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

Optimal. Leaf size=173 \[ \frac{x^3}{e^{2 a} x^4+1}-\frac{3 e^{-3 a/2} \log \left (e^a x^2-\sqrt{2} e^{a/2} x+1\right )}{4 \sqrt{2}}+\frac{3 e^{-3 a/2} \log \left (e^a x^2+\sqrt{2} e^{a/2} x+1\right )}{4 \sqrt{2}}+\frac{3 e^{-3 a/2} \tan ^{-1}\left (1-\sqrt{2} e^{a/2} x\right )}{2 \sqrt{2}}-\frac{3 e^{-3 a/2} \tan ^{-1}\left (\sqrt{2} e^{a/2} x+1\right )}{2 \sqrt{2}}+\frac{x^3}{3} \]

[Out]

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

________________________________________________________________________________________

Rubi [F]  time = 0.0479177, 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 = {} \[ \int x^2 \tanh ^2(a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*Tanh[a + 2*Log[x]]^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.649469, size = 174, normalized size = 1.01 \[ \frac{1}{12} \left (\frac{12 x^3}{e^{2 a} x^4+1}+9 (-1)^{3/4} e^{-3 a/2} \log \left (\sqrt [4]{-1} e^{-3 a/2}-e^{-a} x\right )+9 \sqrt [4]{-1} e^{-3 a/2} \log \left ((-1)^{3/4} e^{-3 a/2}-e^{-a} x\right )-9 (-1)^{3/4} e^{-3 a/2} \log \left (e^{-a} x+\sqrt [4]{-1} e^{-3 a/2}\right )-9 \sqrt [4]{-1} e^{-3 a/2} \log \left (e^{-a} x+(-1)^{3/4} e^{-3 a/2}\right )+4 x^3\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Tanh[a + 2*Log[x]]^2,x]

[Out]

(4*x^3 + (12*x^3)/(1 + E^(2*a)*x^4) + (9*(-1)^(3/4)*Log[(-1)^(1/4)/E^((3*a)/2) - x/E^a])/E^((3*a)/2) + (9*(-1)
^(1/4)*Log[(-1)^(3/4)/E^((3*a)/2) - x/E^a])/E^((3*a)/2) - (9*(-1)^(3/4)*Log[(-1)^(1/4)/E^((3*a)/2) + x/E^a])/E
^((3*a)/2) - (9*(-1)^(1/4)*Log[(-1)^(3/4)/E^((3*a)/2) + x/E^a])/E^((3*a)/2))/12

________________________________________________________________________________________

Maple [C]  time = 0.022, size = 53, normalized size = 0.3 \begin{align*}{\frac{{x}^{3}}{3}}+{\frac{{x}^{3}}{1+{{\rm e}^{2\,a}}{x}^{4}}}-{\frac{3\,{{\rm e}^{-2\,a}}}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\rm e}^{2\,a}}{{\it \_Z}}^{4}+1 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{\it \_R}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.77848, size = 194, normalized size = 1.12 \begin{align*} \frac{1}{3} \, x^{3} - \frac{3}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x e^{a} + \sqrt{2} e^{\left (\frac{1}{2} \, a\right )}\right )} e^{\left (-\frac{1}{2} \, a\right )}\right ) e^{\left (-\frac{3}{2} \, a\right )} - \frac{3}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x e^{a} - \sqrt{2} e^{\left (\frac{1}{2} \, a\right )}\right )} e^{\left (-\frac{1}{2} \, a\right )}\right ) e^{\left (-\frac{3}{2} \, a\right )} + \frac{3}{8} \, \sqrt{2} e^{\left (-\frac{3}{2} \, a\right )} \log \left (x^{2} e^{a} + \sqrt{2} x e^{\left (\frac{1}{2} \, a\right )} + 1\right ) - \frac{3}{8} \, \sqrt{2} e^{\left (-\frac{3}{2} \, a\right )} \log \left (x^{2} e^{a} - \sqrt{2} x e^{\left (\frac{1}{2} \, a\right )} + 1\right ) + \frac{x^{3}}{x^{4} e^{\left (2 \, a\right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 - 3/4*sqrt(2)*arctan(1/2*sqrt(2)*(2*x*e^a + sqrt(2)*e^(1/2*a))*e^(-1/2*a))*e^(-3/2*a) - 3/4*sqrt(2)*ar
ctan(1/2*sqrt(2)*(2*x*e^a - sqrt(2)*e^(1/2*a))*e^(-1/2*a))*e^(-3/2*a) + 3/8*sqrt(2)*e^(-3/2*a)*log(x^2*e^a + s
qrt(2)*x*e^(1/2*a) + 1) - 3/8*sqrt(2)*e^(-3/2*a)*log(x^2*e^a - sqrt(2)*x*e^(1/2*a) + 1) + x^3/(x^4*e^(2*a) + 1
)

