\(\int \frac {\sinh (a+b x) \tanh ^2(a+b x)}{x^2} \, dx\) [389]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.11 (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 = {} \[ \int \frac {\sinh (a+b x) \tanh ^2(a+b x)}{x^2} \, dx=\int \frac {\sinh (a+b x) \tanh ^2(a+b x)}{x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 6.40 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\sinh (a+b x) \tanh ^2(a+b x)}{x^2} \, dx=\int \frac {\sinh (a+b x) \tanh ^2(a+b x)}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.35 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\sinh (a+b x) \tanh ^2(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{3}}{x^{2}} \,d x } \]

[In]

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

[Out]

integral(sech(b*x + a)^2*sinh(b*x + a)^3/x^2, x)

Sympy [N/A]

Not integrable

Time = 10.90 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\sinh (a+b x) \tanh ^2(a+b x)}{x^2} \, dx=\int \frac {\sinh ^{3}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}}{x^{2}}\, dx \]

[In]

integrate(sech(b*x+a)**2*sinh(b*x+a)**3/x**2,x)

[Out]

Integral(sinh(a + b*x)**3*sech(a + b*x)**2/x**2, x)

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 87, normalized size of antiderivative = 4.83 \[ \int \frac {\sinh (a+b x) \tanh ^2(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{3}}{x^{2}} \,d x } \]

[In]

integrate(sech(b*x+a)^2*sinh(b*x+a)^3/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) + 4*in
tegrate(e^(b*x + a)/(b*x^3*e^(2*b*x + 2*a) + b*x^3), x)

Giac [N/A]

Not integrable

Time = 1.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\sinh (a+b x) \tanh ^2(a+b x)}{x^2} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{3}}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(sech(b*x + a)^2*sinh(b*x + a)^3/x^2, x)

Mupad [N/A]

Not integrable

Time = 2.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\sinh (a+b x) \tanh ^2(a+b x)}{x^2} \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^3}{x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2} \,d x \]

[In]

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

[Out]

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