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

Optimal. Leaf size=62 \[ \frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{6 c}-\frac{b \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{6 c^3} \]

[Out]

(b*Sqrt[1 + 1/(c^2*x^2)]*x^2)/(6*c) + (x^3*(a + b*ArcCsch[c*x]))/3 - (b*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/(6*c^3
)

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Rubi [A]  time = 0.0350997, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {6284, 266, 51, 63, 208} \[ \frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{6 c}-\frac{b \tanh ^{-1}\left (\sqrt{\frac{1}{c^2 x^2}+1}\right )}{6 c^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcCsch[c*x]),x]

[Out]

(b*Sqrt[1 + 1/(c^2*x^2)]*x^2)/(6*c) + (x^3*(a + b*ArcCsch[c*x]))/3 - (b*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/(6*c^3
)

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 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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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 x^2 \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b \int \frac{x}{\sqrt{1+\frac{1}{c^2 x^2}}} \, dx}{3 c}\\ &=\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{b \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{6 c}\\ &=\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^2}{6 c}+\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{c^2}}} \, dx,x,\frac{1}{x^2}\right )}{12 c^3}\\ &=\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^2}{6 c}+\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )+\frac{b \operatorname{Subst}\left (\int \frac{1}{-c^2+c^2 x^2} \, dx,x,\sqrt{1+\frac{1}{c^2 x^2}}\right )}{6 c}\\ &=\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^2}{6 c}+\frac{1}{3} x^3 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{b \tanh ^{-1}\left (\sqrt{1+\frac{1}{c^2 x^2}}\right )}{6 c^3}\\ \end{align*}

Mathematica [A]  time = 0.049858, size = 85, normalized size = 1.37 \[ \frac{a x^3}{3}+\frac{b x^2 \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}}{6 c}-\frac{b \log \left (x \left (\sqrt{\frac{c^2 x^2+1}{c^2 x^2}}+1\right )\right )}{6 c^3}+\frac{1}{3} b x^3 \text{csch}^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*ArcCsch[c*x]),x]

[Out]

(a*x^3)/3 + (b*x^2*Sqrt[(1 + c^2*x^2)/(c^2*x^2)])/(6*c) + (b*x^3*ArcCsch[c*x])/3 - (b*Log[x*(1 + Sqrt[(1 + c^2
*x^2)/(c^2*x^2)])])/(6*c^3)

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Maple [A]  time = 0.171, size = 87, normalized size = 1.4 \begin{align*}{\frac{1}{{c}^{3}} \left ({\frac{a{c}^{3}{x}^{3}}{3}}+b \left ({\frac{{c}^{3}{x}^{3}{\rm arccsch} \left (cx\right )}{3}}-{\frac{1}{6\,cx}\sqrt{{c}^{2}{x}^{2}+1} \left ( -cx\sqrt{{c}^{2}{x}^{2}+1}+{\it Arcsinh} \left ( cx \right ) \right ){\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^2*(a+b*arccsch(c*x)),x)

[Out]

1/c^3*(1/3*a*c^3*x^3+b*(1/3*c^3*x^3*arccsch(c*x)-1/6*(c^2*x^2+1)^(1/2)*(-c*x*(c^2*x^2+1)^(1/2)+arcsinh(c*x))/(
(c^2*x^2+1)/c^2/x^2)^(1/2)/c/x))

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Maxima [A]  time = 0.985773, size = 130, normalized size = 2.1 \begin{align*} \frac{1}{3} \, a x^{3} + \frac{1}{12} \,{\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 )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3 + 1/12*(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)*b

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Fricas [B]  time = 2.32475, size = 414, normalized size = 6.68 \begin{align*} \frac{2 \, a c^{3} x^{3} + b c^{2} x^{2} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 2 \, b c^{3} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - 2 \, b c^{3} \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 ) + 2 \,{\left (b c^{3} x^{3} - b c^{3}\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{6 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*a*c^3*x^3 + b*c^2*x^2*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 2*b*c^3*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c
*x + 1) - 2*b*c^3*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) + 2*(b*c^3*x^3 - b*c^3)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)))/c^3

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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 )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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