\(\int \frac {x^3}{(c+a^2 c x^2)^2 \arctan (a x)^{3/2}} \, dx\) [990]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=-\frac {2 x^3}{a c^2 \left (1+a^2 x^2\right ) \sqrt {\arctan (a x)}}+\frac {6 \sqrt {\arctan (a x)}}{a^4 c^2}-\frac {3 \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arctan (a x)}}{\sqrt {\pi }}\right )}{a^4 c^2}+2 a \text {Int}\left (\frac {x^4}{\left (c+a^2 c x^2\right )^2 \sqrt {\arctan (a x)}},x\right ) \]

[Out]

-3*FresnelC(2*arctan(a*x)^(1/2)/Pi^(1/2))*Pi^(1/2)/a^4/c^2-2*x^3/a/c^2/(a^2*x^2+1)/arctan(a*x)^(1/2)+6*arctan(
a*x)^(1/2)/a^4/c^2+2*a*Unintegrable(x^4/(a^2*c*x^2+c)^2/arctan(a*x)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.18 (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 = {} \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=\int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx \]

[In]

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

[Out]

(-2*x^3)/(a*c^2*(1 + a^2*x^2)*Sqrt[ArcTan[a*x]]) + (6*Sqrt[ArcTan[a*x]])/(a^4*c^2) - (3*Sqrt[Pi]*FresnelC[(2*S
qrt[ArcTan[a*x]])/Sqrt[Pi]])/(a^4*c^2) + 2*a*Defer[Int][x^4/((c + a^2*c*x^2)^2*Sqrt[ArcTan[a*x]]), x]

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

Mathematica [N/A]

Not integrable

Time = 6.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=\int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 5.51 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x^3/(a^2*c*x^2+c)^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 [N/A]

Not integrable

Time = 3.83 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=\frac {\int \frac {x^{3}}{a^{4} x^{4} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + 2 a^{2} x^{2} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )} + \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx}{c^{2}} \]

[In]

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

[Out]

Integral(x**3/(a**4*x**4*atan(a*x)**(3/2) + 2*a**2*x**2*atan(a*x)**(3/2) + atan(a*x)**(3/2)), x)/c**2

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^2 \arctan (a x)^{3/2}} \, dx=\int \frac {x^3}{{\mathrm {atan}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

[In]

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

[Out]

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