3.159 \(\int \frac{\tanh ^2(a+2 \log (x))}{x^3} \, dx\)

Optimal. Leaf size=59 \[ -\frac{3 e^{2 a} x^2}{2 \left (e^{2 a} x^4+1\right )}-\frac{1}{2 x^2 \left (e^{2 a} x^4+1\right )}-e^a \tan ^{-1}\left (e^a x^2\right ) \]

[Out]

-1/(2*x^2*(1 + E^(2*a)*x^4)) - (3*E^(2*a)*x^2)/(2*(1 + E^(2*a)*x^4)) - E^a*ArcTan[E^a*x^2]

________________________________________________________________________________________

Rubi [F]  time = 0.0534242, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\tanh ^2(a+2 \log (x))}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tanh[a + 2*Log[x]]^2/x^3,x]

[Out]

Defer[Int][Tanh[a + 2*Log[x]]^2/x^3, x]

Rubi steps

\begin{align*} \int \frac{\tanh ^2(a+2 \log (x))}{x^3} \, dx &=\int \frac{\tanh ^2(a+2 \log (x))}{x^3} \, dx\\ \end{align*}

Mathematica [A]  time = 0.355313, size = 40, normalized size = 0.68 \[ \frac{-\frac{2}{e^{-2 (a+2 \log (x))}+1}-1}{2 x^2}+e^a \tan ^{-1}\left (\frac{e^{-a}}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[a + 2*Log[x]]^2/x^3,x]

[Out]

(-1 - 2/(1 + E^(-2*(a + 2*Log[x]))))/(2*x^2) + E^a*ArcTan[1/(E^a*x^2)]

________________________________________________________________________________________

Maple [C]  time = 0.029, size = 66, normalized size = 1.1 \begin{align*}{\frac{1}{{x}^{2} \left ( 1+{{\rm e}^{2\,a}}{x}^{4} \right ) } \left ( -{\frac{3\,{{\rm e}^{2\,a}}{x}^{4}}{2}}-{\frac{1}{2}} \right ) }+{\frac{\sum _{{\it \_R}={\it RootOf} \left ({{\rm e}^{2\,a}}+{{\it \_Z}}^{2} \right ) }{\it \_R}\,\ln \left ( \left ( -4\,{{\rm e}^{2\,a}}-5\,{{\it \_R}}^{2} \right ){x}^{2}-{\it \_R} \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a+2*ln(x))^2/x^3,x)

[Out]

(-3/2*exp(2*a)*x^4-1/2)/x^2/(1+exp(2*a)*x^4)+1/2*sum(_R*ln((-4*exp(2*a)-5*_R^2)*x^2-_R),_R=RootOf(exp(2*a)+_Z^
2))

________________________________________________________________________________________

Maxima [A]  time = 1.81886, size = 50, normalized size = 0.85 \begin{align*} \arctan \left (\frac{e^{\left (-a\right )}}{x^{2}}\right ) e^{a} - \frac{1}{2 \, x^{2}} - \frac{e^{\left (2 \, a\right )}}{x^{2}{\left (\frac{1}{x^{4}} + e^{\left (2 \, a\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+2*log(x))^2/x^3,x, algorithm="maxima")

[Out]

arctan(e^(-a)/x^2)*e^a - 1/2/x^2 - e^(2*a)/(x^2*(1/x^4 + e^(2*a)))

________________________________________________________________________________________

Fricas [A]  time = 1.88606, size = 122, normalized size = 2.07 \begin{align*} -\frac{3 \, x^{4} e^{\left (2 \, a\right )} + 2 \,{\left (x^{6} e^{\left (3 \, a\right )} + x^{2} e^{a}\right )} \arctan \left (x^{2} e^{a}\right ) + 1}{2 \,{\left (x^{6} e^{\left (2 \, a\right )} + x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+2*log(x))^2/x^3,x, algorithm="fricas")

[Out]

-1/2*(3*x^4*e^(2*a) + 2*(x^6*e^(3*a) + x^2*e^a)*arctan(x^2*e^a) + 1)/(x^6*e^(2*a) + x^2)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{2}{\left (a + 2 \log{\left (x \right )} \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+2*ln(x))**2/x**3,x)

[Out]

Integral(tanh(a + 2*log(x))**2/x**3, x)

________________________________________________________________________________________

Giac [A]  time = 1.40139, size = 53, normalized size = 0.9 \begin{align*} -\arctan \left (x^{2} e^{a}\right ) e^{a} - \frac{3 \, x^{4} e^{\left (2 \, a\right )} + 1}{2 \,{\left (x^{6} e^{\left (2 \, a\right )} + x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+2*log(x))^2/x^3,x, algorithm="giac")

[Out]

-arctan(x^2*e^a)*e^a - 1/2*(3*x^4*e^(2*a) + 1)/(x^6*e^(2*a) + x^2)