Integrand size = 22, antiderivative size = 259 \[ \int \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx=-\frac {3 a \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{16 \sqrt {1+\frac {x^2}{a^2}}}-\frac {3 x^2 \sqrt {a^2+x^2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}{8 a \sqrt {1+\frac {x^2}{a^2}}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {1+\frac {x^2}{a^2}}}+\frac {3 a \sqrt {\frac {\pi }{2}} \sqrt {a^2+x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {1+\frac {x^2}{a^2}}} \]
1/2*x*arcsinh(x/a)^(3/2)*(a^2+x^2)^(1/2)+1/5*a*arcsinh(x/a)^(5/2)*(a^2+x^2 )^(1/2)/(1+x^2/a^2)^(1/2)+3/128*a*erf(2^(1/2)*arcsinh(x/a)^(1/2))*2^(1/2)* Pi^(1/2)*(a^2+x^2)^(1/2)/(1+x^2/a^2)^(1/2)+3/128*a*erfi(2^(1/2)*arcsinh(x/ a)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2+x^2)^(1/2)/(1+x^2/a^2)^(1/2)-3/16*a*(a^2+x ^2)^(1/2)*arcsinh(x/a)^(1/2)/(1+x^2/a^2)^(1/2)-3/8*x^2*(a^2+x^2)^(1/2)*arc sinh(x/a)^(1/2)/a/(1+x^2/a^2)^(1/2)
Time = 0.25 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.51 \[ \int \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx=\frac {a \sqrt {a^2+x^2} \left (15 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+15 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+8 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )} \left (16 \text {arcsinh}\left (\frac {x}{a}\right )^2-15 \cosh \left (2 \text {arcsinh}\left (\frac {x}{a}\right )\right )+20 \text {arcsinh}\left (\frac {x}{a}\right ) \sinh \left (2 \text {arcsinh}\left (\frac {x}{a}\right )\right )\right )\right )}{640 \sqrt {1+\frac {x^2}{a^2}}} \]
(a*Sqrt[a^2 + x^2]*(15*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcSinh[x/a]]] + 15*Sqr t[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcSinh[x/a]]] + 8*Sqrt[ArcSinh[x/a]]*(16*ArcSin h[x/a]^2 - 15*Cosh[2*ArcSinh[x/a]] + 20*ArcSinh[x/a]*Sinh[2*ArcSinh[x/a]]) ))/(640*Sqrt[1 + x^2/a^2])
Time = 0.98 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.75, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6200, 6192, 6198, 6234, 3042, 25, 3793, 2009}
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.
| \(\displaystyle \int \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle -\frac {3 \sqrt {a^2+x^2} \int x \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}dx}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {\sqrt {a^2+x^2} \int \frac {\text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {\frac {x^2}{a^2}+1}}dx}{2 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\) |
| \(\Big \downarrow \) 6192 |
| \(\displaystyle -\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x^2}{a^2}+1} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {\sqrt {a^2+x^2} \int \frac {\text {arcsinh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {\frac {x^2}{a^2}+1}}dx}{2 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle -\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x^2}{a^2}+1} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\) |
| \(\Big \downarrow \) 6234 |
| \(\displaystyle -\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \int \frac {x^2}{a^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}d\text {arcsinh}\left (\frac {x}{a}\right )\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \int -\frac {\sin \left (i \text {arcsinh}\left (\frac {x}{a}\right )\right )^2}{\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}d\text {arcsinh}\left (\frac {x}{a}\right )\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}+\frac {1}{4} a^2 \int \frac {\sin \left (i \text {arcsinh}\left (\frac {x}{a}\right )\right )^2}{\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}d\text {arcsinh}\left (\frac {x}{a}\right )\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle -\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{4} a^2 \int \left (\frac {1}{2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}-\frac {\cosh \left (2 \text {arcsinh}\left (\frac {x}{a}\right )\right )}{2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}}\right )d\text {arcsinh}\left (\frac {x}{a}\right )+\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt {a^2+x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \left (\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )-\sqrt {\text {arcsinh}\left (\frac {x}{a}\right )}\right )\right )}{4 a \sqrt {\frac {x^2}{a^2}+1}}+\frac {a \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x^2}{a^2}+1}}+\frac {1}{2} x \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2}\) |
(x*Sqrt[a^2 + x^2]*ArcSinh[x/a]^(3/2))/2 + (a*Sqrt[a^2 + x^2]*ArcSinh[x/a] ^(5/2))/(5*Sqrt[1 + x^2/a^2]) - (3*Sqrt[a^2 + x^2]*((x^2*Sqrt[ArcSinh[x/a] ])/2 - (a^2*(-Sqrt[ArcSinh[x/a]] + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[x/ a]]])/4 + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[x/a]]])/4))/4))/(4*a*Sqrt[ 1 + x^2/a^2])
3.5.91.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^(m + 1)*((a + b*ArcSinh[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1)) Int [x^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; Free Q[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
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_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 /2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[(a + b*ArcSinh[c*x])^n/Sq rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* x^2]] Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e }, x] && EqQ[e, c^2*d] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
\[\int \operatorname {arcsinh}\left (\frac {x}{a}\right )^{\frac {3}{2}} \sqrt {a^{2}+x^{2}}d x\]
Exception generated. \[ \int \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx=\int \sqrt {a^{2} + x^{2}} \operatorname {asinh}^{\frac {3}{2}}{\left (\frac {x}{a} \right )}\, dx \]
\[ \int \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx=\int { \sqrt {a^{2} + x^{2}} \operatorname {arsinh}\left (\frac {x}{a}\right )^{\frac {3}{2}} \,d x } \]
\[ \int \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx=\int { \sqrt {a^{2} + x^{2}} \operatorname {arsinh}\left (\frac {x}{a}\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \sqrt {a^2+x^2} \text {arcsinh}\left (\frac {x}{a}\right )^{3/2} \, dx=\int {\mathrm {asinh}\left (\frac {x}{a}\right )}^{3/2}\,\sqrt {a^2+x^2} \,d x \]