### 3.388 $$\int \frac{\sinh (a+b x) \tanh ^2(a+b x)}{x} \, dx$$

Optimal. Leaf size=35 $-\text{CannotIntegrate}\left (\frac{\tanh (a+b x) \text{sech}(a+b x)}{x},x\right )+\sinh (a) \text{Chi}(b x)+\cosh (a) \text{Shi}(b x)$

[Out]

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

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Rubi [A]  time = 0.128341, 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 = {} $\int \frac{\sinh (a+b x) \tanh ^2(a+b x)}{x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 11.5038, size = 0, normalized size = 0. $\int \frac{\sinh (a+b x) \tanh ^2(a+b x)}{x} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.082, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm sech} \left (bx+a\right ) \right ) ^{2} \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)^2*sinh(b*x+a)^3/x,x)

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*Ei(-b*x)*e^(-a) + 1/2*Ei(b*x)*e^a + 2*e^(b*x + a)/(b*x*e^(2*b*x + 2*a) + b*x) + 2*integrate(e^(b*x + a)/(
b*x^2*e^(2*b*x + 2*a) + b*x^2), 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 )^{2} \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)^2*sinh(b*x+a)^3/x,x, algorithm="fricas")

[Out]

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

[Out]

Integral(sinh(a + b*x)**3*sech(a + b*x)**2/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 )^{2} \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)^2*sinh(b*x+a)^3/x,x, algorithm="giac")

[Out]

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