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

Optimal. Leaf size=42 \[ -\text{Unintegrable}\left (\frac{\text{sech}(a+b x)}{x^2},x\right )+b \sinh (a) \text{Chi}(b x)+b \cosh (a) \text{Shi}(b x)-\frac{\cosh (a+b x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.106884, 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 = {} \[ \int \frac{\sinh (a+b x) \tanh (a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[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*} \int \frac{\sinh (a+b x) \tanh (a+b x)}{x^2} \, dx &=\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 [A]  time = 8.73568, size = 0, normalized size = 0. \[ \int \frac{\sinh (a+b x) \tanh (a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[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]

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Maple [A]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm sech} \left (bx+a\right ) \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [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 )^{2}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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 )^{2}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{2}{\left (a + b x \right )} \operatorname{sech}{\left (a + b x \right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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 )^{2}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)