\(\int \frac {1}{x^2 (a+b \tan (c+d \sqrt {x}))} \, dx\) [41]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt {x}\right )\right )} \, dx=\text {Int}\left (\frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt {x}\right )\right )},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt {x}\right )\right )} \, dx=\int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt {x}\right )\right )} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 5.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt {x}\right )\right )} \, dx=\int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt {x}\right )\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.46 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt {x}\right )\right )} \, dx=\int { \frac {1}{{\left (b \tan \left (d \sqrt {x} + c\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt {x}\right )\right )} \, dx=\int \frac {1}{x^{2} \left (a + b \tan {\left (c + d \sqrt {x} \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.27 (sec) , antiderivative size = 496, normalized size of antiderivative = 24.80 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt {x}\right )\right )} \, dx=\int { \frac {1}{{\left (b \tan \left (d \sqrt {x} + c\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.68 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt {x}\right )\right )} \, dx=\int { \frac {1}{{\left (b \tan \left (d \sqrt {x} + c\right ) + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 3.66 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (a+b \tan \left (c+d \sqrt {x}\right )\right )} \, dx=\int \frac {1}{x^2\,\left (a+b\,\mathrm {tan}\left (c+d\,\sqrt {x}\right )\right )} \,d x \]

[In]

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

[Out]

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