\(\int \frac {a+b \tan (c+d \sqrt {x})}{x^2} \, dx\) [30]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (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 = {} \[ \int \frac {a+b \tan \left (c+d \sqrt {x}\right )}{x^2} \, dx=\int \frac {a+b \tan \left (c+d \sqrt {x}\right )}{x^2} \, dx \]

[In]

Int[(a + b*Tan[c + d*Sqrt[x]])/x^2,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 12.48 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \tan \left (c+d \sqrt {x}\right )}{x^2} \, dx=\int \frac {a+b \tan \left (c+d \sqrt {x}\right )}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.37 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \tan \left (c+d \sqrt {x}\right )}{x^2} \, dx=\int { \frac {b \tan \left (d \sqrt {x} + c\right ) + a}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.78 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {a+b \tan \left (c+d \sqrt {x}\right )}{x^2} \, dx=\int \frac {a + b \tan {\left (c + d \sqrt {x} \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 72, normalized size of antiderivative = 4.00 \[ \int \frac {a+b \tan \left (c+d \sqrt {x}\right )}{x^2} \, dx=\int { \frac {b \tan \left (d \sqrt {x} + c\right ) + a}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \tan \left (c+d \sqrt {x}\right )}{x^2} \, dx=\int { \frac {b \tan \left (d \sqrt {x} + c\right ) + a}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 4.45 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \tan \left (c+d \sqrt {x}\right )}{x^2} \, dx=\int \frac {a+b\,\mathrm {tan}\left (c+d\,\sqrt {x}\right )}{x^2} \,d x \]

[In]

int((a + b*tan(c + d*x^(1/2)))/x^2,x)

[Out]

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