Integrand size = 8, antiderivative size = 97 \[ \int x \text {arcsinh}(a x)^3 \, dx=-\frac {3 x \sqrt {1+a^2 x^2}}{8 a}+\frac {3 \text {arcsinh}(a x)}{8 a^2}+\frac {3}{4} x^2 \text {arcsinh}(a x)-\frac {3 x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{4 a}+\frac {\text {arcsinh}(a x)^3}{4 a^2}+\frac {1}{2} x^2 \text {arcsinh}(a x)^3 \]
3/8*arcsinh(a*x)/a^2+3/4*x^2*arcsinh(a*x)+1/4*arcsinh(a*x)^3/a^2+1/2*x^2*a rcsinh(a*x)^3-3/8*x*(a^2*x^2+1)^(1/2)/a-3/4*x*arcsinh(a*x)^2*(a^2*x^2+1)^( 1/2)/a
Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int x \text {arcsinh}(a x)^3 \, dx=\frac {-3 a x \sqrt {1+a^2 x^2}+\left (3+6 a^2 x^2\right ) \text {arcsinh}(a x)-6 a x \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2+\left (2+4 a^2 x^2\right ) \text {arcsinh}(a x)^3}{8 a^2} \]
(-3*a*x*Sqrt[1 + a^2*x^2] + (3 + 6*a^2*x^2)*ArcSinh[a*x] - 6*a*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2 + (2 + 4*a^2*x^2)*ArcSinh[a*x]^3)/(8*a^2)
Time = 0.54 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.19, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6191, 6227, 6191, 262, 222, 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.
\(\displaystyle \int x \text {arcsinh}(a x)^3 \, dx\) |
\(\Big \downarrow \) 6191 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^3-\frac {3}{2} a \int \frac {x^2 \text {arcsinh}(a x)^2}{\sqrt {a^2 x^2+1}}dx\) |
\(\Big \downarrow \) 6227 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^3-\frac {3}{2} a \left (-\frac {\int \frac {\text {arcsinh}(a x)^2}{\sqrt {a^2 x^2+1}}dx}{2 a^2}-\frac {\int x \text {arcsinh}(a x)dx}{a}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{2 a^2}\right )\) |
\(\Big \downarrow \) 6191 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^3-\frac {3}{2} a \left (-\frac {\frac {1}{2} x^2 \text {arcsinh}(a x)-\frac {1}{2} a \int \frac {x^2}{\sqrt {a^2 x^2+1}}dx}{a}-\frac {\int \frac {\text {arcsinh}(a x)^2}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{2 a^2}\right )\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^3-\frac {3}{2} a \left (-\frac {\frac {1}{2} x^2 \text {arcsinh}(a x)-\frac {1}{2} a \left (\frac {x \sqrt {a^2 x^2+1}}{2 a^2}-\frac {\int \frac {1}{\sqrt {a^2 x^2+1}}dx}{2 a^2}\right )}{a}-\frac {\int \frac {\text {arcsinh}(a x)^2}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{2 a^2}\right )\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^3-\frac {3}{2} a \left (-\frac {\int \frac {\text {arcsinh}(a x)^2}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{2 a^2}-\frac {\frac {1}{2} x^2 \text {arcsinh}(a x)-\frac {1}{2} a \left (\frac {x \sqrt {a^2 x^2+1}}{2 a^2}-\frac {\text {arcsinh}(a x)}{2 a^3}\right )}{a}\right )\) |
\(\Big \downarrow \) 6198 |
\(\displaystyle \frac {1}{2} x^2 \text {arcsinh}(a x)^3-\frac {3}{2} a \left (-\frac {\text {arcsinh}(a x)^3}{6 a^3}+\frac {x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{2 a^2}-\frac {\frac {1}{2} x^2 \text {arcsinh}(a x)-\frac {1}{2} a \left (\frac {x \sqrt {a^2 x^2+1}}{2 a^2}-\frac {\text {arcsinh}(a x)}{2 a^3}\right )}{a}\right )\) |
