\(\int \frac {1}{x^{3/2} (a+b \csc (c+d \sqrt {x}))} \, dx\) [65]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{x^{3/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx=\text {Int}\left (\frac {1}{x^{3/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (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 = {} \[ \int \frac {1}{x^{3/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx=\int \frac {1}{x^{3/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx \]

[In]

Int[1/(x^(3/2)*(a + b*Csc[c + d*Sqrt[x]])),x]

[Out]

Defer[Int][1/(x^(3/2)*(a + b*Csc[c + d*Sqrt[x]])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^{3/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 5.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^{3/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx=\int \frac {1}{x^{3/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx \]

[In]

Integrate[1/(x^(3/2)*(a + b*Csc[c + d*Sqrt[x]])),x]

[Out]

Integrate[1/(x^(3/2)*(a + b*Csc[c + d*Sqrt[x]])), x]

Maple [N/A] (verified)

Not integrable

Time = 0.36 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82

\[\int \frac {1}{x^{\frac {3}{2}} \left (a +b \csc \left (c +d \sqrt {x}\right )\right )}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x^{3/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx=\int { \frac {1}{{\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )} x^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^{3/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx=\int \frac {1}{x^{\frac {3}{2}} \left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.96 (sec) , antiderivative size = 244, normalized size of antiderivative = 11.09 \[ \int \frac {1}{x^{3/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx=\int { \frac {1}{{\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )} x^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

-2*(a*b*sqrt(x)*integrate((2*b*cos(d*sqrt(x) + c)^2 + a*cos(d*sqrt(x) + c)*sin(2*d*sqrt(x) + 2*c) - a*cos(2*d*
sqrt(x) + 2*c)*sin(d*sqrt(x) + c) + 2*b*sin(d*sqrt(x) + c)^2 + a*sin(d*sqrt(x) + c))/((a^3*cos(2*d*sqrt(x) + 2
*c)^2 + 4*a*b^2*cos(d*sqrt(x) + c)^2 + 4*a^2*b*cos(d*sqrt(x) + c)*sin(2*d*sqrt(x) + 2*c) + a^3*sin(2*d*sqrt(x)
+ 2*c)^2 + 4*a*b^2*sin(d*sqrt(x) + c)^2 + 4*a^2*b*sin(d*sqrt(x) + c) + a^3 - 2*(2*a^2*b*sin(d*sqrt(x) + c) +
a^3)*cos(2*d*sqrt(x) + 2*c))*x^(3/2)), x) + 1)/(a*sqrt(x))

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^{3/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx=\int { \frac {1}{{\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )} x^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 17.96 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{3/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx=\int \frac {1}{x^{3/2}\,\left (a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}\right )} \,d x \]

[In]

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

[Out]

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