\(\int \frac {1}{(a+b \tan (c+d x^2))^2} \, dx\) [22]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {1}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\text {Int}\left (\frac {1}{\left (a+b \tan \left (c+d x^2\right )\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 14, 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 = {} \[ \int \frac {1}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx \]

[In]

Int[(a + b*Tan[c + d*x^2])^(-2),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 6.81 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.18 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.43 \[ \int \frac {1}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

integral(1/(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 = 0.61 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{\left (a + b \tan {\left (c + d x^{2} \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.42 (sec) , antiderivative size = 2550, normalized size of antiderivative = 182.14 \[ \int \frac {1}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(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*cos(2*d*x^2 + 2*c)^2 + a
^8*d*x*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*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*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*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*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 - 2*((a^4*b^4*cos(2*c) - 2*(a^7*b + a^5*b^3)*sin(2*c))*d*x*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*sin(2*d*x^2) - (a^8 + 2*a^6*b^2 + a^4*b^4)*d*x)*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*cos(2*d*x^2) + (a^4*b^4*cos(2*c) - 2*(a^7*b + a^5*b^3)*s
in(2*c))*d*x*sin(2*d*x^2))*sin(2*d*x^2 + 2*c))*integrate(((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 + 2*(a^3*b^3 + a*b^5)*cos(2*c))*cos(2*d*x^2) + (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 - 2*(a^3*b^3 + a*b^5)*sin(2*c))*sin(2*d*x^2) + (4*a^5*b*d*x^2 - a^4*b^2)*sin(
2*d*x^2 + 2*c))/(a^8*d*x^2*cos(2*d*x^2 + 2*c)^2 + a^8*d*x^2*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^2*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^
2*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^2*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)*si
n(2*c))*d*x^2*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^2 - 2*((a^4*b^4*cos(2*c) - 2*
(a^7*b + a^5*b^3)*sin(2*c))*d*x^2*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^2*sin(2
*d*x^2) - (a^8 + 2*a^6*b^2 + a^4*b^4)*d*x^2)*cos(2*d*x^2 + 2*c) - 2*((a^4*b^4*sin(2*c) + 2*(a^7*b + a^5*b^3)*c
os(2*c))*d*x^2*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^2*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*cos(2*d*x^2 + 2*c)^2 + a^8*d*x*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*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*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*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*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 - 2*((
a^4*b^4*cos(2*c) - 2*(a^7*b + a^5*b^3)*sin(2*c))*d*x*cos(2*d*x^2) - (a^4*b^4*sin(2*c) + 2*(a^7*b + a^5*b^3)*co
s(2*c))*d*x*sin(2*d*x^2) - (a^8 + 2*a^6*b^2 + a^4*b^4)*d*x)*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*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*sin(2*d*x^2))
*sin(2*d*x^2 + 2*c))

Giac [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{2} + c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 4.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\left (a+b \tan \left (c+d x^2\right )\right )^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {tan}\left (d\,x^2+c\right )\right )}^2} \,d x \]

[In]

int(1/(a + b*tan(c + d*x^2))^2,x)

[Out]

int(1/(a + b*tan(c + d*x^2))^2, x)