3.5.100 \(\int \frac {\text {csch}^2(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx\) [500]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.05, 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 = {} \begin {gather*} \int \frac {\text {csch}^2(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 12.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {csch}^2(a+b x) \text {sech}^2(a+b x)}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 1.25, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {csch}\left (b x +a \right )^{2} \mathrm {sech}\left (b x +a \right )^{2}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________