3.155 \(\int x \tanh ^2(a+2 \log (x)) \, dx\)

Optimal. Leaf size=40 \[ -e^{-a} \tan ^{-1}\left (e^a x^2\right )+\frac {x^2}{e^{2 a} x^4+1}+\frac {x^2}{2} \]

[Out]

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

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Rubi [F]  time = 0.03, 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 = {} \[ \int x \tanh ^2(a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x*Tanh[a + 2*Log[x]]^2,x]

[Out]

Defer[Int][x*Tanh[a + 2*Log[x]]^2, x]

Rubi steps

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

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Mathematica [A]  time = 0.41, size = 41, normalized size = 1.02 \[ \frac {x^2}{e^{2 (a+2 \log (x))}+1}-e^{-a} \tan ^{-1}\left (e^a x^2\right )+\frac {x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Tanh[a + 2*Log[x]]^2,x]

[Out]

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

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fricas [A]  time = 0.41, size = 50, normalized size = 1.25 \[ \frac {x^{6} e^{\left (3 \, a\right )} + 3 \, x^{2} e^{a} - 2 \, {\left (x^{4} e^{\left (2 \, a\right )} + 1\right )} \arctan \left (x^{2} e^{a}\right )}{2 \, {\left (x^{4} e^{\left (3 \, a\right )} + e^{a}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.12, size = 35, normalized size = 0.88 \[ \frac {1}{2} \, x^{2} - \arctan \left (x^{2} e^{a}\right ) e^{\left (-a\right )} + \frac {x^{2}}{x^{4} e^{\left (2 \, a\right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.09, size = 57, normalized size = 1.42 \[ \frac {x^{2}}{2}+\frac {x^{2}}{1+{\mathrm e}^{2 a} x^{4}}+\frac {i {\mathrm e}^{-a} \ln \left ({\mathrm e}^{a} x^{2}-i\right )}{2}-\frac {i {\mathrm e}^{-a} \ln \left ({\mathrm e}^{a} x^{2}+i\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2+x^2/(1+exp(2*a)*x^4)+1/2*I*exp(-a)*ln(exp(a)*x^2-I)-1/2*I*exp(-a)*ln(exp(a)*x^2+I)

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maxima [A]  time = 0.43, size = 35, normalized size = 0.88 \[ \frac {1}{2} \, x^{2} - \arctan \left (x^{2} e^{a}\right ) e^{\left (-a\right )} + \frac {x^{2}}{x^{4} e^{\left (2 \, a\right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.05, size = 41, normalized size = 1.02 \[ \frac {x^2}{{\mathrm {e}}^{2\,a}\,x^4+1}-\frac {\mathrm {atan}\left (x^2\,\sqrt {{\mathrm {e}}^{2\,a}}\right )}{\sqrt {{\mathrm {e}}^{2\,a}}}+\frac {x^2}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*tanh(a + 2*log(x))^2,x)

[Out]

x^2/(x^4*exp(2*a) + 1) - atan(x^2*exp(2*a)^(1/2))/exp(2*a)^(1/2) + x^2/2

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \tanh ^{2}{\left (a + 2 \log {\relax (x )} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*tanh(a + 2*log(x))**2, x)

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