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

Optimal. Leaf size=21 \[ b \text{Unintegrable}\left (\frac{\text{csch}\left (c+d x^2\right )}{x},x\right )+a \log (x) \]

[Out]

a*Log[x] + b*Unintegrable[Csch[c + d*x^2]/x, x]

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Rubi [A]  time = 0.0183072, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{a+b \text{csch}\left (c+d x^2\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Csch[c + d*x^2])/x,x]

[Out]

a*Log[x] + b*Defer[Int][Csch[c + d*x^2]/x, x]

Rubi steps

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

Mathematica [A]  time = 8.64687, size = 0, normalized size = 0. \[ \int \frac{a+b \text{csch}\left (c+d x^2\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm csch} \left (d{x}^{2}+c\right )}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} 2 \, b \int \frac{1}{x{\left (e^{\left (d x^{2} + c\right )} - e^{\left (-d x^{2} - c\right )}\right )}}\,{d x} + a \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*integrate(1/(x*(e^(d*x^2 + c) - e^(-d*x^2 - c))), x) + a*log(x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{csch}\left (d x^{2} + c\right ) + a}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{csch}{\left (c + d x^{2} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{csch}\left (d x^{2} + c\right ) + a}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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