(x^2*ArcSinh[a*x]^3)/2 - (3*a*((x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(2*a^2 ) - ArcSinh[a*x]^3/(6*a^3) - ((x^2*ArcSinh[a*x])/2 - (a*((x*Sqrt[1 + a^2*x ^2])/(2*a^2) - ArcSinh[a*x]/(2*a^3)))/2)/a))/2
3.1.25.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
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 = 88, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {\frac {\operatorname {arcsinh}\left (a x \right )^{3} \left (a^{2} x^{2}+1\right )}{2}-\frac {3 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a x}{4}-\frac {\operatorname {arcsinh}\left (a x \right )^{3}}{4}+\frac {3 \left (a^{2} x^{2}+1\right ) \operatorname {arcsinh}\left (a x \right )}{4}-\frac {3 a x \sqrt {a^{2} x^{2}+1}}{8}-\frac {3 \,\operatorname {arcsinh}\left (a x \right )}{8}}{a^{2}}\) | \(88\) |
default | \(\frac {\frac {\operatorname {arcsinh}\left (a x \right )^{3} \left (a^{2} x^{2}+1\right )}{2}-\frac {3 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}\, a x}{4}-\frac {\operatorname {arcsinh}\left (a x \right )^{3}}{4}+\frac {3 \left (a^{2} x^{2}+1\right ) \operatorname {arcsinh}\left (a x \right )}{4}-\frac {3 a x \sqrt {a^{2} x^{2}+1}}{8}-\frac {3 \,\operatorname {arcsinh}\left (a x \right )}{8}}{a^{2}}\) | \(88\) |
1/a^2*(1/2*arcsinh(a*x)^3*(a^2*x^2+1)-3/4*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2) *a*x-1/4*arcsinh(a*x)^3+3/4*(a^2*x^2+1)*arcsinh(a*x)-3/8*a*x*(a^2*x^2+1)^( 1/2)-3/8*arcsinh(a*x))
Time = 0.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.15 \[ \int x \text {arcsinh}(a x)^3 \, dx=-\frac {6 \, \sqrt {a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} - 2 \, {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} + 3 \, \sqrt {a^{2} x^{2} + 1} a x - 3 \, {\left (2 \, a^{2} x^{2} + 1\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{8 \, a^{2}} \]
-1/8*(6*sqrt(a^2*x^2 + 1)*a*x*log(a*x + sqrt(a^2*x^2 + 1))^2 - 2*(2*a^2*x^ 2 + 1)*log(a*x + sqrt(a^2*x^2 + 1))^3 + 3*sqrt(a^2*x^2 + 1)*a*x - 3*(2*a^2 *x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1)))/a^2
Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95 \[ \int x \text {arcsinh}(a x)^3 \, dx=\begin {cases} \frac {x^{2} \operatorname {asinh}^{3}{\left (a x \right )}}{2} + \frac {3 x^{2} \operatorname {asinh}{\left (a x \right )}}{4} - \frac {3 x \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{4 a} - \frac {3 x \sqrt {a^{2} x^{2} + 1}}{8 a} + \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{4 a^{2}} + \frac {3 \operatorname {asinh}{\left (a x \right )}}{8 a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((x**2*asinh(a*x)**3/2 + 3*x**2*asinh(a*x)/4 - 3*x*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/(4*a) - 3*x*sqrt(a**2*x**2 + 1)/(8*a) + asinh(a*x)**3/ (4*a**2) + 3*asinh(a*x)/(8*a**2), Ne(a, 0)), (0, True))
\[ \int x \text {arcsinh}(a x)^3 \, dx=\int { x \operatorname {arsinh}\left (a x\right )^{3} \,d x } \]
1/2*x^2*log(a*x + sqrt(a^2*x^2 + 1))^3 - integrate(3/2*(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))^2/(a^3*x^3 + a*x + (a^2*x^2 + 1)^(3/2)), x)
\[ \int x \text {arcsinh}(a x)^3 \, dx=\int { x \operatorname {arsinh}\left (a x\right )^{3} \,d x } \]
Timed out. \[ \int x \text {arcsinh}(a x)^3 \, dx=\int x\,{\mathrm {asinh}\left (a\,x\right )}^3 \,d x \]