Integrand size = 23, antiderivative size = 449 \[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{3/2} \, dx=-\frac {27 c \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)}}{256 a \sqrt {1+a^2 x^2}}-\frac {9 a c x^2 \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)}}{32 \sqrt {1+a^2 x^2}}-\frac {3 c \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{3/2}+\frac {3 c \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{5/2}}{20 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{2048 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{2048 a \sqrt {1+a^2 x^2}}+\frac {3 c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}} \]
1/4*x*(a^2*c*x^2+c)^(3/2)*arcsinh(a*x)^(3/2)+3/8*c*x*arcsinh(a*x)^(3/2)*(a ^2*c*x^2+c)^(1/2)+3/20*c*arcsinh(a*x)^(5/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2 +1)^(1/2)+3/128*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)+3/128*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)+3/2048*c*erf(2*ar csinh(a*x)^(1/2))*Pi^(1/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)+3/2048* 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)-3/32*c*(a^2*x^2+1)^(3/2)*(a^2*c*x^2+c)^(1/2)*arcsinh(a*x)^(1/2)/a-27/2 56*c*(a^2*c*x^2+c)^(1/2)*arcsinh(a*x)^(1/2)/a/(a^2*x^2+1)^(1/2)-9/32*a*c*x ^2*(a^2*c*x^2+c)^(1/2)*arcsinh(a*x)^(1/2)/(a^2*x^2+1)^(1/2)
Time = 0.36 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.41 \[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{3/2} \, dx=\frac {c \sqrt {c+a^2 c x^2} \left (384 \text {arcsinh}(a x)^3-480 \text {arcsinh}(a x) \cosh (2 \text {arcsinh}(a x))+60 \sqrt {2 \pi } \sqrt {\text {arcsinh}(a x)} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+60 \sqrt {2 \pi } \sqrt {\text {arcsinh}(a x)} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+5 \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {5}{2},-4 \text {arcsinh}(a x)\right )-5 \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {5}{2},4 \text {arcsinh}(a x)\right )+640 \text {arcsinh}(a x)^2 \sinh (2 \text {arcsinh}(a x))\right )}{2560 a \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \]
(c*Sqrt[c + a^2*c*x^2]*(384*ArcSinh[a*x]^3 - 480*ArcSinh[a*x]*Cosh[2*ArcSi nh[a*x]] + 60*Sqrt[2*Pi]*Sqrt[ArcSinh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]] ] + 60*Sqrt[2*Pi]*Sqrt[ArcSinh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]] + 5* Sqrt[-ArcSinh[a*x]]*Gamma[5/2, -4*ArcSinh[a*x]] - 5*Sqrt[ArcSinh[a*x]]*Gam ma[5/2, 4*ArcSinh[a*x]] + 640*ArcSinh[a*x]^2*Sinh[2*ArcSinh[a*x]]))/(2560* a*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])
Time = 2.32 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.89, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {6201, 6200, 6192, 6198, 6213, 6206, 3042, 3793, 2009, 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.
| \(\displaystyle \int \text {arcsinh}(a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle -\frac {3 a c \sqrt {a^2 c x^2+c} \int x \left (a^2 x^2+1\right ) \sqrt {\text {arcsinh}(a x)}dx}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \int \sqrt {a^2 c x^2+c} \text {arcsinh}(a x)^{3/2}dx+\frac {1}{4} x \text {arcsinh}(a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle -\frac {3 a c \sqrt {a^2 c x^2+c} \int x \left (a^2 x^2+1\right ) \sqrt {\text {arcsinh}(a x)}dx}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {3 a \sqrt {a^2 c x^2+c} \int x \sqrt {\text {arcsinh}(a x)}dx}{4 \sqrt {a^2 x^2+1}}+\frac {\sqrt {a^2 c x^2+c} \int \frac {\text {arcsinh}(a x)^{3/2}}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 6192 |
| \(\displaystyle -\frac {3 a c \sqrt {a^2 c x^2+c} \int x \left (a^2 x^2+1\right ) \sqrt {\text {arcsinh}(a x)}dx}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {1}{4} a \int \frac {x^2}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\sqrt {a^2 c x^2+c} \int \frac {\text {arcsinh}(a x)^{3/2}}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {3}{4} c \left (-\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {1}{4} a \int \frac {x^2}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}}{5 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\right )-\frac {3 a c \sqrt {a^2 c x^2+c} \int x \left (a^2 x^2+1\right ) \sqrt {\text {arcsinh}(a x)}dx}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {3}{4} c \left (-\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {1}{4} a \int \frac {x^2}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}}{5 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\right )-\frac {3 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \sqrt {\text {arcsinh}(a x)}}{4 a^2}-\frac {\int \frac {\left (a^2 x^2+1\right )^{3/2}}{\sqrt {\text {arcsinh}(a x)}}dx}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 6206 |
| \(\displaystyle \frac {3}{4} c \left (-\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {1}{4} a \int \frac {x^2}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}}{5 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\right )-\frac {3 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \sqrt {\text {arcsinh}(a x)}}{4 a^2}-\frac {\int \frac {\left (a^2 x^2+1\right )^2}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{8 a^2}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} c \left (-\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {1}{4} a \int \frac {x^2}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}}{5 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\right )-\frac {3 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \sqrt {\text {arcsinh}(a x)}}{4 a^2}-\frac {\int \frac {\sin \left (i \text {arcsinh}(a x)+\frac {\pi }{2}\right )^4}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{8 a^2}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {3}{4} c \left (-\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {1}{4} a \int \frac {x^2}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}}{5 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\right )-\frac {3 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \sqrt {\text {arcsinh}(a x)}}{4 a^2}-\frac {\int \left (\frac {\cosh (2 \text {arcsinh}(a x))}{2 \sqrt {\text {arcsinh}(a x)}}+\frac {\cosh (4 \text {arcsinh}(a x))}{8 \sqrt {\text {arcsinh}(a x)}}+\frac {3}{8 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{8 a^2}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{4} c \left (-\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {1}{4} a \int \frac {x^2}{\sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}}{5 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\right )-\frac {3 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \sqrt {\text {arcsinh}(a x)}}{4 a^2}-\frac {\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {3}{4} \sqrt {\text {arcsinh}(a x)}}{8 a^2}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 6234 |
| \(\displaystyle \frac {3}{4} c \left (-\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {\int \frac {a^2 x^2}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a^2}\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}}{5 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\right )-\frac {3 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \sqrt {\text {arcsinh}(a x)}}{4 a^2}-\frac {\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {3}{4} \sqrt {\text {arcsinh}(a x)}}{8 a^2}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} c \left (-\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {\int -\frac {\sin (i \text {arcsinh}(a x))^2}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a^2}\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}}{5 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\right )-\frac {3 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \sqrt {\text {arcsinh}(a x)}}{4 a^2}-\frac {\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {3}{4} \sqrt {\text {arcsinh}(a x)}}{8 a^2}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{4} c \left (-\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}+\frac {\int \frac {\sin (i \text {arcsinh}(a x))^2}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a^2}\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}}{5 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\right )-\frac {3 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \sqrt {\text {arcsinh}(a x)}}{4 a^2}-\frac {\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {3}{4} \sqrt {\text {arcsinh}(a x)}}{8 a^2}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {3}{4} c \left (-\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {\int \left (\frac {1}{2 \sqrt {\text {arcsinh}(a x)}}-\frac {\cosh (2 \text {arcsinh}(a x))}{2 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{4 a^2}+\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}}{5 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\right )-\frac {3 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \sqrt {\text {arcsinh}(a x)}}{4 a^2}-\frac {\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {3}{4} \sqrt {\text {arcsinh}(a x)}}{8 a^2}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \sqrt {\text {arcsinh}(a x)}}{4 a^2}-\frac {\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {3}{4} \sqrt {\text {arcsinh}(a x)}}{8 a^2}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {3 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \sqrt {\text {arcsinh}(a x)}-\frac {\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\sqrt {\text {arcsinh}(a x)}}{4 a^2}\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}}{5 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{3/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}\) |
(x*(c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^(3/2))/4 - (3*a*c*Sqrt[c + a^2*c*x^2 ]*(((1 + a^2*x^2)^2*Sqrt[ArcSinh[a*x]])/(4*a^2) - ((3*Sqrt[ArcSinh[a*x]])/ 4 + (Sqrt[Pi]*Erf[2*Sqrt[ArcSinh[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)/(8*a^2)))/(8*Sqrt[1 + a^2*x^2]) + ( 3*c*((x*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))/2 + (Sqrt[c + a^2*c*x^2]*A rcSinh[a*x]^(5/2))/(5*a*Sqrt[1 + a^2*x^2]) - (3*a*Sqrt[c + a^2*c*x^2]*((x^ 2*Sqrt[ArcSinh[a*x]])/2 - (-Sqrt[ArcSinh[a*x]] + (Sqrt[Pi/2]*Erf[Sqrt[2]*S qrt[ArcSinh[a*x]]])/4 + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]])/4)/( 4*a^2)))/(4*Sqrt[1 + a^2*x^2])))/4
3.5.77.3.1 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_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + (Simp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x ], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[x* (1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]
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[((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_.)*(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]
\[\int \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \operatorname {arcsinh}\left (a x \right )^{\frac {3}{2}}d x\]
Exception generated. \[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{3/2} \, dx=\text {Timed out} \]
\[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{3/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}} \,d x } \]
Exception generated. \[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{3/2} \, dx=\int {\mathrm {asinh}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]