\(\int \frac {1}{x (a+b \csc (c+d x^2))} \, dx\) [21]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {1}{x \left (a+b \csc \left (c+d x^2\right )\right )} \, dx=\text {Int}\left (\frac {1}{x \left (a+b \csc \left (c+d x^2\right )\right )},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

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

[In]

Int[1/(x*(a + b*Csc[c + d*x^2])),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x \left (a+b \csc \left (c+d x^2\right )\right )} \, dx=\int \frac {1}{x \left (a+b \csc \left (c+d x^2\right )\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.12 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x \left (a+b \csc \left (c+d x^2\right )\right )} \, dx=\int { \frac {1}{{\left (b \csc \left (d x^{2} + c\right ) + a\right )} x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.71 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x \left (a+b \csc \left (c+d x^2\right )\right )} \, dx=\int \frac {1}{x \left (a + b \csc {\left (c + d x^{2} \right )}\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 250, normalized size of antiderivative = 13.89 \[ \int \frac {1}{x \left (a+b \csc \left (c+d x^2\right )\right )} \, dx=\int { \frac {1}{{\left (b \csc \left (d x^{2} + c\right ) + a\right )} x} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x \left (a+b \csc \left (c+d x^2\right )\right )} \, dx=\int { \frac {1}{{\left (b \csc \left (d x^{2} + c\right ) + a\right )} x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 20.83 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x \left (a+b \csc \left (c+d x^2\right )\right )} \, dx=\int \frac {1}{x\,\left (a+\frac {b}{\sin \left (d\,x^2+c\right )}\right )} \,d x \]

[In]

int(1/(x*(a + b/sin(c + d*x^2))),x)

[Out]

int(1/(x*(a + b/sin(c + d*x^2))), x)