### 3.375 $$\int \frac{\text{sech}(a+b x) \tanh ^2(a+b x)}{x^2} \, dx$$

Optimal. Leaf size=29 $\text{Unintegrable}\left (\frac{\text{sech}(a+b x)}{x^2},x\right )-\text{Unintegrable}\left (\frac{\text{sech}^3(a+b x)}{x^2},x\right )$

[Out]

Unintegrable[Sech[a + b*x]/x^2, x] - Unintegrable[Sech[a + b*x]^3/x^2, x]

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Rubi [A]  time = 0.0685913, 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 = {} $\int \frac{\text{sech}(a+b x) \tanh ^2(a+b x)}{x^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

Defer[Int][Sech[a + b*x]/x^2, x] - Defer[Int][Sech[a + b*x]^3/x^2, x]

Rubi steps

\begin{align*} \int \frac{\text{sech}(a+b x) \tanh ^2(a+b x)}{x^2} \, dx &=\int \frac{\text{sech}(a+b x)}{x^2} \, dx-\int \frac{\text{sech}^3(a+b x)}{x^2} \, dx\\ \end{align*}

Mathematica [A]  time = 12.8311, size = 0, normalized size = 0. $\int \frac{\text{sech}(a+b x) \tanh ^2(a+b x)}{x^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.195, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm sech} \left (bx+a\right ) \right ) ^{3} \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)^3*sinh(b*x+a)^2/x^2,x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (b x e^{\left (3 \, a\right )} - 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} -{\left (b x e^{a} + 2 \, e^{a}\right )} e^{\left (b x\right )}}{b^{2} x^{3} e^{\left (4 \, b x + 4 \, a\right )} + 2 \, b^{2} x^{3} e^{\left (2 \, b x + 2 \, a\right )} + b^{2} x^{3}} + 2 \, \int \frac{{\left (b^{2} x^{2} e^{a} + 6 \, e^{a}\right )} e^{\left (b x\right )}}{2 \,{\left (b^{2} x^{4} e^{\left (2 \, b x + 2 \, a\right )} + b^{2} x^{4}\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((b*x*e^(3*a) - 2*e^(3*a))*e^(3*b*x) - (b*x*e^a + 2*e^a)*e^(b*x))/(b^2*x^3*e^(4*b*x + 4*a) + 2*b^2*x^3*e^(2*b
*x + 2*a) + b^2*x^3) + 2*integrate(1/2*(b^2*x^2*e^a + 6*e^a)*e^(b*x)/(b^2*x^4*e^(2*b*x + 2*a) + b^2*x^4), 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 )^{3} \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)^3*sinh(b*x+a)^2/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}\left (b x + a\right )^{3} \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)^3*sinh(b*x+a)^2/x^2,x, algorithm="giac")

[Out]

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