Integrand size = 23, antiderivative size = 514 \[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{5/2} \, dx=\frac {225}{512} c x \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)}+\frac {15}{256} c x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)}-\frac {45 c \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2}}{256 a \sqrt {1+a^2 x^2}}-\frac {15 a c x^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2}}{32 \sqrt {1+a^2 x^2}}-\frac {5 c \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2}}{32 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{5/2}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{5/2}+\frac {3 c \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{7/2}}{28 a \sqrt {1+a^2 x^2}}+\frac {15 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erf}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{16384 a \sqrt {1+a^2 x^2}}+\frac {15 c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{256 a \sqrt {1+a^2 x^2}}-\frac {15 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (2 \sqrt {\text {arcsinh}(a x)}\right )}{16384 a \sqrt {1+a^2 x^2}}-\frac {15 c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )}{256 a \sqrt {1+a^2 x^2}} \]
1/4*x*(a^2*c*x^2+c)^(3/2)*arcsinh(a*x)^(5/2)-5/32*c*(a^2*x^2+1)^(3/2)*arcs inh(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/a+3/8*c*x*arcsinh(a*x)^(5/2)*(a^2*c*x^2 +c)^(1/2)-45/256*c*arcsinh(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1 /2)-15/32*a*c*x^2*arcsinh(a*x)^(3/2)*(a^2*c*x^2+c)^(1/2)/(a^2*x^2+1)^(1/2) +3/28*c*arcsinh(a*x)^(7/2)*(a^2*c*x^2+c)^(1/2)/a/(a^2*x^2+1)^(1/2)+15/512* 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)-15/512*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)+15/16384*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)-15/16384*c*erfi(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)+225/512 *c*x*(a^2*c*x^2+c)^(1/2)*arcsinh(a*x)^(1/2)+15/256*c*x*(a^2*x^2+1)*(a^2*c* x^2+c)^(1/2)*arcsinh(a*x)^(1/2)
Time = 0.37 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.39 \[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{5/2} \, dx=\frac {c \sqrt {c+a^2 c x^2} \left (1536 \text {arcsinh}(a x)^4-4480 \text {arcsinh}(a x)^2 \cosh (2 \text {arcsinh}(a x))+420 \sqrt {2 \pi } \sqrt {\text {arcsinh}(a x)} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-420 \sqrt {2 \pi } \sqrt {\text {arcsinh}(a x)} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-7 \sqrt {-\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},-4 \text {arcsinh}(a x)\right )-7 \sqrt {\text {arcsinh}(a x)} \Gamma \left (\frac {7}{2},4 \text {arcsinh}(a x)\right )+3360 \text {arcsinh}(a x) \sinh (2 \text {arcsinh}(a x))+3584 \text {arcsinh}(a x)^3 \sinh (2 \text {arcsinh}(a x))\right )}{14336 a \sqrt {1+a^2 x^2} \sqrt {\text {arcsinh}(a x)}} \]
(c*Sqrt[c + a^2*c*x^2]*(1536*ArcSinh[a*x]^4 - 4480*ArcSinh[a*x]^2*Cosh[2*A rcSinh[a*x]] + 420*Sqrt[2*Pi]*Sqrt[ArcSinh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcSinh[ a*x]]] - 420*Sqrt[2*Pi]*Sqrt[ArcSinh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]] ] - 7*Sqrt[-ArcSinh[a*x]]*Gamma[7/2, -4*ArcSinh[a*x]] - 7*Sqrt[ArcSinh[a*x ]]*Gamma[7/2, 4*ArcSinh[a*x]] + 3360*ArcSinh[a*x]*Sinh[2*ArcSinh[a*x]] + 3 584*ArcSinh[a*x]^3*Sinh[2*ArcSinh[a*x]]))/(14336*a*Sqrt[1 + a^2*x^2]*Sqrt[ ArcSinh[a*x]])
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.
