\(\int \frac {\cosh (a+b x) \coth (a+b x)}{x} \, dx\) [409]

Optimal result

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

[Out]

cosh(a)*Shi(b*x)+Chi(b*x)*sinh(a)+Unintegrable(csch(b*x+a)/x,x)

Rubi [N/A]

Not integrable

Time = 0.06 (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 = {} \[ \int \frac {\cosh (a+b x) \coth (a+b x)}{x} \, dx=\int \frac {\cosh (a+b x) \coth (a+b x)}{x} \, dx \]

[In]

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

[Out]

CoshIntegral[b*x]*Sinh[a] + Cosh[a]*SinhIntegral[b*x] + Defer[Int][Csch[a + b*x]/x, x]

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

Mathematica [N/A]

Not integrable

Time = 8.90 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\cosh (a+b x) \coth (a+b x)}{x} \, dx=\int \frac {\cosh (a+b x) \coth (a+b x)}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.35 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {\cosh (a+b x) \coth (a+b x)}{x} \, dx=\int { \frac {\cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )}{x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 9.38 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {\cosh (a+b x) \coth (a+b x)}{x} \, dx=\int \frac {\cosh ^{2}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {\cosh (a+b x) \coth (a+b x)}{x} \, dx=\int { \frac {\cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )}{x} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {\cosh (a+b x) \coth (a+b x)}{x} \, dx=\int { \frac {\cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {\cosh (a+b x) \coth (a+b x)}{x} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2}{x\,\mathrm {sinh}\left (a+b\,x\right )} \,d x \]

[In]

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

[Out]

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