\(\int \frac {x^2}{(a+b \tan (c+d x^2))^2} \, dx\) [20]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {x^2}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\text {Int}\left (\frac {x^2}{\left (a+b \tan \left (c+d x^2\right )\right )^2},x\right ) \]

[Out]

Unintegrable(x^2/(a+b*tan(d*x^2+c))^2,x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {x^2}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^2}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx \]

[In]

Int[x^2/(a + b*Tan[c + d*x^2])^2,x]

[Out]

Defer[Int][x^2/(a + b*Tan[c + d*x^2])^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 8.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x^2}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^2}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx \]

[In]

Integrate[x^2/(a + b*Tan[c + d*x^2])^2,x]

[Out]

Integrate[x^2/(a + b*Tan[c + d*x^2])^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

int(x^2/(a+b*tan(d*x^2+c))^2,x)

[Out]

int(x^2/(a+b*tan(d*x^2+c))^2,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.11 \[ \int \frac {x^2}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{2}}{{\left (b \tan \left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^2/(a+b*tan(d*x^2+c))^2,x, algorithm="fricas")

[Out]

integral(x^2/(b^2*tan(d*x^2 + c)^2 + 2*a*b*tan(d*x^2 + c) + a^2), x)

Sympy [N/A]

Not integrable

Time = 1.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {x^2}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^{2}}{\left (a + b \tan {\left (c + d x^{2} \right )}\right )^{2}}\, dx \]

[In]

integrate(x**2/(a+b*tan(d*x**2+c))**2,x)

[Out]

Integral(x**2/(a + b*tan(c + d*x**2))**2, x)

Maxima [N/A]

Not integrable

Time = 10.41 (sec) , antiderivative size = 764, normalized size of antiderivative = 42.44 \[ \int \frac {x^2}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{2}}{{\left (b \tan \left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^2/(a+b*tan(d*x^2+c))^2,x, algorithm="maxima")

[Out]

1/3*((a^4 - b^4)*d*x^3*cos(2*d*x^2 + 2*c)^2 + (a^4 - b^4)*d*x^3*sin(2*d*x^2 + 2*c)^2 + (a^4 - b^4)*d*x^3 - 2*(
3*a*b^3*x - (a^4 - 2*a^2*b^2 + b^4)*d*x^3)*cos(2*d*x^2 + 2*c) + 3*((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d*cos(2
*d*x^2 + 2*c)^2 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d*sin(2*d*x^2 + 2*c)^2 + 2*(a^6 + a^4*b^2 - a^2*b^4 - b^
6)*d*cos(2*d*x^2 + 2*c) + 4*(a^5*b + 2*a^3*b^3 + a*b^5)*d*sin(2*d*x^2 + 2*c) + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6)*d)*integrate((2*(4*a^2*b^2*d*x^2 + a*b^3)*cos(2*d*x^2 + 2*c) - (a^2*b^2 - b^4 + 4*(a^3*b - a*b^3)*d*x^2)*
sin(2*d*x^2 + 2*c))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d*cos(2*d*x^2 + 2*c)^2 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)*d*sin(2*d*x^2 + 2*c)^2 + 2*(a^6 + a^4*b^2 - a^2*b^4 - b^6)*d*cos(2*d*x^2 + 2*c) + 4*(a^5*b + 2*a^3*b^3
+ a*b^5)*d*sin(2*d*x^2 + 2*c) + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d), x) + (4*(a^3*b - a*b^3)*d*x^3 + 3*(a^
2*b^2 - b^4)*x)*sin(2*d*x^2 + 2*c))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d*cos(2*d*x^2 + 2*c)^2 + (a^6 + 3*a^4
*b^2 + 3*a^2*b^4 + b^6)*d*sin(2*d*x^2 + 2*c)^2 + 2*(a^6 + a^4*b^2 - a^2*b^4 - b^6)*d*cos(2*d*x^2 + 2*c) + 4*(a
^5*b + 2*a^3*b^3 + a*b^5)*d*sin(2*d*x^2 + 2*c) + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d)

Giac [N/A]

Not integrable

Time = 0.61 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x^2}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{2}}{{\left (b \tan \left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^2/(a+b*tan(d*x^2+c))^2,x, algorithm="giac")

[Out]

integrate(x^2/(b*tan(d*x^2 + c) + a)^2, x)

Mupad [N/A]

Not integrable

Time = 4.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x^2}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^2}{{\left (a+b\,\mathrm {tan}\left (d\,x^2+c\right )\right )}^2} \,d x \]

[In]

int(x^2/(a + b*tan(c + d*x^2))^2,x)

[Out]

int(x^2/(a + b*tan(c + d*x^2))^2, x)