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

Optimal. Leaf size=30 \[ b \text {Int}\left (\frac {\csc \left (c+d \sqrt {x}\right )}{x^{3/2}},x\right )-\frac {2 a}{\sqrt {x}} \]

[Out]

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

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Rubi [A]  time = 0.01, 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 = {} \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 6.35, size = 0, normalized size = 0.00 \[ \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \sqrt {x} \csc \left (d \sqrt {x} + c\right ) + a \sqrt {x}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 2.10, size = 0, normalized size = 0.00 \[ \int \frac {a +b \csc \left (c +d \sqrt {x}\right )}{x^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (b \int \frac {\sin \left (d \sqrt {x} + c\right )}{{\left (\cos \left (d \sqrt {x} + c\right )^{2} + \sin \left (d \sqrt {x} + c\right )^{2} + 2 \, \cos \left (d \sqrt {x} + c\right ) + 1\right )} x^{\frac {3}{2}}}\,{d x} + b \int \frac {\sin \left (d \sqrt {x} + c\right )}{{\left (\cos \left (d \sqrt {x} + c\right )^{2} + \sin \left (d \sqrt {x} + c\right )^{2} - 2 \, \cos \left (d \sqrt {x} + c\right ) + 1\right )} x^{\frac {3}{2}}}\,{d x}\right )} \sqrt {x} - 2 \, a}{\sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*integrate(sin(d*sqrt(x) + c)/((cos(d*sqrt(x) + c)^2 + sin(d*sqrt(x) + c)^2 + 2*cos(d*sqrt(x) + c) + 1)*x^(
3/2)), x) + b*integrate(sin(d*sqrt(x) + c)/((cos(d*sqrt(x) + c)^2 + sin(d*sqrt(x) + c)^2 - 2*cos(d*sqrt(x) + c
) + 1)*x^(3/2)), x))*sqrt(x) - 2*a)/sqrt(x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}}{x^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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