\(\int \frac {\coth (a+b x) \text {csch}(a+b x)}{x} \, dx\) [429]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {\coth (a+b x) \text {csch}(a+b x)}{x} \, dx=\text {Int}\left (\frac {\coth (a+b x) \text {csch}(a+b x)}{x},x\right ) \]

[Out]

CannotIntegrate(coth(b*x+a)*csch(b*x+a)/x,x)

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\coth (a+b x) \text {csch}(a+b x)}{x} \, dx=\int \frac {\coth (a+b x) \text {csch}(a+b x)}{x} \, dx \]

[In]

Int[(Coth[a + b*x]*Csch[a + b*x])/x,x]

[Out]

Defer[Int][(Coth[a + b*x]*Csch[a + b*x])/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\coth (a+b x) \text {csch}(a+b x)}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 22.78 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\coth (a+b x) \text {csch}(a+b x)}{x} \, dx=\int \frac {\coth (a+b x) \text {csch}(a+b x)}{x} \, dx \]

[In]

Integrate[(Coth[a + b*x]*Csch[a + b*x])/x,x]

[Out]

Integrate[(Coth[a + b*x]*Csch[a + b*x])/x, x]

Maple [N/A] (verified)

Not integrable

Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12

\[\int \frac {\cosh \left (b x +a \right ) \operatorname {csch}\left (b x +a \right )^{2}}{x}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {\coth (a+b x) \text {csch}(a+b x)}{x} \, dx=\int { \frac {\cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 9.12 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {\coth (a+b x) \text {csch}(a+b x)}{x} \, dx=\int \frac {\cosh {\left (a + b x \right )} \operatorname {csch}^{2}{\left (a + b x \right )}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 4.94 \[ \int \frac {\coth (a+b x) \text {csch}(a+b x)}{x} \, dx=\int { \frac {\cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.47 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {\coth (a+b x) \text {csch}(a+b x)}{x} \, dx=\int { \frac {\cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {\coth (a+b x) \text {csch}(a+b x)}{x} \, dx=\int \frac {\mathrm {cosh}\left (a+b\,x\right )}{x\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]

[In]

int(cosh(a + b*x)/(x*sinh(a + b*x)^2),x)

[Out]

int(cosh(a + b*x)/(x*sinh(a + b*x)^2), x)