\(\int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx\) [443]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx=\cosh (a) \text {Chi}(b x)+\sinh (a) \text {Shi}(b x)+\text {Int}\left (\frac {\coth (a+b x) \text {csch}(a+b x)}{x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx=\int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx \]

[In]

Int[(Cosh[a + b*x]*Coth[a + b*x]^2)/x,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 18.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx=\int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx \]

[In]

Integrate[(Cosh[a + b*x]*Coth[a + b*x]^2)/x,x]

[Out]

Integrate[(Cosh[a + b*x]*Coth[a + b*x]^2)/x, x]

Maple [N/A] (verified)

Not integrable

Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11

\[\int \frac {\cosh \left (b x +a \right )^{3} \operatorname {csch}\left (b x +a \right )^{2}}{x}d x\]

[In]

int(cosh(b*x+a)^3*csch(b*x+a)^2/x,x)

[Out]

int(cosh(b*x+a)^3*csch(b*x+a)^2/x,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx=\int { \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )^{2}}{x} \,d x } \]

[In]

integrate(cosh(b*x+a)^3*csch(b*x+a)^2/x,x, algorithm="fricas")

[Out]

integral(cosh(b*x + a)^3*csch(b*x + a)^2/x, x)

Sympy [N/A]

Not integrable

Time = 52.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx=\int \frac {\cosh ^{3}{\left (a + b x \right )} \operatorname {csch}^{2}{\left (a + b x \right )}}{x}\, dx \]

[In]

integrate(cosh(b*x+a)**3*csch(b*x+a)**2/x,x)

[Out]

Integral(cosh(a + b*x)**3*csch(a + b*x)**2/x, x)

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 94, normalized size of antiderivative = 5.22 \[ \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx=\int { \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )^{2}}{x} \,d x } \]

[In]

integrate(cosh(b*x+a)^3*csch(b*x+a)^2/x,x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.75 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx=\int { \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )^{2}}{x} \,d x } \]

[In]

integrate(cosh(b*x+a)^3*csch(b*x+a)^2/x,x, algorithm="giac")

[Out]

integrate(cosh(b*x + a)^3*csch(b*x + a)^2/x, x)

Mupad [N/A]

Not integrable

Time = 2.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{x\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]

[In]

int(cosh(a + b*x)^3/(x*sinh(a + b*x)^2),x)

[Out]

int(cosh(a + b*x)^3/(x*sinh(a + b*x)^2), x)