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

Optimal. Leaf size=22 \[ b \text {Int}\left (\frac {\text {csch}\left (c+d x^2\right )}{x},x\right )+a \log (x) \]

[Out]

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

________________________________________________________________________________________

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

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 = 9.65, size = 0, normalized size = 0.00 \[ \int \frac {a+b \text {csch}\left (c+d x^2\right )}{x} \, dx \]

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]

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {csch}\left (d x^{2} + c\right ) + a}{x}, x\right ) \]

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

maple [A]  time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {csch}\left (d \,x^{2}+c \right )}{x}\, dx \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ 2 \, b \int \frac {1}{x {\left (e^{\left (d x^{2} + c\right )} - e^{\left (-d x^{2} - c\right )}\right )}}\,{d x} + a \log \relax (x) \]

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)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/sinh(c + d*x^2))/x,x)

[Out]

int((a + b/sinh(c + d*x^2))/x, x)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________