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

Optimal. Leaf size=20 \[ 4 \text {Int}\left (\frac {\text {csch}^2(2 a+2 b x)}{x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.07, 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 = {} \[ \int \frac {\text {csch}^2(a+b x) \text {sech}^2(a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\text {csch}^2(a+b x) \text {sech}^2(a+b x)}{x} \, dx &=4 \int \frac {\text {csch}^2(2 a+2 b x)}{x} \, dx\\ \end {align*}

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Mathematica [A]  time = 23.96, size = 0, normalized size = 0.00 \[ \int \frac {\text {csch}^2(a+b x) \text {sech}^2(a+b x)}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right )^{2}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.53, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {csch}\left (b x +a \right )^{2} \mathrm {sech}\left (b x +a \right )^{2}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {4}{b x e^{\left (4 \, b x + 4 \, a\right )} - b x} + 16 \, \int \frac {1}{8 \, {\left (b x^{2} e^{\left (2 \, b x + 2 \, a\right )} + b x^{2}\right )}}\,{d x} + 16 \, \int \frac {1}{16 \, {\left (b x^{2} e^{\left (b x + a\right )} + b x^{2}\right )}}\,{d x} - 16 \, \int \frac {1}{16 \, {\left (b x^{2} e^{\left (b x + a\right )} - b x^{2}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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