3.59 \(\int \frac{(a+b \csc (c+d \sqrt{x}))^2}{x^{3/2}} \, dx\)

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{\left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2}{x^{3/2}},x\right ) \]

[Out]

Unintegrable[(a + b*Csc[c + d*Sqrt[x]])^2/x^(3/2), x]

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Rubi [A]  time = 0.0219467, 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 = {} \[ \int \frac{\left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2}{x^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Csc[c + d*Sqrt[x]])^2/x^(3/2),x]

[Out]

Defer[Int][(a + b*Csc[c + d*Sqrt[x]])^2/x^(3/2), x]

Rubi steps

\begin{align*} \int \frac{\left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2}{x^{3/2}} \, dx &=\int \frac{\left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2}{x^{3/2}} \, dx\\ \end{align*}

Mathematica [A]  time = 21.1126, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \csc \left (c+d \sqrt{x}\right )\right )^2}{x^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Csc[c + d*Sqrt[x]])^2/x^(3/2),x]

[Out]

Integrate[(a + b*Csc[c + d*Sqrt[x]])^2/x^(3/2), x]

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Maple [A]  time = 0.19, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\csc \left ( c+d\sqrt{x} \right ) \right ) ^{2}{x}^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*csc(c+d*x^(1/2)))^2/x^(3/2),x)

[Out]

int((a+b*csc(c+d*x^(1/2)))^2/x^(3/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*csc(c+d*x^(1/2)))^2/x^(3/2),x, algorithm="maxima")

[Out]

-(4*b^2*sin(2*d*sqrt(x) + 2*c) - ((d*integrate(2*(a*b*d*sqrt(x)*sin(d*sqrt(x) + c) + b^2*sin(d*sqrt(x) + c))/(
(d*cos(d*sqrt(x) + c)^2 + d*sin(d*sqrt(x) + c)^2 + 2*d*cos(d*sqrt(x) + c) + d)*x^2), x) + d*integrate(2*(a*b*d
*sqrt(x)*sin(d*sqrt(x) + c) - b^2*sin(d*sqrt(x) + c))/((d*cos(d*sqrt(x) + c)^2 + d*sin(d*sqrt(x) + c)^2 - 2*d*
cos(d*sqrt(x) + c) + d)*x^2), x))*cos(2*d*sqrt(x) + 2*c)^2 + (d*integrate(2*(a*b*d*sqrt(x)*sin(d*sqrt(x) + c)
+ b^2*sin(d*sqrt(x) + c))/((d*cos(d*sqrt(x) + c)^2 + d*sin(d*sqrt(x) + c)^2 + 2*d*cos(d*sqrt(x) + c) + d)*x^2)
, x) + d*integrate(2*(a*b*d*sqrt(x)*sin(d*sqrt(x) + c) - b^2*sin(d*sqrt(x) + c))/((d*cos(d*sqrt(x) + c)^2 + d*
sin(d*sqrt(x) + c)^2 - 2*d*cos(d*sqrt(x) + c) + d)*x^2), x))*sin(2*d*sqrt(x) + 2*c)^2 - 2*(d*integrate(2*(a*b*
d*sqrt(x)*sin(d*sqrt(x) + c) + b^2*sin(d*sqrt(x) + c))/((d*cos(d*sqrt(x) + c)^2 + d*sin(d*sqrt(x) + c)^2 + 2*d
*cos(d*sqrt(x) + c) + d)*x^2), x) + d*integrate(2*(a*b*d*sqrt(x)*sin(d*sqrt(x) + c) - b^2*sin(d*sqrt(x) + c))/
((d*cos(d*sqrt(x) + c)^2 + d*sin(d*sqrt(x) + c)^2 - 2*d*cos(d*sqrt(x) + c) + d)*x^2), x))*cos(2*d*sqrt(x) + 2*
c) + d*integrate(2*(a*b*d*sqrt(x)*sin(d*sqrt(x) + c) + b^2*sin(d*sqrt(x) + c))/((d*cos(d*sqrt(x) + c)^2 + d*si
n(d*sqrt(x) + c)^2 + 2*d*cos(d*sqrt(x) + c) + d)*x^2), x) + d*integrate(2*(a*b*d*sqrt(x)*sin(d*sqrt(x) + c) -
b^2*sin(d*sqrt(x) + c))/((d*cos(d*sqrt(x) + c)^2 + d*sin(d*sqrt(x) + c)^2 - 2*d*cos(d*sqrt(x) + c) + d)*x^2),
x))*x + 2*(a^2*d*cos(2*d*sqrt(x) + 2*c)^2 + a^2*d*sin(2*d*sqrt(x) + 2*c)^2 - 2*a^2*d*cos(2*d*sqrt(x) + 2*c) +
a^2*d)*sqrt(x))/((d*cos(2*d*sqrt(x) + 2*c)^2 + d*sin(2*d*sqrt(x) + 2*c)^2 - 2*d*cos(2*d*sqrt(x) + 2*c) + d)*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*csc(c+d*x^(1/2)))^2/x^(3/2),x, algorithm="fricas")

[Out]

integral((b^2*sqrt(x)*csc(d*sqrt(x) + c)^2 + 2*a*b*sqrt(x)*csc(d*sqrt(x) + c) + a^2*sqrt(x))/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 \sqrt{x} \right )}\right )^{2}}{x^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*csc(c+d*x**(1/2)))**2/x**(3/2),x)

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \csc \left (d \sqrt{x} + c\right ) + a\right )}^{2}}{x^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*csc(c+d*x^(1/2)))^2/x^(3/2),x, algorithm="giac")

[Out]

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