\(\int \frac {1}{x^{3/2} (a+b \sec (c+d \sqrt {x}))^2} \, dx\) [69]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{x^{3/2} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Int}\left (\frac {1}{x^{3/2} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2},x\right ) \]

[Out]

Unintegrable(1/x^(3/2)/(a+b*sec(c+d*x^(1/2)))^2,x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {1}{x^{3/2} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{x^{3/2} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx \]

[In]

Int[1/(x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])^2),x]

[Out]

Defer[Int][1/(x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^{3/2} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 44.62 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^{3/2} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{x^{3/2} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx \]

[In]

Integrate[1/(x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])^2),x]

[Out]

Integrate[1/(x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])^2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.56 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82

\[\int \frac {1}{x^{\frac {3}{2}} \left (a +b \sec \left (c +d \sqrt {x}\right )\right )^{2}}d x\]

[In]

int(1/x^(3/2)/(a+b*sec(c+d*x^(1/2)))^2,x)

[Out]

int(1/x^(3/2)/(a+b*sec(c+d*x^(1/2)))^2,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18 \[ \int \frac {1}{x^{3/2} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \sec \left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/x^(3/2)/(a+b*sec(c+d*x^(1/2)))^2,x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 5.92 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{3/2} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{x^{\frac {3}{2}} \left (a + b \sec {\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]

[In]

integrate(1/x**(3/2)/(a+b*sec(c+d*x**(1/2)))**2,x)

[Out]

Integral(1/(x**(3/2)*(a + b*sec(c + d*sqrt(x)))**2), x)

Maxima [F(-1)]

Timed out. \[ \int \frac {1}{x^{3/2} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Timed out} \]

[In]

integrate(1/x^(3/2)/(a+b*sec(c+d*x^(1/2)))^2,x, algorithm="maxima")

[Out]

Timed out

Giac [N/A]

Not integrable

Time = 1.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^{3/2} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \sec \left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/x^(3/2)/(a+b*sec(c+d*x^(1/2)))^2,x, algorithm="giac")

[Out]

integrate(1/((b*sec(d*sqrt(x) + c) + a)^2*x^(3/2)), x)

Mupad [N/A]

Not integrable

Time = 13.75 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{3/2} \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {1}{x^{3/2}\,{\left (a+\frac {b}{\cos \left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \]

[In]

int(1/(x^(3/2)*(a + b/cos(c + d*x^(1/2)))^2),x)

[Out]

int(1/(x^(3/2)*(a + b/cos(c + d*x^(1/2)))^2), x)