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

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{\left (a+b \sec \left (c+d \sqrt{x}\right )\right )^2}{x^2},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0233649, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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 = 19.9603, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \sec \left (c+d \sqrt{x}\right )\right )^2}{x^2} \, dx \]

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]

________________________________________________________________________________________

Maple [A]  time = 0.092, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b\sec \left ( c+d\sqrt{x} \right ) \right ) ^{2}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \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}} \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \sec{\left (c + d \sqrt{x} \right )}\right )^{2}}{x^{2}}\, dx \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sec \left (d \sqrt{x} + c\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}

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)