| \(\displaystyle \int \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \int x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^{3/2}dx}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \int \sqrt {a^2 c x^2+c} \text {arcsinh}(a x)^{5/2}dx+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \int x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^{3/2}dx}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \int x \text {arcsinh}(a x)^{3/2}dx}{4 \sqrt {a^2 x^2+1}}+\frac {\sqrt {a^2 c x^2+c} \int \frac {\text {arcsinh}(a x)^{5/2}}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 6192 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \int x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^{3/2}dx}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\sqrt {a^2 c x^2+c} \int \frac {\text {arcsinh}(a x)^{5/2}}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )-\frac {5 a c \sqrt {a^2 c x^2+c} \int x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^{3/2}dx}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )-\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \int \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}dx}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle \frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )-\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \int \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}dx+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 6200 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx-\frac {1}{4} a \int \frac {x}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 6195 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (-\frac {\int \frac {a x \sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a}+\frac {1}{2} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx+\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )-\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx-\frac {\int \frac {\sinh (2 \text {arcsinh}(a x))}{2 \sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a}+\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )-\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx-\frac {\int \frac {\sinh (2 \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{8 a}+\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )-\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx-\frac {\int -\frac {i \sin (2 i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{8 a}+\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )-\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx+\frac {i \int \frac {\sin (2 i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{8 a}+\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 3789 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx+\frac {i \left (\frac {1}{2} i \int \frac {e^{2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{2} i \int \frac {e^{-2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)\right )}{8 a}+\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx+\frac {i \left (i \int e^{2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}-i \int e^{-2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{8 a}+\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-i \int e^{-2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{8 a}+\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx+\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a}\right )-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \int \frac {x^2 \sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}-\frac {\int \frac {x}{\sqrt {\text {arcsinh}(a x)}}dx}{4 a}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 6195 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}-\frac {\int \frac {a x \sqrt {a^2 x^2+1}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a^3}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (-\frac {\int \frac {\sinh (2 \text {arcsinh}(a x))}{2 \sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{4 a^3}-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (-\frac {\int \frac {\sinh (2 \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{8 a^3}-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (-\frac {\int -\frac {i \sin (2 i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{8 a^3}-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {i \int \frac {\sin (2 i \text {arcsinh}(a x))}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{8 a^3}-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 3789 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {i \left (\frac {1}{2} i \int \frac {e^{2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)-\frac {1}{2} i \int \frac {e^{-2 \text {arcsinh}(a x)}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)\right )}{8 a^3}-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {i \left (i \int e^{2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}-i \int e^{-2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{8 a^3}-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-i \int e^{-2 \text {arcsinh}(a x)}d\sqrt {\text {arcsinh}(a x)}\right )}{8 a^3}-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {3}{4} c \left (-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (-\frac {\int \frac {\sqrt {\text {arcsinh}(a x)}}{\sqrt {a^2 x^2+1}}dx}{2 a^2}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a^3}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a^2 x^2+1}}+\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {1}{8} a \int \frac {x \left (a^2 x^2+1\right )}{\sqrt {\text {arcsinh}(a x)}}dx+\frac {3}{4} \left (\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}+\frac {3}{4} c \left (\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a^3}-\frac {\text {arcsinh}(a x)^{3/2}}{3 a^3}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a^2 x^2+1}}\right )\) |
| \(\Big \downarrow \) 6234 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {\int \frac {a x \left (a^2 x^2+1\right )^{3/2}}{\sqrt {\text {arcsinh}(a x)}}d\text {arcsinh}(a x)}{8 a}+\frac {3}{4} \left (\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}+\frac {3}{4} c \left (\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a^3}-\frac {\text {arcsinh}(a x)^{3/2}}{3 a^3}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a^2 x^2+1}}\right )\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle -\frac {5 a c \sqrt {a^2 c x^2+c} \left (\frac {\left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^{3/2}}{4 a^2}-\frac {3 \left (-\frac {\int \left (\frac {\sinh (2 \text {arcsinh}(a x))}{4 \sqrt {\text {arcsinh}(a x)}}+\frac {\sinh (4 \text {arcsinh}(a x))}{8 \sqrt {\text {arcsinh}(a x)}}\right )d\text {arcsinh}(a x)}{8 a}+\frac {3}{4} \left (\frac {1}{2} x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}+\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a}+\frac {\text {arcsinh}(a x)^{3/2}}{3 a}\right )+\frac {1}{4} x \left (a^2 x^2+1\right )^{3/2} \sqrt {\text {arcsinh}(a x)}\right )}{8 a}\right )}{8 \sqrt {a^2 x^2+1}}+\frac {1}{4} x \text {arcsinh}(a x)^{5/2} \left (a^2 c x^2+c\right )^{3/2}+\frac {3}{4} c \left (\frac {\text {arcsinh}(a x)^{7/2} \sqrt {a^2 c x^2+c}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \text {arcsinh}(a x)^{5/2} \sqrt {a^2 c x^2+c}-\frac {5 a \sqrt {a^2 c x^2+c} \left (\frac {1}{2} x^2 \text {arcsinh}(a x)^{3/2}-\frac {3}{4} a \left (\frac {i \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arcsinh}(a x)}\right )\right )}{8 a^3}-\frac {\text {arcsinh}(a x)^{3/2}}{3 a^3}+\frac {x \sqrt {a^2 x^2+1} \sqrt {\text {arcsinh}(a x)}}{2 a^2}\right )\right )}{4 \sqrt {a^2 x^2+1}}\right )\) |
3.5.81.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
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_)*(x_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 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_.)*(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_.)*((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]
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]
\[\int \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \operatorname {arcsinh}\left (a x \right )^{\frac {5}{2}}d x\]
Exception generated. \[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{5/2} \, dx=\text {Timed out} \]
\[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{5/2} \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (a x\right )^{\frac {5}{2}} \,d x } \]
Exception generated. \[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{5/2} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{5/2} \, dx=\int {\mathrm {asinh}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]