\(\int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x^2} \, dx\) [444]

Optimal result

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

[Out]

CannotIntegrate(coth(b*x+a)*csch(b*x+a)/x^2,x)-cosh(b*x+a)/x+b*cosh(a)*Shi(b*x)+b*Chi(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 = {} \[ \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x^2} \, dx=\int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 14.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x^2} \, dx=\int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x^2} \, dx=\int { \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 70.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x^2} \, dx=\int \frac {\cosh ^{3}{\left (a + b x \right )} \operatorname {csch}^{2}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 102, normalized size of antiderivative = 5.67 \[ \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x^2} \, dx=\int { \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

-1/2*b*e^(-a)*gamma(-1, b*x) + 1/2*b*e^a*gamma(-1, -b*x) - 2*e^(b*x + a)/(b*x^2*e^(2*b*x + 2*a) - b*x^2) - 2*i
ntegrate(1/(b*x^3*e^(b*x + a) + b*x^3), x) - 2*integrate(1/(b*x^3*e^(b*x + a) - b*x^3), x)

Giac [N/A]

Not integrable

Time = 1.63 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x^2} \, dx=\int { \frac {\cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.40 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\cosh (a+b x) \coth ^2(a+b x)}{x^2} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3}{x^2\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]

[In]

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

[Out]

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