3.7.72 \(\int \frac {1}{x (c+a^2 c x^2)^{5/2} \arctan (a x)^3} \, dx\) [672]

3.7.72.1 Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\frac {a x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {a x}{2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}-\frac {\sqrt {c+a^2 c x^2}}{2 a c^3 x \arctan (a x)^2}+\frac {3}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {1}{2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {5 \sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{8 c^2 \sqrt {c+a^2 c x^2}}+\frac {9 \sqrt {1+a^2 x^2} \text {Si}(3 \arctan (a x))}{8 c^2 \sqrt {c+a^2 c x^2}}-\frac {\text {Int}\left (\frac {1}{x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2},x\right )}{2 a c^2} \]

output

1/2*a*x/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^2+3/2/c/(a^2*c*x^2+c)^(3/2)/arct 
an(a*x)+1/2*a*x/c^2/arctan(a*x)^2/(a^2*c*x^2+c)^(1/2)-1/2/c^2/arctan(a*x)/ 
(a^2*c*x^2+c)^(1/2)+5/8*Si(arctan(a*x))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c 
)^(1/2)+9/8*Si(3*arctan(a*x))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)-1/ 
2*(a^2*c*x^2+c)^(1/2)/a/c^3/x/arctan(a*x)^2-1/2*Unintegrable(1/x^2/arctan( 
a*x)^2/(a^2*c*x^2+c)^(1/2),x)/a/c^2
 

3.7.72.2 Mathematica [N/A]

Not integrable

Time = 4.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx \]

input

Integrate[1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]
 

output

Integrate[1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x]
 

3.7.72.3 Rubi [N/A]

Not integrable

Time = 5.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {5501, 5501, 5477, 5437, 5503, 5437, 5499, 5437, 5506, 5505, 3042, 3780, 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.

\(\displaystyle \int \frac {1}{x \arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 5501

\(\displaystyle \frac {\int \frac {1}{x \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}dx}{c}-a^2 \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^3}dx\)

\(\Big \downarrow \) 5501

\(\displaystyle \frac {\frac {\int \frac {1}{x \sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{c}-a^2 \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}dx}{c}-a^2 \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^3}dx\)

\(\Big \downarrow \) 5477

\(\displaystyle \frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{2 a}-\frac {\sqrt {a^2 c x^2+c}}{2 a c x \arctan (a x)^2}}{c}-a^2 \left (\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}dx}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^3}dx\)

\(\Big \downarrow \) 5437

\(\displaystyle \frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{2 a}-\frac {\sqrt {a^2 c x^2+c}}{2 a c x \arctan (a x)^2}}{c}-a^2 \left (\frac {-a \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^3}dx\)

\(\Big \downarrow \) 5503

\(\displaystyle \frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{2 a}-\frac {\sqrt {a^2 c x^2+c}}{2 a c x \arctan (a x)^2}}{c}-a^2 \left (\frac {-a \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx}{2 a}-a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx-\frac {x}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 5437

\(\displaystyle \frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{2 a}-\frac {\sqrt {a^2 c x^2+c}}{2 a c x \arctan (a x)^2}}{c}-a^2 \left (\frac {-a \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx+\frac {-3 a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 5499

\(\displaystyle \frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{2 a}-\frac {\sqrt {a^2 c x^2+c}}{2 a c x \arctan (a x)^2}}{c}-a^2 \left (\frac {-a \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-a \left (\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}dx}{a^2 c}-\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx}{a^2}\right )+\frac {-3 a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 5437

\(\displaystyle \frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{2 a}-\frac {\sqrt {a^2 c x^2+c}}{2 a c x \arctan (a x)^2}}{c}-a^2 \left (\frac {-a \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {-3 a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{2 a}-a \left (\frac {-a \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-3 a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )-\frac {x}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 5506

\(\displaystyle \frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{2 a}-\frac {\sqrt {a^2 c x^2+c}}{2 a c x \arctan (a x)^2}}{c}-a^2 \left (\frac {-\frac {a \sqrt {a^2 x^2+1} \int \frac {x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}dx}{c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {-\frac {3 a \sqrt {a^2 x^2+1} \int \frac {x}{\left (a^2 x^2+1\right )^{5/2} \arctan (a x)}dx}{c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{2 a}-a \left (\frac {-\frac {a \sqrt {a^2 x^2+1} \int \frac {x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}dx}{c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 a \sqrt {a^2 x^2+1} \int \frac {x}{\left (a^2 x^2+1\right )^{5/2} \arctan (a x)}dx}{c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )-\frac {x}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 5505

