\(\int \frac {1}{x (c+a^2 c x^2)^3 \arctan (a x)^{5/2}} \, dx\) [1072]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=-\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {5 \sqrt {2 \pi } \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{3 c^3}+\frac {20 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{3 c^3}+\frac {8 \text {Int}\left (\frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}},x\right )}{3 a^2}+8 \text {Int}\left (\frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}},x\right ) \]

[Out]

-2/3/a/c^3/x/(a^2*x^2+1)^2/arctan(a*x)^(3/2)+20/3*FresnelS(2*arctan(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/c^3+5/3*Fres
nelS(2*2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/c^3+20/3/c^3/(a^2*x^2+1)^2/arctan(a*x)^(1/2)+4/3/a
^2/c^3/x^2/(a^2*x^2+1)^2/arctan(a*x)^(1/2)+8/3*Unintegrable(1/x^3/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x)/a^2+8*U
nintegrable(1/x/(a^2*c*x^2+c)^3/arctan(a*x)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx \]

[In]

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

[Out]

-2/(3*a*c^3*x*(1 + a^2*x^2)^2*ArcTan[a*x]^(3/2)) + 20/(3*c^3*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + 4/(3*a^2*c^3
*x^2*(1 + a^2*x^2)^2*Sqrt[ArcTan[a*x]]) + (5*Sqrt[2*Pi]*FresnelS[2*Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(3*c^3) + (2
0*Sqrt[Pi]*FresnelS[(2*Sqrt[ArcTan[a*x]])/Sqrt[Pi]])/(3*c^3) + (8*Defer[Int][1/(x^3*(c + a^2*c*x^2)^3*Sqrt[Arc
Tan[a*x]]), x])/(3*a^2) + 8*Defer[Int][1/(x*(c + a^2*c*x^2)^3*Sqrt[ArcTan[a*x]]), x]

Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}-\frac {2 \int \frac {1}{x^2 \left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx}{3 a}-\frac {1}{3} (10 a) \int \frac {1}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^{3/2}} \, dx \\ & = -\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+8 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{3 a^2}+\frac {1}{3} \left (80 a^2\right ) \int \frac {x}{\left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx \\ & = -\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+8 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{3 a^2}+\frac {80 \text {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 c^3} \\ & = -\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+8 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{3 a^2}+\frac {80 \text {Subst}\left (\int \left (\frac {\sin (2 x)}{4 \sqrt {x}}+\frac {\sin (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\arctan (a x)\right )}{3 c^3} \\ & = -\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+8 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{3 a^2}+\frac {10 \text {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 c^3}+\frac {20 \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arctan (a x)\right )}{3 c^3} \\ & = -\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+8 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{3 a^2}+\frac {20 \text {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{3 c^3}+\frac {40 \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arctan (a x)}\right )}{3 c^3} \\ & = -\frac {2}{3 a c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)^{3/2}}+\frac {20}{3 c^3 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {4}{3 a^2 c^3 x^2 \left (1+a^2 x^2\right )^2 \sqrt {\arctan (a x)}}+\frac {5 \sqrt {2 \pi } \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{3 c^3}+\frac {20 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{3 c^3}+8 \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx+\frac {8 \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^3 \sqrt {\arctan (a x)}} \, dx}{3 a^2} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.52 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 28.97 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.71 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\frac {\int \frac {1}{a^{6} x^{7} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 3 a^{4} x^{5} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + 3 a^{2} x^{3} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + x \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c^{3}} \]

[In]

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

[Out]

Integral(1/(a**6*x**7*atan(a*x)**(5/2) + 3*a**4*x**5*atan(a*x)**(5/2) + 3*a**2*x**3*atan(a*x)**(5/2) + x*atan(
a*x)**(5/2)), x)/c**3

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 0.67 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^3 \arctan (a x)^{5/2}} \, dx=\int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \]

[In]

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

[Out]

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