\(\int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx\) [417]

Optimal result

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

[Out]

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

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 7.40 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx=\int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.44 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx=\int { \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 26.59 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx=\int \frac {\cosh ^{3}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 72, normalized size of antiderivative = 4.00 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx=\int { \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

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

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\cosh ^2(a+b x) \coth (a+b x)}{x^2} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{x^2\,\mathrm {sinh}\left (a+b\,x\right )} \,d x \]

[In]

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

[Out]

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