[ \frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )}{c}+\frac{1}{2} x^2 \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{b^2 \log (x)}{c^2} \]

[Out]

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

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Rubi [A]  time = 0.0760874, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6286, 5452, 4184, 3475} \[ \frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )}{c}+\frac{1}{2} x^2 \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{b^2 \log (x)}{c^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcCsch[c*x])^2,x]

[Out]

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

Rule 6286

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b
*x)^n*Csch[x]^(m + 1)*Coth[x], x], x, ArcCsch[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] &
& (GtQ[n, 0] || LtQ[m, -1])

Rule 5452

Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Si
mp[((c + d*x)^m*Csch[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x], x] /
; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.137106, size = 87, normalized size = 1.61 \[ \frac{a c x \left (a c x+2 b \sqrt{\frac{1}{c^2 x^2}+1}\right )+2 b c x \text{csch}^{-1}(c x) \left (a c x+b \sqrt{\frac{1}{c^2 x^2}+1}\right )+b^2 c^2 x^2 \text{csch}^{-1}(c x)^2+2 b^2 \log (c x)}{2 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcCsch[c*x])^2,x]

[Out]

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

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Maple [F]  time = 0.19, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b{\rm arccsch} \left (cx\right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arccsch(c*x))^2,x)

[Out]

int(x*(a+b*arccsch(c*x))^2,x)

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Maxima [A]  time = 1.00935, size = 111, normalized size = 2.06 \begin{align*} \frac{1}{2} \, b^{2} x^{2} \operatorname{arcsch}\left (c x\right )^{2} + \frac{1}{2} \, a^{2} x^{2} +{\left (x^{2} \operatorname{arcsch}\left (c x\right ) + \frac{x \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c}\right )} a b +{\left (\frac{x \sqrt{\frac{1}{c^{2} x^{2}} + 1} \operatorname{arcsch}\left (c x\right )}{c} + \frac{\log \left (x\right )}{c^{2}}\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccsch(c*x))^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.45099, size = 524, normalized size = 9.7 \begin{align*} \frac{b^{2} c^{2} x^{2} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + a^{2} c^{2} x^{2} + 2 \, a b c^{2} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - 2 \, a b c^{2} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 2 \, a b c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 2 \, b^{2} \log \left (x\right ) + 2 \,{\left (a b c^{2} x^{2} + b^{2} c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - a b c^{2}\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{2 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccsch(c*x))^2,x, algorithm="fricas")

[Out]

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

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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 )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*acsch(c*x))**2,x)

[Out]

Integral(x*(a + b*acsch(c*x))**2, 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 )}^{2} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccsch(c*x))^2,x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)^2*x, x)