\(\int \frac {1}{x^2 (a+b \tan (c+d x^2))} \, dx\) [18]

Optimal result

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

[Out]

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

[In]

Int[1/(x^2*(a + b*Tan[c + d*x^2])),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.88 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )} \, dx=\int \frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.13 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{2} + c\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.64 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )} \, dx=\int \frac {1}{x^{2} \left (a + b \tan {\left (c + d x^{2} \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 534, normalized size of antiderivative = 29.67 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{2} + c\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{2} + c\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 4.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d x^2\right )\right )} \, dx=\int \frac {1}{x^2\,\left (a+b\,\mathrm {tan}\left (d\,x^2+c\right )\right )} \,d x \]

[In]

int(1/(x^2*(a + b*tan(c + d*x^2))),x)

[Out]

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