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

Optimal. Leaf size=18 \[ 2 \text {Int}\left (\frac {\text {csch}(2 a+2 b x)}{x^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, 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 = {} \[ \int \frac {\text {csch}(a+b x) \text {sech}(a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 16.01, size = 0, normalized size = 0.00 \[ \int \frac {\text {csch}(a+b x) \text {sech}(a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {csch}\left (b x + a\right ) \operatorname {sech}\left (b x + a\right )}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.49, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {csch}\left (b x +a \right ) \mathrm {sech}\left (b x +a \right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{x^2\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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