\(\int \frac {\sinh (a+b x) \tanh (a+b x)}{x^2} \, dx\) [361]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {\sinh (a+b x) \tanh (a+b x)}{x^2} \, dx=-\frac {\cosh (a+b x)}{x}+b \text {Chi}(b x) \sinh (a)+b \cosh (a) \text {Shi}(b x)-\text {Int}\left (\frac {\text {sech}(a+b x)}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (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 = {} \[ \int \frac {\sinh (a+b x) \tanh (a+b x)}{x^2} \, dx=\int \frac {\sinh (a+b x) \tanh (a+b x)}{x^2} \, dx \]

[In]

Int[(Sinh[a + b*x]*Tanh[a + b*x])/x^2,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 4.61 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\sinh (a+b x) \tanh (a+b x)}{x^2} \, dx=\int \frac {\sinh (a+b x) \tanh (a+b x)}{x^2} \, dx \]

[In]

Integrate[(Sinh[a + b*x]*Tanh[a + b*x])/x^2,x]

[Out]

Integrate[(Sinh[a + b*x]*Tanh[a + b*x])/x^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.34 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {\sinh (a+b x) \tanh (a+b x)}{x^2} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

integral(sech(b*x + a)*sinh(b*x + a)^2/x^2, x)

Sympy [N/A]

Not integrable

Time = 1.41 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {\sinh (a+b x) \tanh (a+b x)}{x^2} \, dx=\int \frac {\sinh ^{2}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

integrate(sech(b*x+a)*sinh(b*x+a)**2/x**2,x)

[Out]

Integral(sinh(a + b*x)**2*sech(a + b*x)/x**2, x)

Maxima [N/A]

Not integrable

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

[In]

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

[Out]

integrate(sech(b*x + a)*sinh(b*x + a)^2/x^2, x)

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \frac {\sinh (a+b x) \tanh (a+b x)}{x^2} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{2}}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(sech(b*x + a)*sinh(b*x + a)^2/x^2, x)

Mupad [N/A]

Not integrable

Time = 2.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {\sinh (a+b x) \tanh (a+b x)}{x^2} \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^2}{x^2\,\mathrm {cosh}\left (a+b\,x\right )} \,d x \]

[In]

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

[Out]

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