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

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{\left (a+b \csc \left (c+d x^2\right )\right )^2}{x^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0217219, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (a+b \csc \left (c+d x^2\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Csc[c + d*x^2])^2/x^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 16.7891, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \csc \left (c+d x^2\right )\right )^2}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.237, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\csc \left ( d{x}^{2}+c \right ) \right ) ^{2}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{2}}{x} - \frac{b^{2} \sin \left (2 \, d x^{2} + 2 \, c\right ) - \frac{1}{2} \,{\left (d x^{3} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d x^{3} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} - 2 \, d x^{3} \cos \left (2 \, d x^{2} + 2 \, c\right ) + d x^{3}\right )} \int \frac{{\left (4 \, a b d x^{2} + 3 \, b^{2}\right )} \sin \left (d x^{2} + c\right )}{d x^{4} \cos \left (d x^{2} + c\right )^{2} + d x^{4} \sin \left (d x^{2} + c\right )^{2} + 2 \, d x^{4} \cos \left (d x^{2} + c\right ) + d x^{4}}\,{d x} - \frac{1}{2} \,{\left (d x^{3} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d x^{3} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} - 2 \, d x^{3} \cos \left (2 \, d x^{2} + 2 \, c\right ) + d x^{3}\right )} \int \frac{{\left (4 \, a b d x^{2} - 3 \, b^{2}\right )} \sin \left (d x^{2} + c\right )}{d x^{4} \cos \left (d x^{2} + c\right )^{2} + d x^{4} \sin \left (d x^{2} + c\right )^{2} - 2 \, d x^{4} \cos \left (d x^{2} + c\right ) + d x^{4}}\,{d x}}{d x^{3} \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d x^{3} \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} - 2 \, d x^{3} \cos \left (2 \, d x^{2} + 2 \, c\right ) + d x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \csc \left (d x^{2} + c\right )^{2} + 2 \, a b \csc \left (d x^{2} + c\right ) + a^{2}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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