Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=-\frac {a^3 x}{c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {2 a^3 x}{c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {2 a \sqrt {c+a^2 c x^2}}{c^3 x \arctan (a x)}+\frac {9 a^2 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{4 c^2 \sqrt {c+a^2 c x^2}}+\frac {3 a^2 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(3 \arctan (a x))}{4 c^2 \sqrt {c+a^2 c x^2}}+\frac {\text {Int}\left (\frac {1}{x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2},x\right )}{c^2}+\frac {2 a \text {Int}\left (\frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)},x\right )}{c^2} \]
-a^3*x/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)-2*a^3*x/c^2/arctan(a*x)/(a^2*c*x^ 2+c)^(1/2)+9/4*a^2*Ci(arctan(a*x))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/ 2)+3/4*a^2*Ci(3*arctan(a*x))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)+2*a *(a^2*c*x^2+c)^(1/2)/c^3/x/arctan(a*x)+Unintegrable(1/x^3/arctan(a*x)^2/(a ^2*c*x^2+c)^(1/2),x)/c^2+2*a*Unintegrable(1/x^2/arctan(a*x)/(a^2*c*x^2+c)^ (1/2),x)/c^2
Not integrable
Time = 6.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx \]
Not integrable
Time = 8.46 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {5501, 5501, 5501, 5477, 5440, 5439, 3042, 3783, 5503, 5440, 5439, 3042, 3793, 2009, 5506, 5505, 4906, 2009, 5560}
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.
| \(\displaystyle \int \frac {1}{x^3 \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\int \frac {1}{x^3 \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}dx}{c}-a^2 \int \frac {1}{x \left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \int \frac {1}{x \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}dx}{c}-a^2 \left (\frac {\int \frac {1}{x \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}dx}{c}-a^2 \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx\right )\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \left (\frac {\int \frac {1}{x \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}dx\right )}{c}-a^2 \left (\frac {\frac {\int \frac {1}{x \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}dx}{c}-a^2 \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx\right )\) |
| \(\Big \downarrow \) 5477 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \left (\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}dx}{a}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}dx}{a}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx\right )\) |
| \(\Big \downarrow \) 5440 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \left (\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \int \frac {1}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}dx}{a c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \int \frac {1}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}dx}{a c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx\right )\) |
| \(\Big \downarrow \) 5439 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \left (\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \int \frac {1}{\sqrt {a^2 x^2+1} \arctan (a x)}d\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \int \frac {1}{\sqrt {a^2 x^2+1} \arctan (a x)}d\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \left (\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )}{\arctan (a x)}d\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )}{\arctan (a x)}d\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx\right )\) |
| \(\Big \downarrow \) 3783 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \left (\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx\right )\) |
| \(\Big \downarrow \) 5503 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \left (\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx}{a}-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
| \(\Big \downarrow \) 5440 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \left (\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \int \frac {1}{\left (a^2 x^2+1\right )^{5/2} \arctan (a x)}dx}{a c^2 \sqrt {a^2 c x^2+c}}-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
| \(\Big \downarrow \) 5439 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \left (\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \int \frac {1}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a^2 c^2 \sqrt {a^2 c x^2+c}}-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \left (\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )^3}{\arctan (a x)}d\arctan (a x)}{a^2 c^2 \sqrt {a^2 c x^2+c}}-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \left (\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \int \left (\frac {\cos (3 \arctan (a x))}{4 \arctan (a x)}+\frac {3}{4 \sqrt {a^2 x^2+1} \arctan (a x)}\right )d\arctan (a x)}{a^2 c^2 \sqrt {a^2 c x^2+c}}-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \left (\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
