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

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {1}{x^{5/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2},x\right ) \]

[Out]

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

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*}

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Mathematica [A]  time = 28.50, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{5/2} \left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2} \, dx \]

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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fricas [A]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x}}{b^{2} x^{3} \csc \left (d \sqrt {x} + c\right )^{2} + 2 \, a b x^{3} \csc \left (d \sqrt {x} + c\right ) + a^{2} x^{3}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{\frac {5}{2}}}\,{d x} \]

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)

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maple [A]  time = 3.02, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{\frac {5}{2}} \left (a +b \csc \left (c +d \sqrt {x}\right )\right )^{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{x^{5/2}\,{\left (a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(5/2)*(a + b/sin(c + d*x^(1/2)))^2),x)

[Out]

int(1/(x^(5/2)*(a + b/sin(c + d*x^(1/2)))^2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{\frac {5}{2}} \left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]

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]

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

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