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

Optimal. Leaf size=122 \[ -\frac{b^2 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c x)}\right )}{3 c^3}+\frac{b^2 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c x)}\right )}{3 c^3}+\frac{b x^2 \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )}{3 c}-\frac{2 b \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{3 c^3}+\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{b^2 x}{3 c^2} \]

[Out]

(b^2*x)/(3*c^2) + (b*Sqrt[1 + 1/(c^2*x^2)]*x^2*(a + b*ArcCsch[c*x]))/(3*c) + (x^3*(a + b*ArcCsch[c*x])^2)/3 -
(2*b*(a + b*ArcCsch[c*x])*ArcTanh[E^ArcCsch[c*x]])/(3*c^3) - (b^2*PolyLog[2, -E^ArcCsch[c*x]])/(3*c^3) + (b^2*
PolyLog[2, E^ArcCsch[c*x]])/(3*c^3)

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Rubi [A]  time = 0.132321, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6286, 5452, 4185, 4182, 2279, 2391} \[ -\frac{b^2 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c x)}\right )}{3 c^3}+\frac{b^2 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c x)}\right )}{3 c^3}+\frac{b x^2 \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )}{3 c}-\frac{2 b \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right )}{3 c^3}+\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{b^2 x}{3 c^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcCsch[c*x])^2,x]

[Out]

(b^2*x)/(3*c^2) + (b*Sqrt[1 + 1/(c^2*x^2)]*x^2*(a + b*ArcCsch[c*x]))/(3*c) + (x^3*(a + b*ArcCsch[c*x])^2)/3 -
(2*b*(a + b*ArcCsch[c*x])*ArcTanh[E^ArcCsch[c*x]])/(3*c^3) - (b^2*PolyLog[2, -E^ArcCsch[c*x]])/(3*c^3) + (b^2*
PolyLog[2, E^ArcCsch[c*x]])/(3*c^3)

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 4185

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> -Simp[(b^2*(c + d*x)*Cot[e + f*x]*
(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2
), x], x] - Simp[(b^2*d*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && G
tQ[n, 1] && NeQ[n, 2]

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int x^2 \left (a+b \text{csch}^{-1}(c x)\right )^2 \, dx &=-\frac{\operatorname{Subst}\left (\int (a+b x)^2 \coth (x) \text{csch}^3(x) \, dx,x,\text{csch}^{-1}(c x)\right )}{c^3}\\ &=\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )^2-\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \text{csch}^3(x) \, dx,x,\text{csch}^{-1}(c x)\right )}{3 c^3}\\ &=\frac{b^2 x}{3 c^2}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 c}+\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{b \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\text{csch}^{-1}(c x)\right )}{3 c^3}\\ &=\frac{b^2 x}{3 c^2}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 c}+\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )^2-\frac{2 b \left (a+b \text{csch}^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right )}{3 c^3}-\frac{b^2 \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{3 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text{csch}^{-1}(c x)\right )}{3 c^3}\\ &=\frac{b^2 x}{3 c^2}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 c}+\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )^2-\frac{2 b \left (a+b \text{csch}^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right )}{3 c^3}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{3 c^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\text{csch}^{-1}(c x)}\right )}{3 c^3}\\ &=\frac{b^2 x}{3 c^2}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^2 \left (a+b \text{csch}^{-1}(c x)\right )}{3 c}+\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )^2-\frac{2 b \left (a+b \text{csch}^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c x)}\right )}{3 c^3}-\frac{b^2 \text{Li}_2\left (-e^{\text{csch}^{-1}(c x)}\right )}{3 c^3}+\frac{b^2 \text{Li}_2\left (e^{\text{csch}^{-1}(c x)}\right )}{3 c^3}\\ \end{align*}

Mathematica [A]  time = 1.36635, size = 211, normalized size = 1.73 \[ \frac{b^2 \text{PolyLog}\left (2,-e^{-\text{csch}^{-1}(c x)}\right )-b^2 \text{PolyLog}\left (2,e^{-\text{csch}^{-1}(c x)}\right )+a^2 c^3 x^3+a b c^2 x^2 \sqrt{\frac{1}{c^2 x^2}+1}-\frac{a b c x \sqrt{\frac{1}{c^2 x^2}+1} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+2 a b c^3 x^3 \text{csch}^{-1}(c x)+b^2 c^3 x^3 \text{csch}^{-1}(c x)^2+b^2 c^2 x^2 \sqrt{\frac{1}{c^2 x^2}+1} \text{csch}^{-1}(c x)+b^2 c x+b^2 \text{csch}^{-1}(c x) \log \left (1-e^{-\text{csch}^{-1}(c x)}\right )-b^2 \text{csch}^{-1}(c x) \log \left (e^{-\text{csch}^{-1}(c x)}+1\right )}{3 c^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*(a + b*ArcCsch[c*x])^2,x]

