Integrand size = 26, antiderivative size = 26 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=-\frac {2}{3 a c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}}+\frac {16}{3 c \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c x^2 \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}+\frac {4 \sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {4 \sqrt {\frac {2 \pi }{3}} \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {8 \text {Int}\left (\frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}},x\right )}{3 a^2}+\frac {20}{3} \text {Int}\left (\frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}},x\right ) \] Output:
-2/3/a/c/x/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(3/2)+16/3/c/(a^2*c*x^2+c)^(3/2 )/arctan(a*x)^(1/2)+4/3/a^2/c/x^2/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2)+4* 2^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)*FresnelS(2^(1/2)/Pi^(1/2)*arctan(a*x)^( 1/2))/c^2/(a^2*c*x^2+c)^(1/2)+4/3*6^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)*Fresn elS(6^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))/c^2/(a^2*c*x^2+c)^(1/2)+8/3*Defer( Int)(1/x^3/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(1/2),x)/a^2+20/3*Defer(Int)(1/ x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(1/2),x)
Not integrable
Time = 8.74 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx \] Input:
Integrate[1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]
Output:
Integrate[1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)), x]
Not integrable
Time = 2.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 9, 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.
\(\displaystyle \int \frac {1}{x \arctan (a x)^{5/2} \left (a^2 c x^2+c\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 5503 |
\(\displaystyle -\frac {8}{3} a \int \frac {1}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^{3/2}}dx-\frac {2 \int \frac {1}{x^2 \left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^{3/2}}dx}{3 a}-\frac {2}{3 a c x \arctan (a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}}\) |
\(\Big \downarrow \) 5437 |
\(\displaystyle -\frac {2 \int \frac {1}{x^2 \left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^{3/2}}dx}{3 a}-\frac {8}{3} a \left (-6 a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx-\frac {2}{a c \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {2}{3 a c x \arctan (a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}}\) |
\(\Big \downarrow \) 5503 |
\(\displaystyle -\frac {8}{3} a \left (-6 a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx-\frac {2}{a c \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {2 \left (-10 a \int \frac {1}{x \left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx-\frac {4 \int \frac {1}{x^3 \left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx}{a}-\frac {2}{a c x^2 \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}\right )}{3 a}-\frac {2}{3 a c x \arctan (a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}}\) |
\(\Big \downarrow \) 5506 |
\(\displaystyle -\frac {8}{3} a \left (-\frac {6 a \sqrt {a^2 x^2+1} \int \frac {x}{\left (a^2 x^2+1\right )^{5/2} \sqrt {\arctan (a x)}}dx}{c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{a c \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {2 \left (-10 a \int \frac {1}{x \left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx-\frac {4 \int \frac {1}{x^3 \left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx}{a}-\frac {2}{a c x^2 \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}\right )}{3 a}-\frac {2}{3 a c x \arctan (a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}}\) |
\(\Big \downarrow \) 5505 |
\(\displaystyle -\frac {8}{3} a \left (-\frac {6 \sqrt {a^2 x^2+1} \int \frac {a x}{\left (a^2 x^2+1\right )^{3/2} \sqrt {\arctan (a x)}}d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{a c \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {2 \left (-10 a \int \frac {1}{x \left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx-\frac {4 \int \frac {1}{x^3 \left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx}{a}-\frac {2}{a c x^2 \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}\right )}{3 a}-\frac {2}{3 a c x \arctan (a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle -\frac {8}{3} a \left (-\frac {6 \sqrt {a^2 x^2+1} \int \left (\frac {a x}{4 \sqrt {a^2 x^2+1} \sqrt {\arctan (a x)}}+\frac {\sin (3 \arctan (a x))}{4 \sqrt {\arctan (a x)}}\right )d\arctan (a x)}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{a c \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {2 \left (-10 a \int \frac {1}{x \left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx-\frac {4 \int \frac {1}{x^3 \left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx}{a}-\frac {2}{a c x^2 \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}\right )}{3 a}-\frac {2}{3 a c x \arctan (a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 \left (-10 a \int \frac {1}{x \left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx-\frac {4 \int \frac {1}{x^3 \left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx}{a}-\frac {2}{a c x^2 \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}\right )}{3 a}-\frac {8}{3} a \left (-\frac {6 \sqrt {a^2 x^2+1} \left (\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{a c \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {2}{3 a c x \arctan (a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}}\) |
\(\Big \downarrow \) 5560 |
\(\displaystyle -\frac {2 \left (-10 a \int \frac {1}{x \left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx-\frac {4 \int \frac {1}{x^3 \left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx}{a}-\frac {2}{a c x^2 \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}\right )}{3 a}-\frac {8}{3} a \left (-\frac {6 \sqrt {a^2 x^2+1} \left (\frac {1}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelS}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{a c \sqrt {\arctan (a x)} \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {2}{3 a c x \arctan (a x)^{3/2} \left (a^2 c x^2+c\right )^{3/2}}\) |
Input:
Int[1/(x*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^(5/2)),x]
Output:
$Aborted
Not integrable
Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85
\[\int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{\frac {5}{2}}}d x\]
Input:
int(1/x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(5/2),x)
Output:
int(1/x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(5/2),x)
Exception generated. \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(5/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(1/x/(a**2*c*x**2+c)**(5/2)/atan(a*x)**(5/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(5/2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Exception generated. \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(5/2),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 = 0.62 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \] Input:
int(1/(x*atan(a*x)^(5/2)*(c + a^2*c*x^2)^(5/2)),x)
Output:
int(1/(x*atan(a*x)^(5/2)*(c + a^2*c*x^2)^(5/2)), x)
Not integrable
Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.92 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{5/2}} \, dx=\frac {\sqrt {c}\, \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {atan} \left (a x \right )}}{\mathit {atan} \left (a x \right )^{3} a^{6} x^{7}+3 \mathit {atan} \left (a x \right )^{3} a^{4} x^{5}+3 \mathit {atan} \left (a x \right )^{3} a^{2} x^{3}+\mathit {atan} \left (a x \right )^{3} x}d x \right )}{c^{3}} \] Input:
int(1/x/(a^2*c*x^2+c)^(5/2)/atan(a*x)^(5/2),x)
Output:
(sqrt(c)*int((sqrt(a**2*x**2 + 1)*sqrt(atan(a*x)))/(atan(a*x)**3*a**6*x**7 + 3*atan(a*x)**3*a**4*x**5 + 3*atan(a*x)**3*a**2*x**3 + atan(a*x)**3*x),x ))/c**3