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

Optimal. Leaf size=24 \[ b \text {Int}\left (\frac {\text {csch}\left (c+d \sqrt {x}\right )}{x},x\right )+a \log (x) \]

[Out]

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

Verification is Not applicable to the result.

[In]

Int[(a + b*Csch[c + d*Sqrt[x]])/x,x]

[Out]

a*Log[x] + b*Defer[Int][Csch[c + d*Sqrt[x]]/x, x]

Rubi steps

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

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Mathematica [A]  time = 18.27, size = 0, normalized size = 0.00 \[ \int \frac {a+b \text {csch}\left (c+d \sqrt {x}\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.73, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {csch}\left (c +d \sqrt {x}\right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*integrate(1/(x*e^(d*sqrt(x) + c) + x), x) + b*integrate(1/(x*e^(d*sqrt(x) + c) - x), x) + a*log(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/sinh(c + d*x^(1/2)))/x,x)

[Out]

int((a + b/sinh(c + d*x^(1/2)))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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