\(\int \frac {1}{x (a+b \tan (c+d x^2))} \, dx\) [17]

Optimal result

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

[Out]

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

[In]

Int[1/(x*(a + b*Tan[c + d*x^2])),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.35 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x \left (a+b \tan \left (c+d x^2\right )\right )} \, dx=\int \frac {1}{x \left (a+b \tan \left (c+d x^2\right )\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.14 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x \left (a+b \tan \left (c+d x^2\right )\right )} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{2} + c\right ) + a\right )} x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.76 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x \left (a+b \tan \left (c+d x^2\right )\right )} \, dx=\int \frac {1}{x \left (a + b \tan {\left (c + d x^{2} \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 510, normalized size of antiderivative = 28.33 \[ \int \frac {1}{x \left (a+b \tan \left (c+d x^2\right )\right )} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{2} + c\right ) + a\right )} x} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x \left (a+b \tan \left (c+d x^2\right )\right )} \, dx=\int { \frac {1}{{\left (b \tan \left (d x^{2} + c\right ) + a\right )} x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 4.35 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x \left (a+b \tan \left (c+d x^2\right )\right )} \, dx=\int \frac {1}{x\,\left (a+b\,\mathrm {tan}\left (d\,x^2+c\right )\right )} \,d x \]

[In]

int(1/(x*(a + b*tan(c + d*x^2))),x)

[Out]

int(1/(x*(a + b*tan(c + d*x^2))), x)