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

Optimal. Leaf size=19 \[ \text {Int}\left (\frac {\tanh (a+b x) \text {sech}(a+b x)}{x^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.20, 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 = {} \[ \int \frac {\text {sech}(a+b x) \tanh (a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 8.20, size = 0, normalized size = 0.00 \[ \int \frac {\text {sech}(a+b x) \tanh (a+b x)}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {sech}\left (b x +a \right )^{2} \sinh \left (b x +a \right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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