3.1.44 \(\int \frac {1}{x (a+b \csc (c+d \sqrt {x}))} \, dx\) [44]

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {1}{x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )},x\right ) \]

[Out]

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

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 2.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (a+b \csc \left (c+d \sqrt {x}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {1}{x \left (a +b \csc \left (c +d \sqrt {x}\right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-(2*a*b*integrate((2*b*cos(d*sqrt(x) + c)^2 + a*cos(d*sqrt(x) + c)*sin(2*d*sqrt(x) + 2*c) - a*cos(2*d*sqrt(x)
+ 2*c)*sin(d*sqrt(x) + c) + 2*b*sin(d*sqrt(x) + c)^2 + a*sin(d*sqrt(x) + c))/((a^3*cos(2*d*sqrt(x) + 2*c)^2 +
4*a*b^2*cos(d*sqrt(x) + c)^2 + 4*a^2*b*cos(d*sqrt(x) + c)*sin(2*d*sqrt(x) + 2*c) + a^3*sin(2*d*sqrt(x) + 2*c)^
2 + 4*a*b^2*sin(d*sqrt(x) + c)^2 + 4*a^2*b*sin(d*sqrt(x) + c) + a^3 - 2*(2*a^2*b*sin(d*sqrt(x) + c) + a^3)*cos
(2*d*sqrt(x) + 2*c))*x), x) - log(x))/a

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

[Out]

integral(1/(b*x*csc(d*sqrt(x) + c) + a*x), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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