### 3.347 $$\int \frac{\text{sech}(a+b x) \tanh (a+b x)}{x^2} \, dx$$

Optimal. Leaf size=18 $\text{CannotIntegrate}\left (\frac{\tanh (a+b x) \text{sech}(a+b x)}{x^2},x\right )$

[Out]

CannotIntegrate[(Sech[a + b*x]*Tanh[a + b*x])/x^2, x]

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Rubi [A]  time = 0.196115, 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 = {} $\int \frac{\text{sech}(a+b x) \tanh (a+b x)}{x^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 7.77246, size = 0, normalized size = 0. $\int \frac{\text{sech}(a+b x) \tanh (a+b x)}{x^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm sech} \left (bx+a\right ) \right ) ^{2}\sinh \left ( bx+a \right ) }{{x}^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \, e^{\left (b x + a\right )}}{b x^{2} e^{\left (2 \, b x + 2 \, a\right )} + b x^{2}} - 4 \, \int \frac{e^{\left (b x + a\right )}}{b x^{3} e^{\left (2 \, b x + 2 \, a\right )} + b x^{3}}\,{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)/x^2,x, algorithm="maxima")

[Out]

-2*e^(b*x + a)/(b*x^2*e^(2*b*x + 2*a) + b*x^2) - 4*integrate(e^(b*x + a)/(b*x^3*e^(2*b*x + 2*a) + b*x^3), 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 )}{x^{2}}, 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)/x^2,x, algorithm="fricas")

[Out]

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

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

[Out]

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

[Out]

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