Integrand size = 23, antiderivative size = 264 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\text {arcsinh}(a x)}} \, dx=\frac {3 c \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)}}{4 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{32 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{4 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{32 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{4 a \sqrt {1+a^2 x^2}} \]
1/8*c*erf(2^(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2*c*x^2+c)^(1/2) /a/(a^2*x^2+1)^(1/2)+1/8*c*erfi(2^(1/2)*arcsinh(a*x)^(1/2))*2^(1/2)*Pi^(1/ 2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)+1/32*c*erf(2*arcsinh(a*x)^(1/2) )*Pi^(1/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)+1/32*c*erfi(2*arcsinh(a *x)^(1/2))*Pi^(1/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)+3/4*c*(a^2*c*x ^2+c)^(1/2)*arcsinh(a*x)^(1/2)/a/(a^2*x^2+1)^(1/2)
Time = 0.17 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.53 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\text {arcsinh}(a x)}} \, dx=\frac {c \sqrt {c+a^2 c x^2} \left (\sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-4 \text {arcsinh}(a x)\right )+4 \sqrt {2} \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {1}{2},-2 \text {arcsinh}(a x)\right )+\sqrt {\text {arcsinh}(a x)} \left (24 \sqrt {\text {arcsinh}(a x)}-4 \sqrt {2} \Gamma \left (\frac {1}{2},2 \text {arcsinh}(a x)\right )-\Gamma \left (\frac {1}{2},4 \text {arcsinh}(a x)\right )\right )\right )}{32 a \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \]
(c*Sqrt[c + a^2*c*x^2]*(Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -4*ArcSinh[a*x]] + 4*Sqrt[2]*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -2*ArcSinh[a*x]] + Sqrt[ArcSinh[a *x]]*(24*Sqrt[ArcSinh[a*x]] - 4*Sqrt[2]*Gamma[1/2, 2*ArcSinh[a*x]] - Gamma [1/2, 4*ArcSinh[a*x]])))/(32*a*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])
Time = 0.44 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.53, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6206, 3042, 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.
\(\displaystyle \int \frac {\left (a^2 c x^2+c\right )^{3/2}}{\sqrt {\text {arcsinh}(a x)}} \, dx\) |
\(\Big \downarrow \) 6206 |
\(\displaystyle \frac {c \sqrt {a^2 c x^2+c} \int \frac {\left (a^2 x^2+1\right )^2}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a \sqrt {a^2 x^2+1}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {c \sqrt {a^2 c x^2+c} \int \frac {\sin \left (i \text {arcsinh}(a x)+\frac {\pi }{2}\right )^4}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{a \sqrt {a^2 x^2+1}}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {c \sqrt {a^2 c x^2+c} \int \left (\frac {\cosh (2 \text {arcsinh}(a x))}{2 \sqrt {\text {arcsinh}(a x)}}+\frac {\cosh (4 \text {arcsinh}(a x))}{8 \sqrt {\text {arcsinh}(a x)}}+\frac {3}{8 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{a \sqrt {a^2 x^2+1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c \sqrt {a^2 c x^2+c} \left (\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {3}{4} \sqrt {\text {arcsinh}(a x)}\right )}{a \sqrt {a^2 x^2+1}}\) |
(c*Sqrt[c + a^2*c*x^2]*((3*Sqrt[ArcSinh[a*x]])/4 + (Sqrt[Pi]*Erf[2*Sqrt[Ar cSinh[a*x]]])/32 + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/4 + (Sqrt[ Pi]*Erfi[2*Sqrt[ArcSinh[a*x]]])/32 + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh [a*x]]])/4))/(a*Sqrt[1 + a^2*x^2])
3.5.96.3.1 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_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Subst[Int [x^n*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, 0]
\[\int \frac {\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{\sqrt {\operatorname {arcsinh}\left (a x \right )}}d x\]
Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\text {arcsinh}(a x)}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}{\sqrt {\operatorname {asinh}{\left (a x \right )}}}\, dx \]
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{\sqrt {\operatorname {arsinh}\left (a x\right )}} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{\sqrt {\operatorname {arsinh}\left (a x\right )}} \,d x } \]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\sqrt {\text {arcsinh}(a x)}} \, dx=\int \frac {{\left (c\,a^2\,x^2+c\right )}^{3/2}}{\sqrt {\mathrm {asinh}\left (a\,x\right )}} \,d x \]