\(\int \frac {\sinh ^2(a+b x) \tanh (a+b x)}{x} \, dx\) [381]

Optimal result

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

[Out]

1/2*cosh(2*a)*Shi(2*b*x)+1/2*Chi(2*b*x)*sinh(2*a)-Unintegrable(tanh(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 = {} \[ \int \frac {\sinh ^2(a+b x) \tanh (a+b x)}{x} \, dx=\int \frac {\sinh ^2(a+b x) \tanh (a+b x)}{x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 12.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\sinh ^2(a+b x) \tanh (a+b x)}{x} \, dx=\int \frac {\sinh ^2(a+b x) \tanh (a+b x)}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.42 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.82 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {\sinh ^2(a+b x) \tanh (a+b x)}{x} \, dx=\int \frac {\sinh ^{3}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.56 \[ \int \frac {\sinh ^2(a+b x) \tanh (a+b x)}{x} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{3}}{x} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\sinh ^2(a+b x) \tanh (a+b x)}{x} \, dx=\int { \frac {\operatorname {sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{3}}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {\sinh ^2(a+b x) \tanh (a+b x)}{x} \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^3}{x\,\mathrm {cosh}\left (a+b\,x\right )} \,d x \]

[In]

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

[Out]

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