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

Optimal. Leaf size=22 \[ \text{CannotIntegrate}\left (\frac{\text{csch}^2(a+b x) \text{sech}^3(a+b x)}{x^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.30223, 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 = {} \[ \int \frac{\text{csch}^2(a+b x) \text{sech}^3(a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 33.5094, size = 0, normalized size = 0. \[ \int \frac{\text{csch}^2(a+b x) \text{sech}^3(a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.507, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm csch} \left (bx+a\right ) \right ) ^{2} \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)^2*sech(b*x+a)^3/x^2,x)

[Out]

int(csch(b*x+a)^2*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 \, b x e^{\left (3 \, b x + 3 \, a\right )} +{\left (3 \, b x e^{\left (5 \, a\right )} - 2 \, e^{\left (5 \, a\right )}\right )} e^{\left (5 \, b x\right )} +{\left (3 \, b x e^{a} + 2 \, e^{a}\right )} e^{\left (b x\right )}}{b^{2} x^{3} e^{\left (6 \, b x + 6 \, a\right )} + b^{2} x^{3} e^{\left (4 \, b x + 4 \, a\right )} - b^{2} x^{3} e^{\left (2 \, b x + 2 \, a\right )} - b^{2} x^{3}} - 32 \, \int \frac{3 \,{\left (b^{2} x^{2} e^{a} - 2 \, e^{a}\right )} e^{\left (b x\right )}}{32 \,{\left (b^{2} x^{4} e^{\left (2 \, b x + 2 \, a\right )} + b^{2} x^{4}\right )}}\,{d x} - 32 \, \int \frac{1}{16 \,{\left (b x^{3} e^{\left (b x + a\right )} + b x^{3}\right )}}\,{d x} - 32 \, \int \frac{1}{16 \,{\left (b x^{3} e^{\left (b x + a\right )} - b x^{3}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b*x*e^(3*b*x + 3*a) + (3*b*x*e^(5*a) - 2*e^(5*a))*e^(5*b*x) + (3*b*x*e^a + 2*e^a)*e^(b*x))/(b^2*x^3*e^(6*b
*x + 6*a) + b^2*x^3*e^(4*b*x + 4*a) - b^2*x^3*e^(2*b*x + 2*a) - b^2*x^3) - 32*integrate(3/32*(b^2*x^2*e^a - 2*
e^a)*e^(b*x)/(b^2*x^4*e^(2*b*x + 2*a) + b^2*x^4), x) - 32*integrate(1/16/(b*x^3*e^(b*x + a) + b*x^3), x) - 32*
integrate(1/16/(b*x^3*e^(b*x + 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{csch}\left (b x + a\right )^{2} \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)^2*sech(b*x+a)^3/x^2,x, algorithm="fricas")

[Out]

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

[Out]

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

[Out]

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