3.486 $$\int \frac{\text{csch}(a+b x) \text{sech}^3(a+b x)}{x^2} \, dx$$

Optimal. Leaf size=20 $\text{CannotIntegrate}\left (\frac{\text{csch}(a+b x) \text{sech}^3(a+b x)}{x^2},x\right )$

[Out]

CannotIntegrate[(Csch[a + b*x]*Sech[a + b*x]^3)/x^2, x]

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Rubi [A]  time = 0.254342, 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 = {} $\int \frac{\text{csch}(a+b x) \text{sech}^3(a+b x)}{x^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Int[(Csch[a + b*x]*Sech[a + b*x]^3)/x^2,x]

[Out]

Defer[Int][(Csch[a + b*x]*Sech[a + b*x]^3)/x^2, x]

Rubi steps

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

Mathematica [A]  time = 28.0959, size = 0, normalized size = 0. $\int \frac{\text{csch}(a+b x) \text{sech}^3(a+b x)}{x^2} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Integrate[(Csch[a + b*x]*Sech[a + b*x]^3)/x^2,x]

[Out]

Integrate[(Csch[a + b*x]*Sech[a + b*x]^3)/x^2, x]

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Maple [A]  time = 0.53, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm csch} \left (bx+a\right ) \left ({\rm sech} \left (bx+a\right ) \right ) ^{3}}{{x}^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left ({\left (b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )} - 1\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}} + 16 \, \int \frac{b^{2} x^{2} - 3}{8 \,{\left (b^{2} x^{4} e^{\left (2 \, b x + 2 \, a\right )} + b^{2} x^{4}\right )}}\,{d x} - 16 \, \int \frac{1}{16 \,{\left (x^{2} e^{\left (b x + a\right )} + x^{2}\right )}}\,{d x} + 16 \, \int \frac{1}{16 \,{\left (x^{2} e^{\left (b x + a\right )} - x^{2}\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((b*x*e^(2*a) - e^(2*a))*e^(2*b*x) - 1)/(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) + 16
*integrate(1/8*(b^2*x^2 - 3)/(b^2*x^4*e^(2*b*x + 2*a) + b^2*x^4), x) - 16*integrate(1/16/(x^2*e^(b*x + a) + x^
2), x) + 16*integrate(1/16/(x^2*e^(b*x + a) - x^2), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{csch}\left (b x + a\right ) \operatorname{sech}\left (b x + a\right )^{3}}{x^{2}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}{\left (a + b x \right )} \operatorname{sech}^{3}{\left (a + b x \right )}}{x^{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csch(a + b*x)*sech(a + b*x)**3/x**2, x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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