Integrand size = 8, antiderivative size = 110 \[ \int x \text {arcsinh}(a x)^4 \, dx=\frac {3 x^2}{4}-\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)}{2 a}+\frac {3 \text {arcsinh}(a x)^2}{4 a^2}+\frac {3}{2} x^2 \text {arcsinh}(a x)^2-\frac {x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3}{a}+\frac {\text {arcsinh}(a x)^4}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^4 \]
3/4*x^2+3/4*arcsinh(a*x)^2/a^2+3/2*x^2*arcsinh(a*x)^2+1/4*arcsinh(a*x)^4/a ^2+1/2*x^2*arcsinh(a*x)^4-3/2*x*arcsinh(a*x)*(a^2*x^2+1)^(1/2)/a-x*arcsinh (a*x)^3*(a^2*x^2+1)^(1/2)/a
Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.85 \[ \int x \text {arcsinh}(a x)^4 \, dx=\frac {3 a^2 x^2-6 a x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)+\left (3+6 a^2 x^2\right ) \text {arcsinh}(a x)^2-4 a x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^3+\left (1+2 a^2 x^2\right ) \text {arcsinh}(a x)^4}{4 a^2} \]
(3*a^2*x^2 - 6*a*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + (3 + 6*a^2*x^2)*ArcSin h[a*x]^2 - 4*a*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3 + (1 + 2*a^2*x^2)*ArcSin h[a*x]^4)/(4*a^2)
Time = 0.82 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6191, 6227, 6191, 6198, 6227, 15, 6198}
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.
\(\displaystyle \int x \text {arcsinh}(a x)^4 \, dx\) |
\(\Big \downarrow \) 6191 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^4-2 a \int \frac {x^2 \text {arcsinh}(a x)^3}{\sqrt {a^2 x^2+1}}dx\) |
\(\Big \downarrow \) 6227 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^4-2 a \left (-\frac {\int \frac {\text {arcsinh}(a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2}-\frac {3 \int x \text {arcsinh}(a x)^2dx}{2 a}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 a^2}\right )\) |
\(\Big \downarrow \) 6191 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^4-2 a \left (-\frac {3 \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx\right )}{2 a}-\frac {\int \frac {\text {arcsinh}(a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 a^2}\right )\) |
\(\Big \downarrow \) 6198 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^4-2 a \left (-\frac {3 \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx\right )}{2 a}-\frac {\text {arcsinh}(a x)^4}{8 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 a^2}\right )\) |
\(\Big \downarrow \) 6227 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^4-2 a \left (-\frac {3 \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \left (-\frac {\int \frac {\text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2}-\frac {\int xdx}{2 a}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}\right )\right )}{2 a}-\frac {\text {arcsinh}(a x)^4}{8 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 a^2}\right )\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^4-2 a \left (-\frac {3 \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \left (-\frac {\int \frac {\text {arcsinh}(a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right )}{2 a}-\frac {\text {arcsinh}(a x)^4}{8 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 a^2}\right )\) |
\(\Big \downarrow \) 6198 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^4-2 a \left (-\frac {\text {arcsinh}(a x)^4}{8 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^3}{2 a^2}-\frac {3 \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^2-a \left (-\frac {\text {arcsinh}(a x)^2}{4 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)}{2 a^2}-\frac {x^2}{4 a}\right )\right )}{2 a}\right )\) |
