\(\int \frac {a+b \csc (c+d \sqrt {x})}{x^2} \, dx\) [45]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^2} \, dx=-\frac {a}{x}+b \text {Int}\left (\frac {\csc \left (c+d \sqrt {x}\right )}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^2} \, dx=\int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^2} \, dx \]

[In]

Int[(a + b*Csc[c + d*Sqrt[x]])/x^2,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^2} \, dx=\int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^2} \, dx=\int { \frac {b \csc \left (d \sqrt {x} + c\right ) + a}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.58 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^2} \, dx=\int \frac {a + b \csc {\left (c + d \sqrt {x} \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 111, normalized size of antiderivative = 6.17 \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^2} \, dx=\int { \frac {b \csc \left (d \sqrt {x} + c\right ) + a}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^2} \, dx=\int { \frac {b \csc \left (d \sqrt {x} + c\right ) + a}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^2} \, dx=\int \frac {a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}}{x^2} \,d x \]

[In]

int((a + b/sin(c + d*x^(1/2)))/x^2,x)

[Out]

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