\(\int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx\) [416]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx=\frac {1}{2} \text {Chi}(2 b x) \sinh (2 a)+\frac {1}{2} \cosh (2 a) \text {Shi}(2 b x)+\text {Int}\left (\frac {\coth (a+b x)}{x},x\right ) \]

[Out]

1/2*cosh(2*a)*Shi(2*b*x)+1/2*Chi(2*b*x)*sinh(2*a)+Unintegrable(coth(b*x+a)/x,x)

Rubi [N/A]

Not integrable

Time = 0.08 (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 = {} \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx=\int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx \]

[In]

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

[Out]

(CoshIntegral[2*b*x]*Sinh[2*a])/2 + (Cosh[2*a]*SinhIntegral[2*b*x])/2 + Defer[Int][Coth[a + b*x]/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\coth (a+b x)}{x} \, dx+\int \frac {\cosh (a+b x) \sinh (a+b x)}{x} \, dx \\ & = \int \frac {\coth (a+b x)}{x} \, dx+\int \frac {\sinh (2 a+2 b x)}{2 x} \, dx \\ & = \frac {1}{2} \int \frac {\sinh (2 a+2 b x)}{x} \, dx+\int \frac {\coth (a+b x)}{x} \, dx \\ & = \frac {1}{2} \cosh (2 a) \int \frac {\sinh (2 b x)}{x} \, dx+\frac {1}{2} \sinh (2 a) \int \frac {\cosh (2 b x)}{x} \, dx+\int \frac {\coth (a+b x)}{x} \, dx \\ & = \frac {1}{2} \text {Chi}(2 b x) \sinh (2 a)+\frac {1}{2} \cosh (2 a) \text {Shi}(2 b x)+\int \frac {\coth (a+b x)}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 7.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx=\int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.40 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx=\int { \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )}{x} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 21.80 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx=\int \frac {\cosh ^{3}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.17 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx=\int { \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )}{x} \,d x } \]

[In]

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

[Out]

1/4*Ei(2*b*x)*e^(2*a) - 1/4*Ei(-2*b*x)*e^(-2*a) - integrate(1/(x*e^(b*x + a) + x), x) + integrate(1/(x*e^(b*x
+ a) - x), x) + log(x)

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx=\int { \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{x\,\mathrm {sinh}\left (a+b\,x\right )} \,d x \]

[In]

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

[Out]

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