Integrand size = 24, antiderivative size = 24 \[ \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\frac {x^3}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {x}{2 a^5 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}-\frac {3}{2 a^6 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}+\frac {2}{a^6 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}+\frac {7 \sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{8 a^6 c^2 \sqrt {c+a^2 c x^2}}-\frac {9 \sqrt {1+a^2 x^2} \text {Si}(3 \arctan (a x))}{8 a^6 c^2 \sqrt {c+a^2 c x^2}}+\frac {\text {Int}\left (\frac {x}{\sqrt {c+a^2 c x^2} \arctan (a x)^3},x\right )}{a^4 c^2} \] Output:
1/2*x^3/a^3/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^2+1/2*x/a^5/c^2/(a^2*c*x^2+c )^(1/2)/arctan(a*x)^2-3/2/a^6/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)+2/a^6/c^2/ (a^2*c*x^2+c)^(1/2)/arctan(a*x)+7/8*(a^2*x^2+1)^(1/2)*Si(arctan(a*x))/a^6/ c^2/(a^2*c*x^2+c)^(1/2)-9/8*(a^2*x^2+1)^(1/2)*Si(3*arctan(a*x))/a^6/c^2/(a ^2*c*x^2+c)^(1/2)+Defer(Int)(x/(a^2*c*x^2+c)^(1/2)/arctan(a*x)^3,x)/a^4/c^ 2
Not integrable
Time = 7.51 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx \] Input:
Integrate[x^5/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]
Output:
Integrate[x^5/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3), x]
Not integrable
Time = 3.99 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 14, 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.
| \(\displaystyle \int \frac {x^5}{\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5499 |
| \(\displaystyle \frac {\int \frac {x^3}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}dx}{a^2 c}-\frac {\int \frac {x^3}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^3}dx}{a^2}\) |
| \(\Big \downarrow \) 5477 |
| \(\displaystyle \frac {\int \frac {x^3}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}dx}{a^2 c}-\frac {\frac {3 \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx}{2 a}-\frac {x^3}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 5499 |
| \(\displaystyle \frac {\frac {\int \frac {x}{\sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{a^2 c}-\frac {\int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}dx}{a^2}}{a^2 c}-\frac {\frac {3 \left (\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}dx}{a^2 c}-\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx}{a^2}\right )}{2 a}-\frac {x^3}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 5437 |
| \(\displaystyle \frac {\frac {\int \frac {x}{\sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{a^2 c}-\frac {\int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^3}dx}{a^2}}{a^2 c}-\frac {\frac {3 \left (\frac {-a \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-3 a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )}{2 a}-\frac {x^3}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 5477 |
| \(\displaystyle \frac {\frac {\int \frac {x}{\sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{a^2 c}-\frac {\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}dx}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {\frac {3 \left (\frac {-a \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-3 a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )}{2 a}-\frac {x^3}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 5437 |
| \(\displaystyle \frac {\frac {\int \frac {x}{\sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{a^2 c}-\frac {\frac {-a \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {\frac {3 \left (\frac {-a \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-3 a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )}{2 a}-\frac {x^3}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 5506 |
| \(\displaystyle \frac {\frac {\int \frac {x}{\sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{a^2 c}-\frac {\frac {-\frac {a \sqrt {a^2 x^2+1} \int \frac {x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}dx}{c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {\frac {3 \left (\frac {-\frac {a \sqrt {a^2 x^2+1} \int \frac {x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}dx}{c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 a \sqrt {a^2 x^2+1} \int \frac {x}{\left (a^2 x^2+1\right )^{5/2} \arctan (a x)}dx}{c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )}{2 a}-\frac {x^3}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 5505 |
