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

Optimal. Leaf size=41 \[ -\text{Unintegrable}\left (\frac{\tanh (a+b x)}{x},x\right )+\frac{1}{2} \sinh (2 a) \text{Chi}(2 b x)+\frac{1}{2} \cosh (2 a) \text{Shi}(2 b x) \]

[Out]

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

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Rubi [A]  time = 0.103409, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sinh ^2(a+b x) \tanh (a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[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*} \int \frac{\sinh ^2(a+b x) \tanh (a+b x)}{x} \, dx &=\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 [A]  time = 18.7739, size = 0, normalized size = 0. \[ \int \frac{\sinh ^2(a+b x) \tanh (a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[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]

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Maple [A]  time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm sech} \left (bx+a\right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \,{\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - \frac{1}{4} \,{\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} + 2 \, \int \frac{1}{x e^{\left (2 \, b x + 2 \, a\right )} + x}\,{d x} - \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{3}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{3}{\left (a + b x \right )} \operatorname{sech}{\left (a + b x \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}\left (b x + a\right ) \sinh \left (b x + a\right )^{3}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)