\(\int \frac {x^2}{(c+a^2 c x^2)^{3/2} \arctan (a x)^3} \, dx\) [659]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.26 (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 = {} \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=\int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 5.52 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=\int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 2.98 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.12 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

integral(sqrt(a^2*c*x^2 + c)*x^2/((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*arctan(a*x)^3), x)

Sympy [N/A]

Not integrable

Time = 4.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=\int \frac {x^{2}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 79.78 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.12 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=\int { \frac {x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [N/A]

Not integrable

Time = 0.48 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=\int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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