\(\int \frac {x^2}{(c+a^2 c x^2)^{3/2} \arctan (a x)^{3/2}} \, dx\) [1026]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=-\frac {2 x^2}{a c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}}+\frac {4 \sqrt {2 \pi } \sqrt {1+a^2 x^2} \operatorname {FresnelS}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}+2 a \text {Int}\left (\frac {x^3}{\left (c+a^2 c x^2\right )^{3/2} \sqrt {\arctan (a x)}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

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

[In]

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

[Out]

(-2*x^2)/(a*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]]) + (4*Sqrt[2*Pi]*Sqrt[1 + a^2*x^2]*FresnelS[Sqrt[2/Pi]*Sqr
t[ArcTan[a*x]]])/(a^3*c*Sqrt[c + a^2*c*x^2]) + 2*a*Defer[Int][x^3/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcTan[a*x]]), x
]

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

Mathematica [N/A]

Not integrable

Time = 3.66 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 3.99 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(x^2/(a^2*c*x^2+c)^(3/2)/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 = 32.65 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\int \frac {x^{2}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 209.27 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

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

[In]

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

[Out]

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