| \(\displaystyle \frac {\frac {\int \frac {x}{\sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{a^2 c}-\frac {\frac {-\frac {\sqrt {a^2 x^2+1} \int \frac {a x}{\sqrt {a^2 x^2+1} \arctan (a x)}d\arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {\frac {3 \left (\frac {-\frac {\sqrt {a^2 x^2+1} \int \frac {a x}{\sqrt {a^2 x^2+1} \arctan (a x)}d\arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \frac {a x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )}{2 a}-\frac {x^3}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {x}{\sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{a^2 c}-\frac {\frac {-\frac {\sqrt {a^2 x^2+1} \int \frac {\sin (\arctan (a x))}{\arctan (a x)}d\arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {\frac {3 \left (\frac {-\frac {\sqrt {a^2 x^2+1} \int \frac {\sin (\arctan (a x))}{\arctan (a x)}d\arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \frac {a x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )}{2 a}-\frac {x^3}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle \frac {\frac {\int \frac {x}{\sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{a^2 c}-\frac {\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {\frac {3 \left (\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \frac {a x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )}{2 a}-\frac {x^3}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 4906 |
| \(\displaystyle \frac {\frac {\int \frac {x}{\sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{a^2 c}-\frac {\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {\frac {3 \left (\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \int \left (\frac {a x}{4 \sqrt {a^2 x^2+1} \arctan (a x)}+\frac {\sin (3 \arctan (a x))}{4 \arctan (a x)}\right )d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )}{2 a}-\frac {x^3}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\int \frac {x}{\sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{a^2 c}-\frac {\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {\frac {3 \left (\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \left (\frac {1}{4} \text {Si}(\arctan (a x))+\frac {1}{4} \text {Si}(3 \arctan (a x))\right )}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )}{2 a}-\frac {x^3}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
| \(\Big \downarrow \) 5560 |
| \(\displaystyle \frac {\frac {\int \frac {x}{\sqrt {a^2 c x^2+c} \arctan (a x)^3}dx}{a^2 c}-\frac {\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}}{a^2}}{a^2 c}-\frac {\frac {3 \left (\frac {-\frac {\sqrt {a^2 x^2+1} \text {Si}(\arctan (a x))}{a c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 \sqrt {a^2 x^2+1} \left (\frac {1}{4} \text {Si}(\arctan (a x))+\frac {1}{4} \text {Si}(3 \arctan (a x))\right )}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )}{2 a}-\frac {x^3}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\) |
Input:
Int[x^5/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]
Output:
$Aborted
Not integrable
Time = 3.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
\[\int \frac {x^{5}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{3}}d x\]
Input:
int(x^5/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x)
Output:
int(x^5/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x)
Not integrable
Time = 0.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.58 \[ \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {x^{5}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}} \,d x } \] Input:
integrate(x^5/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="fricas")
Output:
integral(sqrt(a^2*c*x^2 + c)*x^5/((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3 *x^2 + c^3)*arctan(a*x)^3), x)
Not integrable
Time = 9.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {x^{5}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \] Input:
integrate(x**5/(a**2*c*x**2+c)**(5/2)/atan(a*x)**3,x)
Output:
Integral(x**5/((c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**3), x)
Not integrable
Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {x^{5}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}} \,d x } \] Input:
integrate(x^5/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="maxima")
Output:
integrate(x^5/((a^2*c*x^2 + c)^(5/2)*arctan(a*x)^3), x)
Exception generated. \[ \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^5/(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
Not integrable
Time = 0.77 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {x^5}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \] Input:
int(x^5/(atan(a*x)^3*(c + a^2*c*x^2)^(5/2)),x)
Output:
int(x^5/(atan(a*x)^3*(c + a^2*c*x^2)^(5/2)), x)
Not integrable
Time = 0.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.38 \[ \int \frac {x^5}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\frac {\int \frac {x^{5}}{\sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3} a^{4} x^{4}+2 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3} a^{2} x^{2}+\sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3}}d x}{\sqrt {c}\, c^{2}} \] Input:
int(x^5/(a^2*c*x^2+c)^(5/2)/atan(a*x)^3,x)
Output:
int(x**5/(sqrt(a**2*x**2 + 1)*atan(a*x)**3*a**4*x**4 + 2*sqrt(a**2*x**2 + 1)*atan(a*x)**3*a**2*x**2 + sqrt(a**2*x**2 + 1)*atan(a*x)**3),x)/(sqrt(c)* c**2)