Integrand size = 24, antiderivative size = 525 \[ \int \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx=\frac {27 a^3 \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{256 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {9 a x^2 \sqrt {a^2-x^2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{32 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 \left (a^2-x^2\right )^{5/2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{32 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3}{8} a^2 x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}-\frac {3 a^3 \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{20 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {erf}\left (2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}-\frac {3 a^3 \sqrt {\pi } \sqrt {a^2-x^2} \text {erfi}\left (2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{2048 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}}+\frac {3 a^3 \sqrt {\frac {\pi }{2}} \sqrt {a^2-x^2} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{64 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \]
1/4*x*(a^2-x^2)^(3/2)*arccosh(x/a)^(3/2)+3/8*a^2*x*arccosh(x/a)^(3/2)*(a^2 -x^2)^(1/2)-3/20*a^3*arccosh(x/a)^(5/2)*(a^2-x^2)^(1/2)/(-1+x/a)^(1/2)/(1+ x/a)^(1/2)+3/128*a^3*erf(2^(1/2)*arccosh(x/a)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2 -x^2)^(1/2)/(-1+x/a)^(1/2)/(1+x/a)^(1/2)+3/128*a^3*erfi(2^(1/2)*arccosh(x/ a)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2-x^2)^(1/2)/(-1+x/a)^(1/2)/(1+x/a)^(1/2)-3/ 2048*a^3*erf(2*arccosh(x/a)^(1/2))*Pi^(1/2)*(a^2-x^2)^(1/2)/(-1+x/a)^(1/2) /(1+x/a)^(1/2)-3/2048*a^3*erfi(2*arccosh(x/a)^(1/2))*Pi^(1/2)*(a^2-x^2)^(1 /2)/(-1+x/a)^(1/2)/(1+x/a)^(1/2)+3/32*(a^2-x^2)^(5/2)*arccosh(x/a)^(1/2)/a /(-1+x/a)^(1/2)/(1+x/a)^(1/2)+27/256*a^3*(a^2-x^2)^(1/2)*arccosh(x/a)^(1/2 )/(-1+x/a)^(1/2)/(1+x/a)^(1/2)-9/32*a*x^2*(a^2-x^2)^(1/2)*arccosh(x/a)^(1/ 2)/(-1+x/a)^(1/2)/(1+x/a)^(1/2)
Time = 0.59 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.42 \[ \int \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx=\frac {a^4 \sqrt {a^2-x^2} \left (-384 \text {arccosh}\left (\frac {x}{a}\right )^3-480 \text {arccosh}\left (\frac {x}{a}\right ) \cosh \left (2 \text {arccosh}\left (\frac {x}{a}\right )\right )+60 \sqrt {2 \pi } \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )+60 \sqrt {2 \pi } \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-5 \sqrt {-\text {arccosh}\left (\frac {x}{a}\right )} \Gamma \left (\frac {5}{2},-4 \text {arccosh}\left (\frac {x}{a}\right )\right )+5 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )} \Gamma \left (\frac {5}{2},4 \text {arccosh}\left (\frac {x}{a}\right )\right )+640 \text {arccosh}\left (\frac {x}{a}\right )^2 \sinh \left (2 \text {arccosh}\left (\frac {x}{a}\right )\right )\right )}{2560 \sqrt {\frac {-a+x}{a+x}} (a+x) \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}} \]
(a^4*Sqrt[a^2 - x^2]*(-384*ArcCosh[x/a]^3 - 480*ArcCosh[x/a]*Cosh[2*ArcCos h[x/a]] + 60*Sqrt[2*Pi]*Sqrt[ArcCosh[x/a]]*Erf[Sqrt[2]*Sqrt[ArcCosh[x/a]]] + 60*Sqrt[2*Pi]*Sqrt[ArcCosh[x/a]]*Erfi[Sqrt[2]*Sqrt[ArcCosh[x/a]]] - 5*S qrt[-ArcCosh[x/a]]*Gamma[5/2, -4*ArcCosh[x/a]] + 5*Sqrt[ArcCosh[x/a]]*Gamm a[5/2, 4*ArcCosh[x/a]] + 640*ArcCosh[x/a]^2*Sinh[2*ArcCosh[x/a]]))/(2560*S qrt[(-a + x)/(a + x)]*(a + x)*Sqrt[ArcCosh[x/a]])
Time = 5.06 (sec) , antiderivative size = 447, normalized size of antiderivative = 0.85, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6312, 25, 27, 6310, 6299, 6308, 6327, 6329, 6322, 3042, 3793, 2009, 6368, 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.
