3.40 \(\int \frac {(a+b \sec (c+d \sqrt {x}))^2}{x^2} \, dx\)

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2}{x^2},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx &=\int \frac {\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 21.03, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \sec \left (d \sqrt {x} + c\right )^{2} + 2 \, a b \sec \left (d \sqrt {x} + c\right ) + a^{2}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sec \left (d \sqrt {x} + c\right ) + a\right )}^{2}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 1.40, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \sec \left (c +d \sqrt {x}\right )\right )^{2}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, {\left (d \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + d\right )} x^{2} \int \frac {3 \, b^{2} \sqrt {x} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + 2 \, {\left (a b d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) \cos \left (d \sqrt {x} + c\right ) + a b d \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) \sin \left (d \sqrt {x} + c\right ) + a b d \cos \left (d \sqrt {x} + c\right )\right )} x}{{\left (d \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + d\right )} x^{3}}\,{d x} + 4 \, b^{2} \sqrt {x} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) - {\left (a^{2} d \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + a^{2} d \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, a^{2} d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + a^{2} d\right )} x}{{\left (d \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + d\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,\sqrt {x}\right )}\right )}^2}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \sec {\left (c + d \sqrt {x} \right )}\right )^{2}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________