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

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {x^2}{a+b \csc \left (c+d x^2\right )},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^2}{a+b \csc \left (c+d x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \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\\ \end {align*}

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Mathematica [A]
time = 1.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2}{a+b \csc \left (c+d x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{a +b \csc \left (d \,x^{2}+c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{a + b \csc {\left (c + d x^{2} \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {x^2}{a+\frac {b}{\sin \left (d\,x^2+c\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b/sin(c + d*x^2)),x)

[Out]

int(x^2/(a + b/sin(c + d*x^2)), x)

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