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

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {\text {csch}^2(a+b x) \text {sech}(a+b x)}{x^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.21, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\text {csch}^2(a+b x) \text {sech}(a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 26.00, size = 0, normalized size = 0.00 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}(a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right )}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError >> type

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maple [A]  time = 1.10, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {csch}\left (b x +a \right )^{2} \mathrm {sech}\left (b x +a \right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, e^{\left (b x + a\right )}}{b x^{2} e^{\left (2 \, b x + 2 \, a\right )} - b x^{2}} - 8 \, \int \frac {e^{\left (b x + a\right )}}{4 \, {\left (x^{2} e^{\left (2 \, b x + 2 \, a\right )} + x^{2}\right )}}\,{d x} - 8 \, \int \frac {1}{4 \, {\left (b x^{3} e^{\left (b x + a\right )} + b x^{3}\right )}}\,{d x} - 8 \, \int \frac {1}{4 \, {\left (b x^{3} e^{\left (b x + a\right )} - b x^{3}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{x^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*cosh(a + b*x)*sinh(a + b*x)^2),x)

[Out]

int(1/(x^2*cosh(a + b*x)*sinh(a + b*x)^2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}^{2}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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