Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\frac {a}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {a}{2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}-\frac {3 a^2 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {a^2 x}{2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {7 a \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{8 c^2 \sqrt {c+a^2 c x^2}}+\frac {9 a \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(3 \arctan (a x))}{8 c^2 \sqrt {c+a^2 c x^2}}+\frac {\text {Int}\left (\frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3},x\right )}{c^2} \] Output:
1/2*a/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^2+1/2*a/c^2/(a^2*c*x^2+c)^(1/2)/ar ctan(a*x)^2-3/2*a^2*x/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)-1/2*a^2*x/c^2/(a^2 *c*x^2+c)^(1/2)/arctan(a*x)+7/8*a*(a^2*x^2+1)^(1/2)*Ci(arctan(a*x))/c^2/(a ^2*c*x^2+c)^(1/2)+9/8*a*(a^2*x^2+1)^(1/2)*Ci(3*arctan(a*x))/c^2/(a^2*c*x^2 +c)^(1/2)+Defer(Int)(1/x^2/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^3,x)/c^2
Not integrable
Time = 3.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx \] Input:
Integrate[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]
Output:
Integrate[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x]
Not integrable
Time = 5.59 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 21, 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.
| \(\displaystyle \int \frac {1}{x^2 \arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\int \frac {1}{x^2 \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}dx}{c}-a^2 \int \frac {1}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^3}dx\) |
| \(\Big \downarrow \) 5437 |
| \(\displaystyle \frac {\int \frac {1}{x^2 \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {3}{2} a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \int \frac {1}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {3}{2} a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 5437 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {1}{2} a \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}dx-\frac {1}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {3}{2} a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 5477 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {1}{2} a \left (\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}dx}{a}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )-\frac {1}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {3}{2} a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 5440 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {1}{2} a \left (\frac {\sqrt {a^2 x^2+1} \int \frac {1}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}dx}{a c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )-\frac {1}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {3}{2} a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 5439 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {1}{2} a \left (\frac {\sqrt {a^2 x^2+1} \int \frac {1}{\sqrt {a^2 x^2+1} \arctan (a x)}d\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )-\frac {1}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {3}{2} a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {1}{2} a \left (\frac {\sqrt {a^2 x^2+1} \int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )}{\arctan (a x)}d\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )-\frac {1}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {3}{2} a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 3783 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {1}{2} a \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )-\frac {1}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {3}{2} a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 5503 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {1}{2} a \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )-\frac {1}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {3}{2} a \left (\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx}{a}-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 5440 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {1}{2} a \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )-\frac {1}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {3}{2} a \left (\frac {\sqrt {a^2 x^2+1} \int \frac {1}{\left (a^2 x^2+1\right )^{5/2} \arctan (a x)}dx}{a c^2 \sqrt {a^2 c x^2+c}}-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 5439 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {1}{2} a \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )-\frac {1}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {3}{2} a \left (\frac {\sqrt {a^2 x^2+1} \int \frac {1}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a^2 c^2 \sqrt {a^2 c x^2+c}}-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {1}{2} a \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )-\frac {1}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {3}{2} a \left (\frac {\sqrt {a^2 x^2+1} \int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )^3}{\arctan (a x)}d\arctan (a x)}{a^2 c^2 \sqrt {a^2 c x^2+c}}-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {1}{2} a \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )-\frac {1}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {3}{2} a \left (\frac {\sqrt {a^2 x^2+1} \int \left (\frac {\cos (3 \arctan (a x))}{4 \arctan (a x)}+\frac {3}{4 \sqrt {a^2 x^2+1} \arctan (a x)}\right )d\arctan (a x)}{a^2 c^2 \sqrt {a^2 c x^2+c}}-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {1}{2} a \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )-\frac {1}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {3}{2} a \left (-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 5506 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {1}{2} a \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )-\frac {1}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {3}{2} a \left (-\frac {2 a \sqrt {a^2 x^2+1} \int \frac {x^2}{\left (a^2 x^2+1\right )^{5/2} \arctan (a x)}dx}{c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 5505 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {1}{2} a \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )-\frac {1}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {3}{2} a \left (-\frac {2 \sqrt {a^2 x^2+1} \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {1}{2} a \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )-\frac {1}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {3}{2} a \left (-\frac {2 \sqrt {a^2 x^2+1} \int \left (\frac {1}{4 \sqrt {a^2 x^2+1} \arctan (a x)}-\frac {\cos (3 \arctan (a x))}{4 \arctan (a x)}\right )d\arctan (a x)}{a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {1}{2} a \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )-\frac {1}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {3}{2} a \left (-\frac {2 \sqrt {a^2 x^2+1} \left (\frac {1}{4} \operatorname {CosIntegral}(\arctan (a x))-\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 5560 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \left (-\frac {1}{2} a \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )-\frac {1}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {3}{2} a \left (-\frac {2 \sqrt {a^2 x^2+1} \left (\frac {1}{4} \operatorname {CosIntegral}(\arctan (a x))-\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
Input:
Int[1/(x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]
Output:
$Aborted
Not integrable
Time = 2.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {1}{x^{2} \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{3}}d x\]
Input:
int(1/x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x)
Output:
int(1/x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x)
Not integrable
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.62 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )^{3}} \,d x } \] Input:
integrate(1/x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="fricas")
Output:
integral(sqrt(a^2*c*x^2 + c)/((a^6*c^3*x^8 + 3*a^4*c^3*x^6 + 3*a^2*c^3*x^4 + c^3*x^2)*arctan(a*x)^3), x)
Not integrable
Time = 20.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {1}{x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \] Input:
integrate(1/x**2/(a**2*c*x**2+c)**(5/2)/atan(a*x)**3,x)
Output:
Integral(1/(x**2*(c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**3), x)
Not integrable
Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )^{3}} \,d x } \] Input:
integrate(1/x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="maxima")
Output:
integrate(1/((a^2*c*x^2 + c)^(5/2)*x^2*arctan(a*x)^3), x)
Not integrable
Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )^{3}} \,d x } \] Input:
integrate(1/x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="giac")
Output:
integrate(1/((a^2*c*x^2 + c)^(5/2)*x^2*arctan(a*x)^3), x)
Not integrable
Time = 0.76 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {1}{x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \] Input:
int(1/(x^2*atan(a*x)^3*(c + a^2*c*x^2)^(5/2)),x)
Output:
int(1/(x^2*atan(a*x)^3*(c + a^2*c*x^2)^(5/2)), x)
Not integrable
Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.33 \[ \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\frac {\int \frac {1}{\sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3} a^{4} x^{6}+2 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3} a^{2} x^{4}+\sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3} x^{2}}d x}{\sqrt {c}\, c^{2}} \] Input:
int(1/x^2/(a^2*c*x^2+c)^(5/2)/atan(a*x)^3,x)
Output:
int(1/(sqrt(a**2*x**2 + 1)*atan(a*x)**3*a**4*x**6 + 2*sqrt(a**2*x**2 + 1)* atan(a*x)**3*a**2*x**4 + sqrt(a**2*x**2 + 1)*atan(a*x)**3*x**2),x)/(sqrt(c )*c**2)