Integrand size = 10, antiderivative size = 80 \[ \int x^2 \text {arcsinh}(a x)^2 \, dx=-\frac {4 x}{9 a^2}+\frac {2 x^3}{27}+\frac {4 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{9 a^3}-\frac {2 x^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{9 a}+\frac {1}{3} x^3 \text {arcsinh}(a x)^2 \] Output:
-4/9*x/a^2+2/27*x^3+4/9*(a^2*x^2+1)^(1/2)*arcsinh(a*x)/a^3-2/9*x^2*(a^2*x^ 2+1)^(1/2)*arcsinh(a*x)/a+1/3*x^3*arcsinh(a*x)^2
Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.74 \[ \int x^2 \text {arcsinh}(a x)^2 \, dx=\frac {1}{27} \left (2 x \left (-\frac {6}{a^2}+x^2\right )-\frac {6 \left (-2+a^2 x^2\right ) \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{a^3}+9 x^3 \text {arcsinh}(a x)^2\right ) \] Input:
Integrate[x^2*ArcSinh[a*x]^2,x]
Output:
(2*x*(-6/a^2 + x^2) - (6*(-2 + a^2*x^2)*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/a^ 3 + 9*x^3*ArcSinh[a*x]^2)/27
Time = 0.41 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6191, 6227, 15, 6213, 24}
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.
\(\displaystyle \int x^2 \text {arcsinh}(a x)^2 \, dx\) |
\(\Big \downarrow \) 6191 |
\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^2-\frac {2}{3} a \int \frac {x^3 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx\) |
\(\Big \downarrow \) 6227 |
\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^2-\frac {2}{3} a \left (-\frac {2 \int \frac {x \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{3 a^2}-\frac {\int x^2dx}{3 a}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{3 a^2}\right )\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^2-\frac {2}{3} a \left (-\frac {2 \int \frac {x \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{3 a^2}-\frac {x^3}{9 a}\right )\) |
\(\Big \downarrow \) 6213 |
\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^2-\frac {2}{3} a \left (-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{a^2}-\frac {\int 1dx}{a}\right )}{3 a^2}+\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{3 a^2}-\frac {x^3}{9 a}\right )\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {1}{3} x^3 \text {arcsinh}(a x)^2-\frac {2}{3} a \left (\frac {x^2 \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{3 a^2}-\frac {2 \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{a^2}-\frac {x}{a}\right )}{3 a^2}-\frac {x^3}{9 a}\right )\) |
Input:
Int[x^2*ArcSinh[a*x]^2,x]
Output:
(x^3*ArcSinh[a*x]^2)/3 - (2*a*(-1/9*x^3/a + (x^2*Sqrt[1 + a^2*x^2]*ArcSinh [a*x])/(3*a^2) - (2*(-(x/a) + (Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/a^2))/(3*a^ 2)))/3
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int [(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] ) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ m, 1] && NeQ[m + 2*p + 1, 0]
Time = 0.84 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {\frac {x^{3} a^{3} \operatorname {arcsinh}\left (x a \right )^{2}}{3}+\frac {4 \,\operatorname {arcsinh}\left (x a \right ) \sqrt {a^{2} x^{2}+1}}{9}-\frac {2 x^{2} a^{2} \operatorname {arcsinh}\left (x a \right ) \sqrt {a^{2} x^{2}+1}}{9}-\frac {4 x a}{9}+\frac {2 x^{3} a^{3}}{27}}{a^{3}}\) | \(72\) |
default | \(\frac {\frac {x^{3} a^{3} \operatorname {arcsinh}\left (x a \right )^{2}}{3}+\frac {4 \,\operatorname {arcsinh}\left (x a \right ) \sqrt {a^{2} x^{2}+1}}{9}-\frac {2 x^{2} a^{2} \operatorname {arcsinh}\left (x a \right ) \sqrt {a^{2} x^{2}+1}}{9}-\frac {4 x a}{9}+\frac {2 x^{3} a^{3}}{27}}{a^{3}}\) | \(72\) |
