\(\int \frac {x^2}{(a+b \sec (c+d x^2))^2} \, dx\) [26]

Optimal result

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

[Out]

Unintegrable(x^2/(a+b*sec(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 = {} \[ \int \frac {x^2}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^2}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx \]

[In]

Int[x^2/(a + b*Sec[c + d*x^2])^2,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 6.73 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x^2}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^2}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx \]

[In]

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

[Out]

Integrate[x^2/(a + b*Sec[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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.11 \[ \int \frac {x^2}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{2}}{{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {x^2}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^{2}}{\left (a + b \sec {\left (c + d x^{2} \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.89 (sec) , antiderivative size = 1261, normalized size of antiderivative = 70.06 \[ \int \frac {x^2}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{2}}{{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x^2}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {x^{2}}{{\left (b \sec \left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 13.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {x^2}{\left (a+b \sec \left (c+d x^2\right )\right )^2} \, dx=\int \frac {x^2}{{\left (a+\frac {b}{\cos \left (d\,x^2+c\right )}\right )}^2} \,d x \]

[In]

int(x^2/(a + b/cos(c + d*x^2))^2,x)

[Out]

int(x^2/(a + b/cos(c + d*x^2))^2, x)