3.1.59 \(\int \frac {(a+b \csc (c+d \sqrt {x}))^2}{x^{3/2}} \, dx\) [59]

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}},x\right ) \]

[Out]

Unintegrable((a+b*csc(c+d*x^(1/2)))^2/x^(3/2),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 = {} \begin {gather*} \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 17.02, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \csc \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx \end {gather*}

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 = 0.16, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \csc \left (c +d \sqrt {x}\right )\right )^{2}}{x^{\frac {3}{2}}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )^{2}}{x^{\frac {3}{2}}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\left (a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}\right )}^2}{x^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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