\(\int \frac {1}{x^2 (a+b \tan (c+d x^2))^2} \, dx\) [24]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\text {Int}\left (\frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )^2},x\right ) \]

[Out]

Unintegrable(1/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 = {} \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx \]

[In]

Int[1/(x^2*(a + b*Tan[c + d*x^2])^2),x]

[Out]

Defer[Int][1/(x^2*(a + b*Tan[c + d*x^2])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 9.76 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx \]

[In]

Integrate[1/(x^2*(a + b*Tan[c + d*x^2])^2),x]

[Out]

Integrate[1/(x^2*(a + b*Tan[c + d*x^2])^2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

\[\int \frac {1}{x^{2} {\left (a +b \tan \left (d \,x^{2}+c \right )\right )}^{2}}d x\]

[In]

int(1/x^2/(a+b*tan(d*x^2+c))^2,x)

[Out]

int(1/x^2/(a+b*tan(d*x^2+c))^2,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.44 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{2} + c\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(a+b*tan(d*x^2+c))^2,x, algorithm="fricas")

[Out]

integral(1/(b^2*x^2*tan(d*x^2 + c)^2 + 2*a*b*x^2*tan(d*x^2 + c) + a^2*x^2), x)

Sympy [N/A]

Not integrable

Time = 0.99 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{x^{2} \left (a + b \tan {\left (c + d x^{2} \right )}\right )^{2}}\, dx \]

[In]

integrate(1/x**2/(a+b*tan(d*x**2+c))**2,x)

[Out]

Integral(1/(x**2*(a + b*tan(c + d*x**2))**2), x)

Maxima [N/A]

Not integrable

Time = 1.41 (sec) , antiderivative size = 2599, normalized size of antiderivative = 144.39 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{2} + c\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(a+b*tan(d*x^2+c))^2,x, algorithm="maxima")

[Out]

