3.71 \(\int \frac{1}{x^{5/2} (a+b \csc (c+d \sqrt{x}))^2} \, dx\)

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{1}{x^{5/2} \left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2},x\right ) \]

[Out]

Unintegrable[1/(x^(5/2)*(a + b*Csc[c + d*Sqrt[x]])^2), x]

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Rubi [A]  time = 0.0239914, 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 = {} \[ \int \frac{1}{x^{5/2} \left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^(5/2)*(a + b*Csc[c + d*Sqrt[x]])^2),x]

[Out]

Defer[Int][1/(x^(5/2)*(a + b*Csc[c + d*Sqrt[x]])^2), x]

Rubi steps

\begin{align*} \int \frac{1}{x^{5/2} \left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2} \, dx &=\int \frac{1}{x^{5/2} \left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2} \, dx\\ \end{align*}

Mathematica [A]  time = 28.1993, size = 0, normalized size = 0. \[ \int \frac{1}{x^{5/2} \left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^(5/2)*(a + b*Csc[c + d*Sqrt[x]])^2),x]

[Out]

Integrate[1/(x^(5/2)*(a + b*Csc[c + d*Sqrt[x]])^2), x]

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Maple [A]  time = 0.176, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-{\frac{5}{2}}} \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^(5/2)/(a+b*csc(c+d*x^(1/2)))^2,x)

[Out]

int(1/x^(5/2)/(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^(5/2)/(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{\sqrt{x}}{b^{2} x^{3} \csc \left (d \sqrt{x} + c\right )^{2} + 2 \, a b x^{3} \csc \left (d \sqrt{x} + c\right ) + a^{2} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(a+b*csc(c+d*x^(1/2)))^2,x, algorithm="fricas")

[Out]

integral(sqrt(x)/(b^2*x^3*csc(d*sqrt(x) + c)^2 + 2*a*b*x^3*csc(d*sqrt(x) + c) + a^2*x^3), x)

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Sympy [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**(5/2)/(a+b*csc(c+d*x**(1/2)))**2,x)

[Out]

Timed out

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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^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(a+b*csc(c+d*x^(1/2)))^2,x, algorithm="giac")

[Out]

integrate(1/((b*csc(d*sqrt(x) + c) + a)^2*x^(5/2)), x)