| \(\Big \downarrow \) 5506 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \left (\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {2 a \sqrt {a^2 x^2+1} \int \frac {x^2}{\left (a^2 x^2+1\right )^{5/2} \arctan (a x)}dx}{c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
| \(\Big \downarrow \) 5505 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \left (\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {2 \sqrt {a^2 x^2+1} \int \frac {a^2 x^2}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \left (\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {2 \sqrt {a^2 x^2+1} \int \left (\frac {1}{4 \sqrt {a^2 x^2+1} \arctan (a x)}-\frac {\cos (3 \arctan (a x))}{4 \arctan (a x)}\right )d\arctan (a x)}{a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \left (\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {2 \sqrt {a^2 x^2+1} \left (\frac {1}{4} \operatorname {CosIntegral}(\arctan (a x))-\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
| \(\Big \downarrow \) 5560 |
| \(\displaystyle \frac {\frac {\int \frac {1}{x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{c}-a^2 \left (\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )\right )}{c}-a^2 \left (\frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)}dx}{a}-\frac {\sqrt {a^2 c x^2+c}}{a c x \arctan (a x)}}{c}-a^2 \left (\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-\frac {2 \sqrt {a^2 x^2+1} \left (\frac {1}{4} \operatorname {CosIntegral}(\arctan (a x))-\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )\right )\) |
3.6.95.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
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[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ Symbol] :> Simp[d^q/c Subst[Int[(a + b*x)^p/Cos[x]^(2*(q + 1)), x], x, Ar cTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*( q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ Symbol] :> Simp[d^(q + 1/2)*(Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]) Int[(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*(q + 1), 0] && !(IntegerQ[q] || GtQ[d, 0])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_.)*((d_) + (e_. )*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x^2)^(q + 1)*((a + b*ArcT an[c*x])^(p + 1)/(b*c*d*(p + 1))), x] - Simp[f*(m/(b*c*(p + 1))) Int[(f*x )^(m - 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 2, 0] && LtQ[p, -1 ]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 )^(q_), x_Symbol] :> Simp[1/d Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c *x])^p, x], x] - Simp[e/d Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2* q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[x^m*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1))), x] + (-Simp[c*((m + 2*q + 2)/(b*(p + 1))) Int[x^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x] - Simp[m/(b*c*(p + 1)) Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x]) /; F reeQ[{a, b, c, d, e, m}, x] && EqQ[e, c^2*d] && IntegerQ[m] && LtQ[q, -1] & & LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[d^q/c^(m + 1) Subst[Int[(a + b*x)^p*(Sin[x]^m/ Cos[x]^(m + 2*(q + 1))), x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p }, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q ] || GtQ[d, 0])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[d^(q + 1/2)*(Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]) Int[x^m*(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && !(I ntegerQ[q] || GtQ[d, 0])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Unintegrab le[u*(a + b*ArcTan[c*x])^p, x] /; FreeQ[{a, b, c, p}, x] && (EqQ[u, 1] || M atchQ[u, ((d_.) + (e_.)*x)^(q_.) /; FreeQ[{d, e, q}, x]] || MatchQ[u, ((f_. )*x)^(m_.)*((d_.) + (e_.)*x)^(q_.) /; FreeQ[{d, e, f, m, q}, x]] || MatchQ[ u, ((d_.) + (e_.)*x^2)^(q_.) /; FreeQ[{d, e, q}, x]] || MatchQ[u, ((f_.)*x) ^(m_.)*((d_.) + (e_.)*x^2)^(q_.) /; FreeQ[{d, e, f, m, q}, x]])
Not integrable
Time = 21.62 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {1}{x^{3} \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{2}}d x\]
Not integrable
Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.62 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{3} \arctan \left (a x\right )^{2}} \,d x } \]
integral(sqrt(a^2*c*x^2 + c)/((a^6*c^3*x^9 + 3*a^4*c^3*x^7 + 3*a^2*c^3*x^5 + c^3*x^3)*arctan(a*x)^2), x)
Not integrable
Time = 24.58 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {1}{x^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}}\, dx \]
Not integrable
Time = 0.37 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{3} \arctan \left (a x\right )^{2}} \,d x } \]
Exception generated. \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\text {Exception raised: TypeError} \]
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
Not integrable
Time = 0.49 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx=\int \frac {1}{x^3\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]