-((a^6 + a^4*b^2)*d*x^2*cos(2*d*x^2 + 2*c)^2 + (a^6 + a^4*b^2)*d*x^2*sin(2*d*x^2 + 2*c)^2 + (a^6 + a^4*b^2 - a
^2*b^4 - b^6)*d*x^2 + (b^6*sin(2*c) - ((4*a^4*b^2 + 5*a^2*b^4 - b^6)*cos(2*c) - 2*(a^5*b - 2*a*b^5)*sin(2*c))*
d*x^2 + 2*(a^3*b^3 + a*b^5)*cos(2*c))*cos(2*d*x^2) - (((a^2*b^4 + b^6)*cos(2*c) - 2*(a^5*b + 2*a^3*b^3 + a*b^5
)*sin(2*c))*d*x^2*cos(2*d*x^2) - (2*(a^5*b + 2*a^3*b^3 + a*b^5)*cos(2*c) + (a^2*b^4 + b^6)*sin(2*c))*d*x^2*sin
(2*d*x^2) - (2*a^6 + 2*a^4*b^2 + 3*a^2*b^4 + b^6)*d*x^2)*cos(2*d*x^2 + 2*c) + (a^8*d*x^3*cos(2*d*x^2 + 2*c)^2
+ a^8*d*x^3*sin(2*d*x^2 + 2*c)^2 + ((4*a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*cos(2*c)^2 + (4*a^6*b^2 + 8*a^4*
b^4 + 4*a^2*b^6 + b^8)*sin(2*c)^2)*d*x^3*cos(2*d*x^2)^2 + ((4*a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*cos(2*c)^
2 + (4*a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*sin(2*c)^2)*d*x^3*sin(2*d*x^2)^2 - 2*((a^4*b^4 + 2*a^2*b^6 + b^8
)*cos(2*c) - 2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*sin(2*c))*d*x^3*cos(2*d*x^2) + 2*(2*(a^7*b + 3*a^5*b^3
+ 3*a^3*b^5 + a*b^7)*cos(2*c) + (a^4*b^4 + 2*a^2*b^6 + b^8)*sin(2*c))*d*x^3*sin(2*d*x^2) + (a^8 + 4*a^6*b^2 +
6*a^4*b^4 + 4*a^2*b^6 + b^8)*d*x^3 - 2*((a^4*b^4*cos(2*c) - 2*(a^7*b + a^5*b^3)*sin(2*c))*d*x^3*cos(2*d*x^2) -
(a^4*b^4*sin(2*c) + 2*(a^7*b + a^5*b^3)*cos(2*c))*d*x^3*sin(2*d*x^2) - (a^8 + 2*a^6*b^2 + a^4*b^4)*d*x^3)*cos
(2*d*x^2 + 2*c) - 2*((a^4*b^4*sin(2*c) + 2*(a^7*b + a^5*b^3)*cos(2*c))*d*x^3*cos(2*d*x^2) + (a^4*b^4*cos(2*c)
- 2*(a^7*b + a^5*b^3)*sin(2*c))*d*x^3*sin(2*d*x^2))*sin(2*d*x^2 + 2*c))*integrate(((3*b^6*sin(2*c) - 4*(a*b^5*
sin(2*c) + 2*(a^4*b^2 + a^2*b^4)*cos(2*c))*d*x^2 + 6*(a^3*b^3 + a*b^5)*cos(2*c))*cos(2*d*x^2) + (3*b^6*cos(2*c
) - 4*(a*b^5*cos(2*c) - 2*(a^4*b^2 + a^2*b^4)*sin(2*c))*d*x^2 - 6*(a^3*b^3 + a*b^5)*sin(2*c))*sin(2*d*x^2) + (
4*a^5*b*d*x^2 - 3*a^4*b^2)*sin(2*d*x^2 + 2*c))/(a^8*d*x^4*cos(2*d*x^2 + 2*c)^2 + a^8*d*x^4*sin(2*d*x^2 + 2*c)^
2 + ((4*a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*cos(2*c)^2 + (4*a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*sin(2*c)
^2)*d*x^4*cos(2*d*x^2)^2 + ((4*a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*cos(2*c)^2 + (4*a^6*b^2 + 8*a^4*b^4 + 4*
a^2*b^6 + b^8)*sin(2*c)^2)*d*x^4*sin(2*d*x^2)^2 - 2*((a^4*b^4 + 2*a^2*b^6 + b^8)*cos(2*c) - 2*(a^7*b + 3*a^5*b
^3 + 3*a^3*b^5 + a*b^7)*sin(2*c))*d*x^4*cos(2*d*x^2) + 2*(2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*cos(2*c) +
(a^4*b^4 + 2*a^2*b^6 + b^8)*sin(2*c))*d*x^4*sin(2*d*x^2) + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d*
x^4 - 2*((a^4*b^4*cos(2*c) - 2*(a^7*b + a^5*b^3)*sin(2*c))*d*x^4*cos(2*d*x^2) - (a^4*b^4*sin(2*c) + 2*(a^7*b +
a^5*b^3)*cos(2*c))*d*x^4*sin(2*d*x^2) - (a^8 + 2*a^6*b^2 + a^4*b^4)*d*x^4)*cos(2*d*x^2 + 2*c) - 2*((a^4*b^4*s
in(2*c) + 2*(a^7*b + a^5*b^3)*cos(2*c))*d*x^4*cos(2*d*x^2) + (a^4*b^4*cos(2*c) - 2*(a^7*b + a^5*b^3)*sin(2*c))
*d*x^4*sin(2*d*x^2))*sin(2*d*x^2 + 2*c)), x) + (b^6*cos(2*c) + (2*(a^5*b - 2*a*b^5)*cos(2*c) + (4*a^4*b^2 + 5*
a^2*b^4 - b^6)*sin(2*c))*d*x^2 - 2*(a^3*b^3 + a*b^5)*sin(2*c))*sin(2*d*x^2) + (2*a^5*b*d*x^2 - a^4*b^2 - (2*(a
^5*b + 2*a^3*b^3 + a*b^5)*cos(2*c) + (a^2*b^4 + b^6)*sin(2*c))*d*x^2*cos(2*d*x^2) - ((a^2*b^4 + b^6)*cos(2*c)
- 2*(a^5*b + 2*a^3*b^3 + a*b^5)*sin(2*c))*d*x^2*sin(2*d*x^2))*sin(2*d*x^2 + 2*c))/(a^8*d*x^3*cos(2*d*x^2 + 2*c
)^2 + a^8*d*x^3*sin(2*d*x^2 + 2*c)^2 + ((4*a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*cos(2*c)^2 + (4*a^6*b^2 + 8*
a^4*b^4 + 4*a^2*b^6 + b^8)*sin(2*c)^2)*d*x^3*cos(2*d*x^2)^2 + ((4*a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*cos(2
*c)^2 + (4*a^6*b^2 + 8*a^4*b^4 + 4*a^2*b^6 + b^8)*sin(2*c)^2)*d*x^3*sin(2*d*x^2)^2 - 2*((a^4*b^4 + 2*a^2*b^6 +
b^8)*cos(2*c) - 2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*sin(2*c))*d*x^3*cos(2*d*x^2) + 2*(2*(a^7*b + 3*a^5*
b^3 + 3*a^3*b^5 + a*b^7)*cos(2*c) + (a^4*b^4 + 2*a^2*b^6 + b^8)*sin(2*c))*d*x^3*sin(2*d*x^2) + (a^8 + 4*a^6*b^
2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*d*x^3 - 2*((a^4*b^4*cos(2*c) - 2*(a^7*b + a^5*b^3)*sin(2*c))*d*x^3*cos(2*d*x^
2) - (a^4*b^4*sin(2*c) + 2*(a^7*b + a^5*b^3)*cos(2*c))*d*x^3*sin(2*d*x^2) - (a^8 + 2*a^6*b^2 + a^4*b^4)*d*x^3)
*cos(2*d*x^2 + 2*c) - 2*((a^4*b^4*sin(2*c) + 2*(a^7*b + a^5*b^3)*cos(2*c))*d*x^3*cos(2*d*x^2) + (a^4*b^4*cos(2
*c) - 2*(a^7*b + a^5*b^3)*sin(2*c))*d*x^3*sin(2*d*x^2))*sin(2*d*x^2 + 2*c))

Giac [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{2} + c\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(a+b*tan(d*x^2+c))^2,x, algorithm="giac")

[Out]

integrate(1/((b*tan(d*x^2 + c) + a)^2*x^2), x)

Mupad [N/A]

Not integrable

Time = 4.38 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{x^2\,{\left (a+b\,\mathrm {tan}\left (d\,x^2+c\right )\right )}^2} \,d x \]

[In]

int(1/(x^2*(a + b*tan(c + d*x^2))^2),x)

[Out]

int(1/(x^2*(a + b*tan(c + d*x^2))^2), x)