\(\int \frac {1}{x (c+a^2 c x^2)^{5/2} \arctan (a x)^{3/2}} \, dx\) [1038]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx=-\frac {2}{a c x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}-\frac {6 \sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {\frac {2 \pi }{3}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \text {Int}\left (\frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}},x\right )}{a} \]

[Out]

-2/3*FresnelC(6^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*6^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)-6
*FresnelC(2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*2^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)-2/a/c
/x/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^(1/2)-2*Unintegrable(1/x^2/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^(1/2),x)/a

Rubi [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx \]

[In]

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

[Out]

-2/(a*c*x*(c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]) - (6*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[
ArcTan[a*x]]])/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Sqrt[(2*Pi)/3]*Sqrt[1 + a^2*x^2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan
[a*x]]])/(c^2*Sqrt[c + a^2*c*x^2]) - (2*Defer[Int][1/(x^2*(c + a^2*c*x^2)^(5/2)*Sqrt[ArcTan[a*x]]), x])/a

Rubi steps \begin{align*} \text {integral}& = -\frac {2}{a c x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{a}-(8 a) \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx \\ & = -\frac {2}{a c x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{a}-\frac {\left (8 a \sqrt {1+a^2 x^2}\right ) \int \frac {1}{\left (1+a^2 x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2}{a c x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{a}-\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos ^3(x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2}{a c x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{a}-\frac {\left (8 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {3 \cos (x)}{4 \sqrt {x}}+\frac {\cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2}{a c x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{a}-\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2}{a c x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{a}-\frac {\left (4 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (12 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {2}{a c x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}}-\frac {6 \sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {\frac {2 \pi }{3}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^{5/2} \sqrt {\arctan (a x)}} \, dx}{a} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 6.84 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 2.74 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^{3/2}} \, dx=\int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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