Integrand size = 24, antiderivative size = 24 \[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{5/2}} \, dx=\text {Int}\left (\frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{5/2}},x\right ) \] Output:
Defer(Int)(arccosh(x/a)^(3/2)/(a^2-x^2)^(5/2),x)
Not integrable
Time = 3.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{5/2}} \, dx=\int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{5/2}} \, dx \] Input:
Integrate[ArcCosh[x/a]^(3/2)/(a^2 - x^2)^(5/2),x]
Output:
Integrate[ArcCosh[x/a]^(3/2)/(a^2 - x^2)^(5/2), x]
Not integrable
Time = 2.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
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.
| \(\displaystyle \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {2 \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{3/2}}dx}{3 a^2}+\frac {\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \int \frac {a^4 x \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{(a-x)^2 (a+x)^2}dx}{2 a^5 \sqrt {a^2-x^2}}+\frac {x \text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{3 a^2 \left (a^2-x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \int \frac {x \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{(a-x)^2 (a+x)^2}dx}{2 a \sqrt {a^2-x^2}}+\frac {2 \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{3/2}}dx}{3 a^2}+\frac {x \text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{3 a^2 \left (a^2-x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6314 |
| \(\displaystyle \frac {\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \int \frac {x \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{(a-x)^2 (a+x)^2}dx}{2 a \sqrt {a^2-x^2}}+\frac {2 \left (\frac {3 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \int \frac {a^2 x \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{a^2-x^2}dx}{2 a^3 \sqrt {a^2-x^2}}+\frac {x \text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{a^2 \sqrt {a^2-x^2}}\right )}{3 a^2}+\frac {x \text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{3 a^2 \left (a^2-x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \int \frac {x \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{(a-x)^2 (a+x)^2}dx}{2 a \sqrt {a^2-x^2}}+\frac {2 \left (\frac {3 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \int \frac {x \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{a^2-x^2}dx}{2 a \sqrt {a^2-x^2}}+\frac {x \text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{a^2 \sqrt {a^2-x^2}}\right )}{3 a^2}+\frac {x \text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{3 a^2 \left (a^2-x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \int \frac {x \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^2}dx}{2 a \sqrt {a^2-x^2}}+\frac {2 \left (\frac {3 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \int \frac {x \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{a^2-x^2}dx}{2 a \sqrt {a^2-x^2}}+\frac {x \text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{a^2 \sqrt {a^2-x^2}}\right )}{3 a^2}+\frac {x \text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{3 a^2 \left (a^2-x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6329 |
| \(\displaystyle \frac {2 \left (\frac {3 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \int \frac {x \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{a^2-x^2}dx}{2 a \sqrt {a^2-x^2}}+\frac {x \text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{a^2 \sqrt {a^2-x^2}}\right )}{3 a^2}+\frac {\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \left (\frac {\int \frac {1}{\left (\frac {x}{a}-1\right )^{3/2} \left (\frac {x}{a}+1\right )^{3/2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}dx}{4 a^3}+\frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{2 \left (a^2-x^2\right )}\right )}{2 a \sqrt {a^2-x^2}}+\frac {x \text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{3 a^2 \left (a^2-x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6326 |
| \(\displaystyle \frac {2 \left (\frac {3 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \int \frac {x \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{a^2-x^2}dx}{2 a \sqrt {a^2-x^2}}+\frac {x \text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{a^2 \sqrt {a^2-x^2}}\right )}{3 a^2}+\frac {\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \left (\frac {\int \frac {1}{\left (\frac {x}{a}-1\right )^{3/2} \left (\frac {x}{a}+1\right )^{3/2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}dx}{4 a^3}+\frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{2 \left (a^2-x^2\right )}\right )}{2 a \sqrt {a^2-x^2}}+\frac {x \text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{3 a^2 \left (a^2-x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 6375 |
