\(\int \frac {(a+b \csc (c+d \sqrt {x}))^2}{x^2} \, dx\) [40]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx=\text {Int}\left (\frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx=\int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 85.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx=\int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.44 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.44 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx=\int \frac {\left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )^{2}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.01 (sec) , antiderivative size = 388, normalized size of antiderivative = 19.40 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2}}{x^{2}} \,d x } \]

[In]

integrate((a+b*csc(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*a
*b*d*x*sin(d*sqrt(x) + c) + 3*b^2*sqrt(x)*sin(d*sqrt(x) + c))/((d*cos(d*sqrt(x) + c)^2 + d*sin(d*sqrt(x) + c)^
2 + 2*d*cos(d*sqrt(x) + c) + d)*x^3), 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*integrate(-(2*a*b*d*x*sin(d*sqrt(x) + c) - 3*b^2*sqrt(x)*sin(d*sqrt(x) + c))/((d*c
os(d*sqrt(x) + c)^2 + d*sin(d*sqrt(x) + c)^2 - 2*d*cos(d*sqrt(x) + c) + d)*x^3), x) - 4*b^2*sqrt(x)*sin(2*d*sq
rt(x) + 2*c) - (a^2*d*cos(2*d*sqrt(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
)

Giac [N/A]

Not integrable

Time = 0.60 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 18.80 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^2} \, dx=\int \frac {{\left (a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}\right )}^2}{x^2} \,d x \]

[In]

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

[Out]

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