\(\int \frac {x^2}{a+b \csc (c+d x^2)} \, dx\) [19]

Optimal result

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

[Out]

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

[In]

Int[x^2/(a + b*Csc[c + d*x^2]),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 4.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x^2}{a+b \csc \left (c+d x^2\right )} \, dx=\int \frac {x^2}{a+b \csc \left (c+d x^2\right )} \, dx \]

[In]

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

[Out]

Integrate[x^2/(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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x^2}{a+b \csc \left (c+d x^2\right )} \, dx=\int { \frac {x^{2}}{b \csc \left (d x^{2} + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {x^2}{a+b \csc \left (c+d x^2\right )} \, dx=\int \frac {x^{2}}{a + b \csc {\left (c + d x^{2} \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 253, normalized size of antiderivative = 14.06 \[ \int \frac {x^2}{a+b \csc \left (c+d x^2\right )} \, dx=\int { \frac {x^{2}}{b \csc \left (d x^{2} + c\right ) + a} \,d x } \]

[In]

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

[Out]

1/3*(x^3 - 6*a*b*integrate((2*b*x^2*cos(d*x^2 + c)^2 + a*x^2*cos(d*x^2 + c)*sin(2*d*x^2 + 2*c) - a*x^2*cos(2*d
*x^2 + 2*c)*sin(d*x^2 + c) + 2*b*x^2*sin(d*x^2 + c)^2 + a*x^2*sin(d*x^2 + c))/(a^3*cos(2*d*x^2 + 2*c)^2 + 4*a*
b^2*cos(d*x^2 + c)^2 + 4*a^2*b*cos(d*x^2 + c)*sin(2*d*x^2 + 2*c) + a^3*sin(2*d*x^2 + 2*c)^2 + 4*a*b^2*sin(d*x^
2 + c)^2 + 4*a^2*b*sin(d*x^2 + c) + a^3 - 2*(2*a^2*b*sin(d*x^2 + c) + a^3)*cos(2*d*x^2 + 2*c)), x))/a

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {x^2}{a+b \csc \left (c+d x^2\right )} \, dx=\int { \frac {x^{2}}{b \csc \left (d x^{2} + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 18.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {x^2}{a+b \csc \left (c+d x^2\right )} \, dx=\int \frac {x^2}{a+\frac {b}{\sin \left (d\,x^2+c\right )}} \,d x \]

[In]

int(x^2/(a + b/sin(c + d*x^2)),x)

[Out]

int(x^2/(a + b/sin(c + d*x^2)), x)