\(\int \frac {(a+b \csc (c+d \sqrt {x}))^2}{x^{3/2}} \, dx\) [59]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (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 = {} \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx=\int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 76.76 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx=\int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.46 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.09 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx=\int { \frac {{\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2}}{x^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.98 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx=\int \frac {\left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )^{2}}{x^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.15 (sec) , antiderivative size = 794, normalized size of antiderivative = 36.09 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx=\int { \frac {{\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2}}{x^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

-(4*b^2*sin(2*d*sqrt(x) + 2*c) - ((d*integrate(2*(a*b*d*sqrt(x)*sin(d*sqrt(x) + c) + b^2*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) + d*integrate(2*(a*b*d
*sqrt(x)*sin(d*sqrt(x) + c) - b^2*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))*cos(2*d*sqrt(x) + 2*c)^2 + (d*integrate(2*(a*b*d*sqrt(x)*sin(d*sqrt(x) + c)
+ b^2*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) + d*integrate(2*(a*b*d*sqrt(x)*sin(d*sqrt(x) + c) - b^2*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))*sin(2*d*sqrt(x) + 2*c)^2 - 2*(d*integrate(2*(a*b*
d*sqrt(x)*sin(d*sqrt(x) + c) + b^2*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) + d*integrate(2*(a*b*d*sqrt(x)*sin(d*sqrt(x) + c) - b^2*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))*cos(2*d*sqrt(x) + 2*
c) + d*integrate(2*(a*b*d*sqrt(x)*sin(d*sqrt(x) + c) + b^2*sin(d*sqrt(x) + c))/((d*cos(d*sqrt(x) + c)^2 + d*si
n(d*sqrt(x) + c)^2 + 2*d*cos(d*sqrt(x) + c) + d)*x^2), x) + d*integrate(2*(a*b*d*sqrt(x)*sin(d*sqrt(x) + c) -
b^2*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))*x + 2*(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)*sqrt(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 = 0.37 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx=\int { \frac {{\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2}}{x^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 18.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx=\int \frac {{\left (a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}\right )}^2}{x^{3/2}} \,d x \]

[In]

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

[Out]

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