[Out]

(b^2*c*x + a*b*c^2*Sqrt[1 + 1/(c^2*x^2)]*x^2 + a^2*c^3*x^3 + b^2*c^2*Sqrt[1 + 1/(c^2*x^2)]*x^2*ArcCsch[c*x] +
2*a*b*c^3*x^3*ArcCsch[c*x] + b^2*c^3*x^3*ArcCsch[c*x]^2 - (a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*ArcSinh[c*x])/Sqrt[1
+ c^2*x^2] + b^2*ArcCsch[c*x]*Log[1 - E^(-ArcCsch[c*x])] - b^2*ArcCsch[c*x]*Log[1 + E^(-ArcCsch[c*x])] + b^2*P
olyLog[2, -E^(-ArcCsch[c*x])] - b^2*PolyLog[2, E^(-ArcCsch[c*x])])/(3*c^3)

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Maple [F]  time = 0.184, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \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^2*(a+b*arccsch(c*x))^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a^{2} x^{3} + \frac{1}{6} \,{\left (4 \, x^{3} \operatorname{arcsch}\left (c x\right ) + \frac{\frac{2 \, \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c^{2}{\left (\frac{1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac{\log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac{\log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} a b + \frac{1}{3} \,{\left (x^{3} \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right )^{2} - 3 \, \int -\frac{3 \, c^{2} x^{4} \log \left (c\right )^{2} + 3 \, x^{2} \log \left (c\right )^{2} + 3 \,{\left (c^{2} x^{4} + x^{2}\right )} \log \left (x\right )^{2} + 6 \,{\left (c^{2} x^{4} \log \left (c\right ) + x^{2} \log \left (c\right )\right )} \log \left (x\right ) - 2 \,{\left (3 \, c^{2} x^{4} \log \left (c\right ) + 3 \, x^{2} \log \left (c\right ) + 3 \,{\left (c^{2} x^{4} + x^{2}\right )} \log \left (x\right ) +{\left (c^{2} x^{4}{\left (3 \, \log \left (c\right ) + 1\right )} + 3 \, x^{2} \log \left (c\right ) + 3 \,{\left (c^{2} x^{4} + x^{2}\right )} \log \left (x\right )\right )} \sqrt{c^{2} x^{2} + 1}\right )} \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right ) + 3 \,{\left (c^{2} x^{4} \log \left (c\right )^{2} + x^{2} \log \left (c\right )^{2} +{\left (c^{2} x^{4} + x^{2}\right )} \log \left (x\right )^{2} + 2 \,{\left (c^{2} x^{4} \log \left (c\right ) + x^{2} \log \left (c\right )\right )} \log \left (x\right )\right )} \sqrt{c^{2} x^{2} + 1}}{3 \,{\left (c^{2} x^{2} +{\left (c^{2} x^{2} + 1\right )}^{\frac{3}{2}} + 1\right )}}\,{d x}\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*x^3 + 1/6*(4*x^3*arccsch(c*x) + (2*sqrt(1/(c^2*x^2) + 1)/(c^2*(1/(c^2*x^2) + 1) - c^2) - log(sqrt(1/(c
^2*x^2) + 1) + 1)/c^2 + log(sqrt(1/(c^2*x^2) + 1) - 1)/c^2)/c)*a*b + 1/3*(x^3*log(sqrt(c^2*x^2 + 1) + 1)^2 - 3
*integrate(-1/3*(3*c^2*x^4*log(c)^2 + 3*x^2*log(c)^2 + 3*(c^2*x^4 + x^2)*log(x)^2 + 6*(c^2*x^4*log(c) + x^2*lo
g(c))*log(x) - 2*(3*c^2*x^4*log(c) + 3*x^2*log(c) + 3*(c^2*x^4 + x^2)*log(x) + (c^2*x^4*(3*log(c) + 1) + 3*x^2
*log(c) + 3*(c^2*x^4 + x^2)*log(x))*sqrt(c^2*x^2 + 1))*log(sqrt(c^2*x^2 + 1) + 1) + 3*(c^2*x^4*log(c)^2 + x^2*
log(c)^2 + (c^2*x^4 + x^2)*log(x)^2 + 2*(c^2*x^4*log(c) + x^2*log(c))*log(x))*sqrt(c^2*x^2 + 1))/(c^2*x^2 + (c
^2*x^2 + 1)^(3/2) + 1), x))*b^2

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} x^{2} \operatorname{arcsch}\left (c x\right )^{2} + 2 \, a b x^{2} \operatorname{arcsch}\left (c x\right ) + a^{2} x^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \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**2*(a+b*acsch(c*x))**2,x)

[Out]

Integral(x**2*(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^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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