\(\int \frac {(a+b \csc (c+d \sqrt {x}))^2}{x} \, dx\) [39]

Optimal result

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

[Out]

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

[In]

Int[(a + b*Csc[c + d*Sqrt[x]])^2/x,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 137.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x} \, dx=\int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.43 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2}}{x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 8.33 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x} \, dx=\int \frac {\left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )^{2}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.83 (sec) , antiderivative size = 386, normalized size of antiderivative = 19.30 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2}}{x} \,d x } \]

[In]

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

[Out]

-(4*b^2*sqrt(x)*sin(2*d*sqrt(x) + 2*c) - (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*integrate((2*a*b*d*x*sin(d*sqrt(x) + c) + b^2*sqrt(x)*sin(d*sqrt(x) + c))/((d*cos(d*sq
rt(x) + c)^2 + d*sin(d*sqrt(x) + c)^2 + 2*d*cos(d*sqrt(x) + c) + d)*x^2), 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*integrate(-(2*a*b*d*x*sin(d*sqrt(x) + c) - b^2*s
qrt(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^2
), x) - (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*log(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)

Giac [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x} \, dx=\int { \frac {{\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2}}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 18.79 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x} \, dx=\int \frac {{\left (a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}\right )}^2}{x} \,d x \]

[In]

int((a + b/sin(c + d*x^(1/2)))^2/x,x)

[Out]

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