________________________________________________________________________________________

Fricas [A]  time = 2.1458, size = 693, normalized size = 4.01 \begin{align*} \frac{8 \, x^{7} e^{\left (2 \, a\right )} + 32 \, x^{3} + 36 \,{\left (\sqrt{2} x^{4} e^{\left (2 \, a\right )} + \sqrt{2}\right )} \arctan \left (-\sqrt{2} x e^{\left (\frac{1}{2} \, a\right )} + \sqrt{2} \sqrt{\sqrt{2} x e^{\left (-\frac{1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}} e^{\left (\frac{1}{2} \, a\right )} - 1\right ) e^{\left (-\frac{3}{2} \, a\right )} + 36 \,{\left (\sqrt{2} x^{4} e^{\left (2 \, a\right )} + \sqrt{2}\right )} \arctan \left (-\sqrt{2} x e^{\left (\frac{1}{2} \, a\right )} + \sqrt{2} \sqrt{-\sqrt{2} x e^{\left (-\frac{1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}} e^{\left (\frac{1}{2} \, a\right )} + 1\right ) e^{\left (-\frac{3}{2} \, a\right )} + 9 \,{\left (\sqrt{2} x^{4} e^{\left (2 \, a\right )} + \sqrt{2}\right )} e^{\left (-\frac{3}{2} \, a\right )} \log \left (\sqrt{2} x e^{\left (-\frac{1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) - 9 \,{\left (\sqrt{2} x^{4} e^{\left (2 \, a\right )} + \sqrt{2}\right )} e^{\left (-\frac{3}{2} \, a\right )} \log \left (-\sqrt{2} x e^{\left (-\frac{1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right )}{24 \,{\left (x^{4} e^{\left (2 \, a\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(8*x^7*e^(2*a) + 32*x^3 + 36*(sqrt(2)*x^4*e^(2*a) + sqrt(2))*arctan(-sqrt(2)*x*e^(1/2*a) + sqrt(2)*sqrt(s
qrt(2)*x*e^(-1/2*a) + x^2 + e^(-a))*e^(1/2*a) - 1)*e^(-3/2*a) + 36*(sqrt(2)*x^4*e^(2*a) + sqrt(2))*arctan(-sqr
t(2)*x*e^(1/2*a) + sqrt(2)*sqrt(-sqrt(2)*x*e^(-1/2*a) + x^2 + e^(-a))*e^(1/2*a) + 1)*e^(-3/2*a) + 9*(sqrt(2)*x
^4*e^(2*a) + sqrt(2))*e^(-3/2*a)*log(sqrt(2)*x*e^(-1/2*a) + x^2 + e^(-a)) - 9*(sqrt(2)*x^4*e^(2*a) + sqrt(2))*
e^(-3/2*a)*log(-sqrt(2)*x*e^(-1/2*a) + x^2 + e^(-a)))/(x^4*e^(2*a) + 1)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.18396, size = 188, normalized size = 1.09 \begin{align*} \frac{1}{3} \, x^{3} - \frac{3}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} e^{\left (-\frac{1}{2} \, a\right )} + 2 \, x\right )} e^{\left (\frac{1}{2} \, a\right )}\right ) e^{\left (-\frac{3}{2} \, a\right )} - \frac{3}{4} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} e^{\left (-\frac{1}{2} \, a\right )} - 2 \, x\right )} e^{\left (\frac{1}{2} \, a\right )}\right ) e^{\left (-\frac{3}{2} \, a\right )} + \frac{3}{8} \, \sqrt{2} e^{\left (-\frac{3}{2} \, a\right )} \log \left (\sqrt{2} x e^{\left (-\frac{1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) - \frac{3}{8} \, \sqrt{2} e^{\left (-\frac{3}{2} \, a\right )} \log \left (-\sqrt{2} x e^{\left (-\frac{1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) + \frac{x^{3}}{x^{4} e^{\left (2 \, a\right )} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 - 3/4*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*e^(-1/2*a) + 2*x)*e^(1/2*a))*e^(-3/2*a) - 3/4*sqrt(2)*arctan
(-1/2*sqrt(2)*(sqrt(2)*e^(-1/2*a) - 2*x)*e^(1/2*a))*e^(-3/2*a) + 3/8*sqrt(2)*e^(-3/2*a)*log(sqrt(2)*x*e^(-1/2*
a) + x^2 + e^(-a)) - 3/8*sqrt(2)*e^(-3/2*a)*log(-sqrt(2)*x*e^(-1/2*a) + x^2 + e^(-a)) + x^3/(x^4*e^(2*a) + 1)