(x^2*ArcSinh[a*x]^4)/2 - 2*a*((x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(2*a^2) - ArcSinh[a*x]^4/(8*a^3) - (3*((x^2*ArcSinh[a*x]^2)/2 - a*(-1/4*x^2/a + ( x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(2*a^2) - ArcSinh[a*x]^2/(4*a^3))))/(2*a ))
3.1.36.3.1 Defintions of rubi rules used
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_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c ^2*d] && NeQ[n, -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 = 0.03 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {arcsinh}\left (a x \right )^{4} \left (a^{2} x^{2}+1\right )}{2}-\operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}\, a x -\frac {\operatorname {arcsinh}\left (a x \right )^{4}}{4}+\frac {3 \operatorname {arcsinh}\left (a x \right )^{2} \left (a^{2} x^{2}+1\right )}{2}-\frac {3 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x}{2}-\frac {3 \operatorname {arcsinh}\left (a x \right )^{2}}{4}+\frac {3 a^{2} x^{2}}{4}+\frac {3}{4}}{a^{2}}\) | \(105\) |
default | \(\frac {\frac {\operatorname {arcsinh}\left (a x \right )^{4} \left (a^{2} x^{2}+1\right )}{2}-\operatorname {arcsinh}\left (a x \right )^{3} \sqrt {a^{2} x^{2}+1}\, a x -\frac {\operatorname {arcsinh}\left (a x \right )^{4}}{4}+\frac {3 \operatorname {arcsinh}\left (a x \right )^{2} \left (a^{2} x^{2}+1\right )}{2}-\frac {3 \,\operatorname {arcsinh}\left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x}{2}-\frac {3 \operatorname {arcsinh}\left (a x \right )^{2}}{4}+\frac {3 a^{2} x^{2}}{4}+\frac {3}{4}}{a^{2}}\) | \(105\) |
1/a^2*(1/2*arcsinh(a*x)^4*(a^2*x^2+1)-arcsinh(a*x)^3*(a^2*x^2+1)^(1/2)*a*x -1/4*arcsinh(a*x)^4+3/2*arcsinh(a*x)^2*(a^2*x^2+1)-3/2*arcsinh(a*x)*(a^2*x ^2+1)^(1/2)*a*x-3/4*arcsinh(a*x)^2+3/4*a^2*x^2+3/4)
Time = 0.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.25 \[ \int x \text {arcsinh}(a x)^4 \, dx=-\frac {4 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{4} - 3 \, a^{2} x^{2} + 6 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 3 \, {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{4 \, a^{2}} \]
-1/4*(4*sqrt(a^2*x^2 + 1)*a*x*log(a*x + sqrt(a^2*x^2 + 1))^3 - (2*a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1))^4 - 3*a^2*x^2 + 6*sqrt(a^2*x^2 + 1)*a*x* log(a*x + sqrt(a^2*x^2 + 1)) - 3*(2*a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1))^2)/a^2
Time = 0.35 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95 \[ \int x \text {arcsinh}(a x)^4 \, dx=\begin {cases} \frac {x^{2} \operatorname {asinh}^{4}{\left (a x \right )}}{2} + \frac {3 x^{2} \operatorname {asinh}^{2}{\left (a x \right )}}{2} + \frac {3 x^{2}}{4} - \frac {x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a x \right )}}{a} - \frac {3 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}{\left (a x \right )}}{2 a} + \frac {\operatorname {asinh}^{4}{\left (a x \right )}}{4 a^{2}} + \frac {3 \operatorname {asinh}^{2}{\left (a x \right )}}{4 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((x**2*asinh(a*x)**4/2 + 3*x**2*asinh(a*x)**2/2 + 3*x**2/4 - x*sq rt(a**2*x**2 + 1)*asinh(a*x)**3/a - 3*x*sqrt(a**2*x**2 + 1)*asinh(a*x)/(2* a) + asinh(a*x)**4/(4*a**2) + 3*asinh(a*x)**2/(4*a**2), Ne(a, 0)), (0, Tru e))
\[ \int x \text {arcsinh}(a x)^4 \, dx=\int { x \operatorname {arsinh}\left (a x\right )^{4} \,d x } \]
1/2*x^2*log(a*x + sqrt(a^2*x^2 + 1))^4 - integrate(2*(a^3*x^4 + sqrt(a^2*x ^2 + 1)*a^2*x^3 + a*x^2)*log(a*x + sqrt(a^2*x^2 + 1))^3/(a^3*x^3 + a*x + ( a^2*x^2 + 1)^(3/2)), x)
Exception generated. \[ \int x \text {arcsinh}(a x)^4 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int x \text {arcsinh}(a x)^4 \, dx=\int x\,{\mathrm {asinh}\left (a\,x\right )}^4 \,d x \]