\(\int \frac {1}{a+b \tan (c+d x^2)} \, dx\) [16]

Optimal result

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

[Out]

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

[In]

Int[(a + b*Tan[c + d*x^2])^(-1),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.41 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{a+b \tan \left (c+d x^2\right )} \, dx=\int \frac {1}{a+b \tan \left (c+d x^2\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.13 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{a+b \tan \left (c+d x^2\right )} \, dx=\int { \frac {1}{b \tan \left (d x^{2} + c\right ) + a} \,d x } \]

[In]

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

[Out]

integral(1/(b*tan(d*x^2 + c) + a), x)

Sympy [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a+b \tan \left (c+d x^2\right )} \, dx=\int \frac {1}{a + b \tan {\left (c + d x^{2} \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 187, normalized size of antiderivative = 13.36 \[ \int \frac {1}{a+b \tan \left (c+d x^2\right )} \, dx=\int { \frac {1}{b \tan \left (d x^{2} + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{a+b \tan \left (c+d x^2\right )} \, dx=\int { \frac {1}{b \tan \left (d x^{2} + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.56 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{a+b \tan \left (c+d x^2\right )} \, dx=\int \frac {1}{a+b\,\mathrm {tan}\left (d\,x^2+c\right )} \,d x \]

[In]

int(1/(a + b*tan(c + d*x^2)),x)

[Out]

int(1/(a + b*tan(c + d*x^2)), x)