| \(\displaystyle \int \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 6312 |
| \(\displaystyle \frac {3}{4} a^2 \int \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}dx+\frac {3 a \sqrt {a^2-x^2} \int -\frac {(a-x) x (a+x) \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{a^2}dx}{8 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{4} a^2 \int \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}dx-\frac {3 a \sqrt {a^2-x^2} \int \frac {(a-x) x (a+x) \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{a^2}dx}{8 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{4} a^2 \int \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}dx-\frac {3 \sqrt {a^2-x^2} \int (a-x) x (a+x) \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}dx}{8 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\) |
| \(\Big \downarrow \) 6310 |
| \(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2-x^2} \int x \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}dx}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {\sqrt {a^2-x^2} \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}dx}{2 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\right )-\frac {3 \sqrt {a^2-x^2} \int (a-x) x (a+x) \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}dx}{8 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\) |
| \(\Big \downarrow \) 6299 |
| \(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {\sqrt {a^2-x^2} \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}dx}{2 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\right )-\frac {3 \sqrt {a^2-x^2} \int (a-x) x (a+x) \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}dx}{8 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\right )-\frac {3 \sqrt {a^2-x^2} \int (a-x) x (a+x) \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}dx}{8 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\right )-\frac {3 \sqrt {a^2-x^2} \int x \left (a^2-x^2\right ) \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}dx}{8 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\) |
| \(\Big \downarrow \) 6329 |
| \(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\right )-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{8} a^3 \int \frac {\left (\frac {x}{a}-1\right )^{3/2} \left (\frac {x}{a}+1\right )^{3/2}}{\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}dx-\frac {1}{4} \left (a^2-x^2\right )^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{8 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\) |
| \(\Big \downarrow \) 6322 |
| \(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\right )-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{8} a^4 \int \frac {\left (\frac {x}{a}-1\right )^2 \left (\frac {x}{a}+1\right )^2}{\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}d\text {arccosh}\left (\frac {x}{a}\right )-\frac {1}{4} \left (a^2-x^2\right )^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{8 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\right )-\frac {3 \sqrt {a^2-x^2} \left (-\frac {1}{4} \left (a^2-x^2\right )^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}+\frac {1}{8} a^4 \int \frac {\sin \left (i \text {arccosh}\left (\frac {x}{a}\right )\right )^4}{\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}d\text {arccosh}\left (\frac {x}{a}\right )\right )}{8 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\right )-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{8} a^4 \int \left (-\frac {\cosh \left (2 \text {arccosh}\left (\frac {x}{a}\right )\right )}{2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}+\frac {\cosh \left (4 \text {arccosh}\left (\frac {x}{a}\right )\right )}{8 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}+\frac {3}{8 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}\right )d\text {arccosh}\left (\frac {x}{a}\right )-\frac {1}{4} \left (a^2-x^2\right )^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{8 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {\int \frac {x^2}{\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}dx}{4 a}\right )}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\right )+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{8} a^4 \left (\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )+\frac {3}{4} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-\frac {1}{4} \left (a^2-x^2\right )^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{8 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}\) |
| \(\Big \downarrow \) 6368 |
| \(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \int \frac {x^2}{a^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}d\text {arccosh}\left (\frac {x}{a}\right )\right )}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\right )+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{8} a^4 \left (\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )+\frac {3}{4} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-\frac {1}{4} \left (a^2-x^2\right )^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{8 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \int \frac {\sin \left (i \text {arccosh}\left (\frac {x}{a}\right )+\frac {\pi }{2}\right )^2}{\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}d\text {arccosh}\left (\frac {x}{a}\right )\right )}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\right )+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{8} a^4 \left (\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )+\frac {3}{4} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-\frac {1}{4} \left (a^2-x^2\right )^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{8 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \int \left (\frac {\cosh \left (2 \text {arccosh}\left (\frac {x}{a}\right )\right )}{2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}+\frac {1}{2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}\right )d\text {arccosh}\left (\frac {x}{a}\right )\right )}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\right )+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{8} a^4 \left (\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )+\frac {3}{4} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-\frac {1}{4} \left (a^2-x^2\right )^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{8 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{4} a^2 \left (-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{2} x^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}-\frac {1}{4} a^2 \left (\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )+\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )+\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )\right )}{4 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}-\frac {a \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}+\frac {1}{2} x \sqrt {a^2-x^2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}\right )+\frac {1}{4} x \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2}-\frac {3 \sqrt {a^2-x^2} \left (\frac {1}{8} a^4 \left (\frac {1}{32} \sqrt {\pi } \text {erf}\left (2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erf}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )+\frac {1}{32} \sqrt {\pi } \text {erfi}\left (2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-\frac {1}{4} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )+\frac {3}{4} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )-\frac {1}{4} \left (a^2-x^2\right )^2 \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}\right )}{8 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1}}\) |
(x*(a^2 - x^2)^(3/2)*ArcCosh[x/a]^(3/2))/4 - (3*Sqrt[a^2 - x^2]*(-1/4*((a^ 2 - x^2)^2*Sqrt[ArcCosh[x/a]]) + (a^4*((3*Sqrt[ArcCosh[x/a]])/4 + (Sqrt[Pi ]*Erf[2*Sqrt[ArcCosh[x/a]]])/32 - (Sqrt[Pi/2]*Erf[Sqrt[2]*Sqrt[ArcCosh[x/a ]]])/4 + (Sqrt[Pi]*Erfi[2*Sqrt[ArcCosh[x/a]]])/32 - (Sqrt[Pi/2]*Erfi[Sqrt[ 2]*Sqrt[ArcCosh[x/a]]])/4))/8))/(8*a*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) + (3*a^ 2*((x*Sqrt[a^2 - x^2]*ArcCosh[x/a]^(3/2))/2 - (a*Sqrt[a^2 - x^2]*ArcCosh[x /a]^(5/2))/(5*Sqrt[-1 + x/a]*Sqrt[1 + x/a]) - (3*Sqrt[a^2 - x^2]*((x^2*Sqr t[ArcCosh[x/a]])/2 - (a^2*(Sqrt[ArcCosh[x/a]] + (Sqrt[Pi/2]*Erf[Sqrt[2]*Sq rt[ArcCosh[x/a]]])/4 + (Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[x/a]]])/4))/4 ))/(4*a*Sqrt[-1 + x/a]*Sqrt[1 + x/a])))/4
3.4.97.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^(m + 1)*((a + b*ArcCosh[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1)) Int [x^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x ], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 ] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcCosh[c*x])^n/2), x] + (-Simp[( 1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[(a + b*ArcC osh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Simp[b*c*(n/2)*Simp[Sq rt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])] Int[x*(a + b*ArcCosh[c*x])^ (n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n , 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcCosh[c*x])^n/(2*p + 1)), x] + (Simp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x ], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p )] Int[x*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( (d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c))*Simp[(d1 + e1*x)^p/ (1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p] Subst[Int[x^n*Sinh[-a/b + x /b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2 , e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[2*p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 *d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 , e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x ])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x _))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))* Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p] Subst[In t[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c *x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[ e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]
\[\int \left (a^{2}-x^{2}\right )^{\frac {3}{2}} \operatorname {arccosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}d x\]
Exception generated. \[ \int \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx=\text {Timed out} \]
\[ \int \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx=\int { {\left (a^{2} - x^{2}\right )}^{\frac {3}{2}} \operatorname {arcosh}\left (\frac {x}{a}\right )^{\frac {3}{2}} \,d x } \]
\[ \int \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx=\int { {\left (a^{2} - x^{2}\right )}^{\frac {3}{2}} \operatorname {arcosh}\left (\frac {x}{a}\right )^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int \left (a^2-x^2\right )^{3/2} \text {arccosh}\left (\frac {x}{a}\right )^{3/2} \, dx=\int {\mathrm {acosh}\left (\frac {x}{a}\right )}^{3/2}\,{\left (a^2-x^2\right )}^{3/2} \,d x \]