3.49 \(\int \frac{1}{x (a+b \csc (c+d \sqrt{x}))^2} \, dx\)

Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{1}{x \left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0243545, 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 = {} \[ \int \frac{1}{x \left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 40.5823, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.161, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \left ( a+b\csc \left ( c+d\sqrt{x} \right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b^{2} x \csc \left (d \sqrt{x} + c\right )^{2} + 2 \, a b x \csc \left (d \sqrt{x} + c\right ) + a^{2} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (a + b \csc{\left (c + d \sqrt{x} \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \csc \left (d \sqrt{x} + c\right ) + a\right )}^{2} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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