\(\displaystyle \frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{2 a}-\frac {\sqrt {a^2 c x^2+c}}{2 a c x \arctan (a x)^2}}{c}-a^2 \left (\frac {-\frac {\sqrt {a^2 x^2+1} \int \frac {a x}{\sqrt {a^2 x^2+1} \arctan (a x)}d\arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \frac {a x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{2 a}-a \left (\frac {-\frac {\sqrt {a^2 x^2+1} \int \frac {a x}{\sqrt {a^2 x^2+1} \arctan (a x)}d\arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \frac {a x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )-\frac {x}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{2 a}-\frac {\sqrt {a^2 c x^2+c}}{2 a c x \arctan (a x)^2}}{c}-a^2 \left (\frac {-\frac {\sqrt {a^2 x^2+1} \int \frac {\sin (\arctan (a x))}{\arctan (a x)}d\arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \frac {a x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{2 a}-a \left (\frac {-\frac {\sqrt {a^2 x^2+1} \int \frac {\sin (\arctan (a x))}{\arctan (a x)}d\arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \frac {a x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )-\frac {x}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 3780

\(\displaystyle \frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{2 a}-\frac {\sqrt {a^2 c x^2+c}}{2 a c x \arctan (a x)^2}}{c}-a^2 \left (\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-a \left (\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \frac {a x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )+\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \frac {a x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{2 a}-\frac {\sqrt {a^2 c x^2+c}}{2 a c x \arctan (a x)^2}}{c}-a^2 \left (\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-a \left (\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \left (\frac {a x}{4 \sqrt {a^2 x^2+1} \arctan (a x)}+\frac {\sin (3 \arctan (a x))}{4 \arctan (a x)}\right )d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )+\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \left (\frac {a x}{4 \sqrt {a^2 x^2+1} \arctan (a x)}+\frac {\sin (3 \arctan (a x))}{4 \arctan (a x)}\right )d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{2 a}-\frac {\sqrt {a^2 c x^2+c}}{2 a c x \arctan (a x)^2}}{c}-a^2 \left (\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-a \left (\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \left (\frac {1}{4} \text {Si}(\arctan (a x))+\frac {1}{4} \text {Si}(3 \arctan (a x))\right )}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )+\frac {-\frac {3 \sqrt {a^2 x^2+1} \left (\frac {1}{4} \text {Si}(\arctan (a x))+\frac {1}{4} \text {Si}(3 \arctan (a x))\right )}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 5560

\(\displaystyle \frac {\frac {-\frac {\int \frac {1}{x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}dx}{2 a}-\frac {\sqrt {a^2 c x^2+c}}{2 a c x \arctan (a x)^2}}{c}-a^2 \left (\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (-a \left (\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \left (\frac {1}{4} \text {Si}(\arctan (a x))+\frac {1}{4} \text {Si}(3 \arctan (a x))\right )}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )+\frac {-\frac {3 \sqrt {a^2 x^2+1} \left (\frac {1}{4} \text {Si}(\arctan (a x))+\frac {1}{4} \text {Si}(3 \arctan (a x))\right )}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\right )\)

input

Int[1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]
 

output

$Aborted
 

3.7.72.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

rule 4906

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]
 

rule 5437

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_S 
ymbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 
 1))), x] - Simp[2*c*((q + 1)/(b*(p + 1)))   Int[x*(d + e*x^2)^q*(a + b*Arc 
Tan[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && 
LtQ[q, -1] && LtQ[p, -1]
 

rule 5477

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 
]
 

rule 5499

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 
)^(q_), x_Symbol] :> Simp[1/e   Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*Ar 
cTan[c*x])^p, x], x] - Simp[d/e   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] && IGtQ[m, 1] && NeQ[p, -1]
 

rule 5501

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]
 

rule 5503

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]
 

rule 5505

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])
 

rule 5506

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])
 

rule 5560

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]])
 

3.7.72.4 Maple [N/A] (verified)

Not integrable

Time = 14.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

\[\int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{3}}d x\]

input

int(1/x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x)
 

output

int(1/x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x)
 

3.7.72.5 Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.54 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x \arctan \left (a x\right )^{3}} \,d x } \]

input

integrate(1/x/(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^7 + 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 
 + c^3*x)*arctan(a*x)^3), x)
 

3.7.72.6 Sympy [N/A]

Not integrable

Time = 21.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {1}{x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \]

input

integrate(1/x/(a**2*c*x**2+c)**(5/2)/atan(a*x)**3,x)
 

output

Integral(1/(x*(c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**3), x)
 

3.7.72.7 Maxima [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x \arctan \left (a x\right )^{3}} \,d x } \]

input

integrate(1/x/(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*arctan(a*x)^3), x)
 

3.7.72.8 Giac [F(-2)]

Exception generated. \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\text {Exception raised: TypeError} \]

input

integrate(1/x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="giac")
 

output

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
 

3.7.72.9 Mupad [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

input

int(1/(x*atan(a*x)^3*(c + a^2*c*x^2)^(5/2)),x)
 

output

int(1/(x*atan(a*x)^3*(c + a^2*c*x^2)^(5/2)), x)