\(\int \frac {a+b \sec (c+d \sqrt {x})}{x} \, dx\) [34]

Optimal result

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

[Out]

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

[In]

Int[(a + b*Sec[c + d*Sqrt[x]])/x,x]

[Out]

a*Log[x] + b*Defer[Int][Sec[c + d*Sqrt[x]]/x, x]

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

Mathematica [N/A]

Not integrable

Time = 3.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \sec \left (c+d \sqrt {x}\right )}{x} \, dx=\int \frac {a+b \sec \left (c+d \sqrt {x}\right )}{x} \, dx \]

[In]

Integrate[(a + b*Sec[c + d*Sqrt[x]])/x,x]

[Out]

Integrate[(a + b*Sec[c + d*Sqrt[x]])/x, x]

Maple [N/A] (verified)

Not integrable

Time = 0.53 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \sec \left (c+d \sqrt {x}\right )}{x} \, dx=\int { \frac {b \sec \left (d \sqrt {x} + c\right ) + a}{x} \,d x } \]

[In]

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

[Out]

integral((b*sec(d*sqrt(x) + c) + a)/x, x)

Sympy [N/A]

Not integrable

Time = 1.61 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \sec \left (c+d \sqrt {x}\right )}{x} \, dx=\int \frac {a + b \sec {\left (c + d \sqrt {x} \right )}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.67 (sec) , antiderivative size = 106, normalized size of antiderivative = 5.89 \[ \int \frac {a+b \sec \left (c+d \sqrt {x}\right )}{x} \, dx=\int { \frac {b \sec \left (d \sqrt {x} + c\right ) + a}{x} \,d x } \]

[In]

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

[Out]

2*b*integrate((cos(2*d*sqrt(x) + 2*c)*cos(d*sqrt(x) + c) + sin(2*d*sqrt(x) + 2*c)*sin(d*sqrt(x) + c) + cos(d*s
qrt(x) + 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), x) + a*
log(x)

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \sec \left (c+d \sqrt {x}\right )}{x} \, dx=\int { \frac {b \sec \left (d \sqrt {x} + c\right ) + a}{x} \,d x } \]

[In]

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

[Out]

integrate((b*sec(d*sqrt(x) + c) + a)/x, x)

Mupad [N/A]

Not integrable

Time = 13.80 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \sec \left (c+d \sqrt {x}\right )}{x} \, dx=\int \frac {a+\frac {b}{\cos \left (c+d\,\sqrt {x}\right )}}{x} \,d x \]

[In]

int((a + b/cos(c + d*x^(1/2)))/x,x)

[Out]

int((a + b/cos(c + d*x^(1/2)))/x, x)