3.44 \(\int \frac {1}{x (a+b \text {csch}(c+d \sqrt {x}))} \, dx\)

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

[Out]

Unintegrable(1/x/(a+b*csch(c+d*x^(1/2))),x)

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Rubi [A]  time = 0.03, 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 = {} \[ \int \frac {1}{x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 5.32, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b x \operatorname {csch}\left (d \sqrt {x} + c\right ) + a x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.71, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a +b \,\mathrm {csch}\left (c +d \sqrt {x}\right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b*integrate(e^(d*sqrt(x) + c)/(a^2*x*e^(2*d*sqrt(x) + 2*c) + 2*a*b*x*e^(d*sqrt(x) + c) - a^2*x), x) + log(x
)/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*(a + b*csch(c + d*sqrt(x)))), x)

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