3.49 \(\int \frac {1}{x (a+b \text {csch}(c+d \sqrt {x}))^2} \, dx\)

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {1}{x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2},x\right ) \]

[Out]

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

________________________________________________________________________________________

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 = {} \[ \int \frac {1}{x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x*(a + b*Csch[c + d*Sqrt[x]])^2),x]

[Out]

Defer[Int][1/(x*(a + b*Csch[c + d*Sqrt[x]])^2), x]

Rubi steps

\begin {align*} \int \frac {1}{x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx &=\int \frac {1}{x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 141.86, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x*(a + b*Csch[c + d*Sqrt[x]])^2),x]

[Out]

Integrate[1/(x*(a + b*Csch[c + d*Sqrt[x]])^2), x]

________________________________________________________________________________________

fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{2} x \operatorname {csch}\left (d \sqrt {x} + c\right )^{2} + 2 \, a b x \operatorname {csch}\left (d \sqrt {x} + c\right ) + a^{2} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 1.04, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a +b \,\mathrm {csch}\left (c +d \sqrt {x}\right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {4 \, {\left (b^{3} \sqrt {x} e^{\left (d \sqrt {x} + c\right )} - a b^{2} \sqrt {x}\right )}}{{\left (a^{5} d e^{\left (2 \, c\right )} + a^{3} b^{2} d e^{\left (2 \, c\right )}\right )} x e^{\left (2 \, d \sqrt {x}\right )} + 2 \, {\left (a^{4} b d e^{c} + a^{2} b^{3} d e^{c}\right )} x e^{\left (d \sqrt {x}\right )} - {\left (a^{5} d + a^{3} b^{2} d\right )} x} + \frac {\log \relax (x)}{a^{2}} + \int -\frac {2 \, {\left (a b^{2} \sqrt {x} - {\left (b^{3} \sqrt {x} e^{c} - {\left (2 \, a^{2} b d e^{c} + b^{3} d e^{c}\right )} x\right )} e^{\left (d \sqrt {x}\right )}\right )}}{{\left (a^{5} d e^{\left (2 \, c\right )} + a^{3} b^{2} d e^{\left (2 \, c\right )}\right )} x^{2} e^{\left (2 \, d \sqrt {x}\right )} + 2 \, {\left (a^{4} b d e^{c} + a^{2} b^{3} d e^{c}\right )} x^{2} e^{\left (d \sqrt {x}\right )} - {\left (a^{5} d + a^{3} b^{2} d\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*(b^3*sqrt(x)*e^(d*sqrt(x) + c) - a*b^2*sqrt(x))/((a^5*d*e^(2*c) + a^3*b^2*d*e^(2*c))*x*e^(2*d*sqrt(x)) + 2*(
a^4*b*d*e^c + a^2*b^3*d*e^c)*x*e^(d*sqrt(x)) - (a^5*d + a^3*b^2*d)*x) + log(x)/a^2 + integrate(-2*(a*b^2*sqrt(
x) - (b^3*sqrt(x)*e^c - (2*a^2*b*d*e^c + b^3*d*e^c)*x)*e^(d*sqrt(x)))/((a^5*d*e^(2*c) + a^3*b^2*d*e^(2*c))*x^2
*e^(2*d*sqrt(x)) + 2*(a^4*b*d*e^c + a^2*b^3*d*e^c)*x^2*e^(d*sqrt(x)) - (a^5*d + a^3*b^2*d)*x^2), x)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{x\,{\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________