orering | \(\frac {\left (19 a^{4} x^{4}-24 a^{2} x^{2}-48\right ) \operatorname {arcsinh}\left (x a \right )^{2}}{27 a^{4} x}-\frac {\left (6 a^{4} x^{4}-17 a^{2} x^{2}-30\right ) \left (2 x \operatorname {arcsinh}\left (x a \right )^{2}+\frac {2 x^{2} \operatorname {arcsinh}\left (x a \right ) a}{\sqrt {a^{2} x^{2}+1}}\right )}{27 x^{2} a^{4}}+\frac {\left (a^{2} x^{2}-6\right ) \left (a^{2} x^{2}+1\right ) \left (2 \operatorname {arcsinh}\left (x a \right )^{2}+\frac {8 x \,\operatorname {arcsinh}\left (x a \right ) a}{\sqrt {a^{2} x^{2}+1}}+\frac {2 a^{2} x^{2}}{a^{2} x^{2}+1}-\frac {2 x^{3} \operatorname {arcsinh}\left (x a \right ) a^{3}}{\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{27 x \,a^{4}}\) | \(187\) |
Input:
int(x^2*arcsinh(x*a)^2,x,method=_RETURNVERBOSE)
Output:
1/a^3*(1/3*x^3*a^3*arcsinh(x*a)^2+4/9*arcsinh(x*a)*(a^2*x^2+1)^(1/2)-2/9*x ^2*a^2*arcsinh(x*a)*(a^2*x^2+1)^(1/2)-4/9*x*a+2/27*x^3*a^3)
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.02 \[ \int x^2 \text {arcsinh}(a x)^2 \, dx=\frac {9 \, a^{3} x^{3} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 2 \, a^{3} x^{3} - 6 \, \sqrt {a^{2} x^{2} + 1} {\left (a^{2} x^{2} - 2\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 12 \, a x}{27 \, a^{3}} \] Input:
integrate(x^2*arcsinh(a*x)^2,x, algorithm="fricas")
Output:
1/27*(9*a^3*x^3*log(a*x + sqrt(a^2*x^2 + 1))^2 + 2*a^3*x^3 - 6*sqrt(a^2*x^ 2 + 1)*(a^2*x^2 - 2)*log(a*x + sqrt(a^2*x^2 + 1)) - 12*a*x)/a^3
Time = 0.24 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.95 \[ \int x^2 \text {arcsinh}(a x)^2 \, dx=\begin {cases} \frac {x^{3} \operatorname {asinh}^{2}{\left (a x \right )}}{3} + \frac {2 x^{3}}{27} - \frac {2 x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{9 a} - \frac {4 x}{9 a^{2}} + \frac {4 \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{9 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x**2*asinh(a*x)**2,x)
Output:
Piecewise((x**3*asinh(a*x)**2/3 + 2*x**3/27 - 2*x**2*sqrt(a**2*x**2 + 1)*a sinh(a*x)/(9*a) - 4*x/(9*a**2) + 4*sqrt(a**2*x**2 + 1)*asinh(a*x)/(9*a**3) , Ne(a, 0)), (0, True))
Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.88 \[ \int x^2 \text {arcsinh}(a x)^2 \, dx=\frac {1}{3} \, x^{3} \operatorname {arsinh}\left (a x\right )^{2} - \frac {2}{9} \, a {\left (\frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{a^{2}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arsinh}\left (a x\right ) + \frac {2 \, {\left (a^{2} x^{3} - 6 \, x\right )}}{27 \, a^{2}} \] Input:
integrate(x^2*arcsinh(a*x)^2,x, algorithm="maxima")
Output:
1/3*x^3*arcsinh(a*x)^2 - 2/9*a*(sqrt(a^2*x^2 + 1)*x^2/a^2 - 2*sqrt(a^2*x^2 + 1)/a^4)*arcsinh(a*x) + 2/27*(a^2*x^3 - 6*x)/a^2
Exception generated. \[ \int x^2 \text {arcsinh}(a x)^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^2*arcsinh(a*x)^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int x^2 \text {arcsinh}(a x)^2 \, dx=\int x^2\,{\mathrm {asinh}\left (a\,x\right )}^2 \,d x \] Input:
int(x^2*asinh(a*x)^2,x)
Output:
int(x^2*asinh(a*x)^2, x)
\[ \int x^2 \text {arcsinh}(a x)^2 \, dx=\int \mathit {asinh} \left (a x \right )^{2} x^{2}d x \] Input:
int(x^2*asinh(a*x)^2,x)
Output:
int(asinh(a*x)**2*x**2,x)