Integrand size = 12, antiderivative size = 62 \[ \int x^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^2}{6 c}+\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b \text {arctanh}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{6 c^3} \]
1/3*x^3*(a+b*arccsch(c*x))-1/6*b*arctanh((1+1/c^2/x^2)^(1/2))/c^3+1/6*b*x^ 2*(1+1/c^2/x^2)^(1/2)/c
Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.37 \[ \int x^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {a x^3}{3}+\frac {b x^2 \sqrt {\frac {1+c^2 x^2}{c^2 x^2}}}{6 c}+\frac {1}{3} b x^3 \text {csch}^{-1}(c x)-\frac {b \log \left (x \left (1+\sqrt {\frac {1+c^2 x^2}{c^2 x^2}}\right )\right )}{6 c^3} \]
(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)
Time = 0.22 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6838, 798, 52, 73, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx\) |
\(\Big \downarrow \) 6838 |
\(\displaystyle \frac {b \int \frac {x}{\sqrt {1+\frac {1}{c^2 x^2}}}dx}{3 c}+\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b \int \frac {x^4}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x^2}}{6 c}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b \left (x^2 \left (-\sqrt {\frac {1}{c^2 x^2}+1}\right )-\frac {\int \frac {x^2}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x^2}}{2 c^2}\right )}{6 c}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b \left (x^2 \left (-\sqrt {\frac {1}{c^2 x^2}+1}\right )-\int \frac {1}{\frac {c^2}{x^4}-c^2}d\sqrt {1+\frac {1}{c^2 x^2}}\right )}{6 c}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b \left (\frac {\text {arctanh}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right )}{c^2}-x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{6 c}\) |
(x^3*(a + b*ArcCsch[c*x]))/3 - (b*(-(Sqrt[1 + 1/(c^2*x^2)]*x^2) + ArcTanh[ Sqrt[1 + 1/(c^2*x^2)]]/c^2))/(6*c)
3.1.5.3.1 Defintions of rubi rules used
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] - Simp[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] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Si mp[(d*x)^(m + 1)*((a + b*ArcCsch[c*x])/(d*(m + 1))), x] + Simp[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]
Time = 0.33 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.34
method | result | size |
parts | \(\frac {a \,x^{3}}{3}+\frac {b \left (\frac {c^{3} x^{3} \operatorname {arccsch}\left (c x \right )}{3}-\frac {\sqrt {c^{2} x^{2}+1}\, \left (-c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right )}{6 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x c}\right )}{c^{3}}\) | \(83\) |
derivativedivides | \(\frac {\frac {a \,c^{3} x^{3}}{3}+b \left (\frac {c^{3} x^{3} \operatorname {arccsch}\left (c x \right )}{3}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (c x \sqrt {c^{2} x^{2}+1}-\operatorname {arcsinh}\left (c x \right )\right )}{6 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\) | \(88\) |
default | \(\frac {\frac {a \,c^{3} x^{3}}{3}+b \left (\frac {c^{3} x^{3} \operatorname {arccsch}\left (c x \right )}{3}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (c x \sqrt {c^{2} x^{2}+1}-\operatorname {arcsinh}\left (c x \right )\right )}{6 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\) | \(88\) |
1/3*a*x^3+b/c^3*(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)/x/c)
Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (52) = 104\).
Time = 0.26 (sec) , antiderivative size = 186, normalized size of antiderivative = 3.00 \[ \int x^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {2 \, a c^{3} x^{3} + b c^{2} x^{2} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 2 \, b c^{3} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - 2 \, b c^{3} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + b \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) + 2 \, {\left (b c^{3} x^{3} - b c^{3}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{6 \, c^{3}} \]
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
\[ \int x^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int x^{2} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )\, dx \]
Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.55 \[ \int x^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {1}{3} \, a x^{3} + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arcsch}\left (c x\right ) + \frac {\frac {2 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b \]
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
\[ \int x^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int { {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int x^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]