| \(\displaystyle \frac {2 \left (\frac {3 \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \int \frac {x \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{a^2-x^2}dx}{2 a \sqrt {a^2-x^2}}+\frac {x \text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{a^2 \sqrt {a^2-x^2}}\right )}{3 a^2}+\frac {\sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \left (\frac {\int \frac {1}{\left (\frac {x}{a}-1\right )^{3/2} \left (\frac {x}{a}+1\right )^{3/2} \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}dx}{4 a^3}+\frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{2 \left (a^2-x^2\right )}\right )}{2 a \sqrt {a^2-x^2}}+\frac {x \text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{3 a^2 \left (a^2-x^2\right )^{3/2}}\) |
Input:
Int[ArcCosh[x/a]^(3/2)/(a^2 - x^2)^(5/2),x]
Output:
$Aborted
Not integrable
Time = 0.44 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83
\[\int \frac {\operatorname {arccosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}}{\left (a^{2}-x^{2}\right )^{\frac {5}{2}}}d x\]
Input:
int(arccosh(x/a)^(3/2)/(a^2-x^2)^(5/2),x)
Output:
int(arccosh(x/a)^(3/2)/(a^2-x^2)^(5/2),x)
Exception generated. \[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(arccosh(x/a)^(3/2)/(a^2-x^2)^(5/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(acosh(x/a)**(3/2)/(a**2-x**2)**(5/2),x)
Output:
Timed out
Not integrable
Time = 0.37 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{5/2}} \, dx=\int { \frac {\operatorname {arcosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}}{{\left (a^{2} - x^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(arccosh(x/a)^(3/2)/(a^2-x^2)^(5/2),x, algorithm="maxima")
Output:
integrate(arccosh(x/a)^(3/2)/(a^2 - x^2)^(5/2), x)
Not integrable
Time = 3.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{5/2}} \, dx=\int { \frac {\operatorname {arcosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}}{{\left (a^{2} - x^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(arccosh(x/a)^(3/2)/(a^2-x^2)^(5/2),x, algorithm="giac")
Output:
integrate(arccosh(x/a)^(3/2)/(a^2 - x^2)^(5/2), x)
Not integrable
Time = 2.61 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{5/2}} \, dx=\int \frac {{\mathrm {acosh}\left (\frac {x}{a}\right )}^{3/2}}{{\left (a^2-x^2\right )}^{5/2}} \,d x \] Input:
int(acosh(x/a)^(3/2)/(a^2 - x^2)^(5/2),x)
Output:
int(acosh(x/a)^(3/2)/(a^2 - x^2)^(5/2), x)
Not integrable
Time = 0.77 (sec) , antiderivative size = 301, normalized size of antiderivative = 12.54 \[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {a +x}\, \sqrt {-a +x}\, \sqrt {a^{2}-x^{2}}\, \sqrt {\mathit {acosh} \left (\frac {x}{a}\right )}\, \mathit {acosh} \left (\frac {x}{a}\right )^{2}-5 \left (\int \frac {\sqrt {a^{2}-x^{2}}\, \sqrt {\mathit {acosh} \left (\frac {x}{a}\right )}\, \mathit {acosh} \left (\frac {x}{a}\right ) x^{4}}{a^{6}-3 a^{4} x^{2}+3 a^{2} x^{4}-x^{6}}d x \right ) a^{2}+5 \left (\int \frac {\sqrt {a^{2}-x^{2}}\, \sqrt {\mathit {acosh} \left (\frac {x}{a}\right )}\, \mathit {acosh} \left (\frac {x}{a}\right ) x^{4}}{a^{6}-3 a^{4} x^{2}+3 a^{2} x^{4}-x^{6}}d x \right ) x^{2}+10 \left (\int \frac {\sqrt {a^{2}-x^{2}}\, \sqrt {\mathit {acosh} \left (\frac {x}{a}\right )}\, \mathit {acosh} \left (\frac {x}{a}\right ) x^{2}}{a^{6}-3 a^{4} x^{2}+3 a^{2} x^{4}-x^{6}}d x \right ) a^{4}-10 \left (\int \frac {\sqrt {a^{2}-x^{2}}\, \sqrt {\mathit {acosh} \left (\frac {x}{a}\right )}\, \mathit {acosh} \left (\frac {x}{a}\right ) x^{2}}{a^{6}-3 a^{4} x^{2}+3 a^{2} x^{4}-x^{6}}d x \right ) a^{2} x^{2}}{5 a^{4} \left (a^{2}-x^{2}\right )} \] Input:
int(acosh(x/a)^(3/2)/(a^2-x^2)^(5/2),x)
Output:
(2*sqrt(a + x)*sqrt( - a + x)*sqrt(a**2 - x**2)*sqrt(acosh(x/a))*acosh(x/a )**2 - 5*int((sqrt(a**2 - x**2)*sqrt(acosh(x/a))*acosh(x/a)*x**4)/(a**6 - 3*a**4*x**2 + 3*a**2*x**4 - x**6),x)*a**2 + 5*int((sqrt(a**2 - x**2)*sqrt( acosh(x/a))*acosh(x/a)*x**4)/(a**6 - 3*a**4*x**2 + 3*a**2*x**4 - x**6),x)* x**2 + 10*int((sqrt(a**2 - x**2)*sqrt(acosh(x/a))*acosh(x/a)*x**2)/(a**6 - 3*a**4*x**2 + 3*a**2*x**4 - x**6),x)*a**4 - 10*int((sqrt(a**2 - x**2)*sqr t(acosh(x/a))*acosh(x/a)*x**2)/(a**6 - 3*a**4*x**2 + 3*a**2*x**4 - x**6),x )*a**2*x**2)/(5*a**4*(a**2 - x**2))