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

Optimal. Leaf size=147 \[ \frac{e^{a/2} \log \left (e^a x^2-\sqrt{2} e^{a/2} x+1\right )}{2 \sqrt{2}}-\frac{e^{a/2} \log \left (e^a x^2+\sqrt{2} e^{a/2} x+1\right )}{2 \sqrt{2}}-\frac{e^{a/2} \tan ^{-1}\left (1-\sqrt{2} e^{a/2} x\right )}{\sqrt{2}}+\frac{e^{a/2} \tan ^{-1}\left (\sqrt{2} e^{a/2} x+1\right )}{\sqrt{2}}+\frac{1}{x} \]

[Out]

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

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Rubi [F]  time = 0.0237645, 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 = {} \[ \int \frac{\tanh (a+2 \log (x))}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tanh[a + 2*Log[x]]/x^2,x]

[Out]

Defer[Int][Tanh[a + 2*Log[x]]/x^2, x]

Rubi steps

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

Mathematica [C]  time = 0.159208, size = 59, normalized size = 0.4 \[ \frac{2-x (\sinh (a)+\cosh (a))^2 \text{RootSum}\left [-\text{$\#$1}^4 \sinh (a)+\text{$\#$1}^4 \cosh (a)+\sinh (a)+\cosh (a)\& ,\frac{\log \left (\frac{1}{x}-\text{$\#$1}\right )+\log (x)}{\text{$\#$1}^3}\& \right ]}{2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[a + 2*Log[x]]/x^2,x]

[Out]

(2 - x*RootSum[Cosh[a] + Sinh[a] + Cosh[a]*#1^4 - Sinh[a]*#1^4 & , (Log[x] + Log[x^(-1) - #1])/#1^3 & ]*(Cosh[
a] + Sinh[a])^2)/(2*x)

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Maple [C]  time = 0.027, size = 42, normalized size = 0.3 \begin{align*}{x}^{-1}+{\frac{\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}+{{\rm e}^{2\,a}} \right ) }{\it \_R}\,\ln \left ( \left ( 5\,{{\it \_R}}^{4}+4\,{{\rm e}^{2\,a}} \right ) x-{{\it \_R}}^{3} \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.90063, size = 169, normalized size = 1.15 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} e^{\left (\frac{1}{2} \, a\right )} + \frac{2}{x}\right )} e^{\left (-\frac{1}{2} \, a\right )}\right ) e^{\left (\frac{1}{2} \, a\right )} - \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} e^{\left (\frac{1}{2} \, a\right )} - \frac{2}{x}\right )} e^{\left (-\frac{1}{2} \, a\right )}\right ) e^{\left (\frac{1}{2} \, a\right )} - \frac{1}{4} \, \sqrt{2} e^{\left (\frac{1}{2} \, a\right )} \log \left (\frac{\sqrt{2} e^{\left (\frac{1}{2} \, a\right )}}{x} + \frac{1}{x^{2}} + e^{a}\right ) + \frac{1}{4} \, \sqrt{2} e^{\left (\frac{1}{2} \, a\right )} \log \left (-\frac{\sqrt{2} e^{\left (\frac{1}{2} \, a\right )}}{x} + \frac{1}{x^{2}} + e^{a}\right ) + \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.19939, size = 597, normalized size = 4.06 \begin{align*} -\frac{4 \, \sqrt{2} x \arctan \left (-{\left (\sqrt{2} x e^{\left (\frac{5}{2} \, a\right )} - \sqrt{2} \sqrt{x^{2} e^{\left (4 \, a\right )} + \sqrt{2} x e^{\left (\frac{7}{2} \, a\right )} + e^{\left (3 \, a\right )}} e^{\left (\frac{1}{2} \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (-2 \, a\right )}\right ) e^{\left (\frac{1}{2} \, a\right )} + 4 \, \sqrt{2} x \arctan \left (-{\left (\sqrt{2} x e^{\left (\frac{5}{2} \, a\right )} - \sqrt{2} \sqrt{x^{2} e^{\left (4 \, a\right )} - \sqrt{2} x e^{\left (\frac{7}{2} \, a\right )} + e^{\left (3 \, a\right )}} e^{\left (\frac{1}{2} \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (-2 \, a\right )}\right ) e^{\left (\frac{1}{2} \, a\right )} + \sqrt{2} x e^{\left (\frac{1}{2} \, a\right )} \log \left (x^{2} e^{\left (4 \, a\right )} + \sqrt{2} x e^{\left (\frac{7}{2} \, a\right )} + e^{\left (3 \, a\right )}\right ) - \sqrt{2} x e^{\left (\frac{1}{2} \, a\right )} \log \left (x^{2} e^{\left (4 \, a\right )} - \sqrt{2} x e^{\left (\frac{7}{2} \, a\right )} + e^{\left (3 \, a\right )}\right ) - 4}{4 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*sqrt(2)*x*arctan(-(sqrt(2)*x*e^(5/2*a) - sqrt(2)*sqrt(x^2*e^(4*a) + sqrt(2)*x*e^(7/2*a) + e^(3*a))*e^(
1/2*a) + e^(2*a))*e^(-2*a))*e^(1/2*a) + 4*sqrt(2)*x*arctan(-(sqrt(2)*x*e^(5/2*a) - sqrt(2)*sqrt(x^2*e^(4*a) -
sqrt(2)*x*e^(7/2*a) + e^(3*a))*e^(1/2*a) - e^(2*a))*e^(-2*a))*e^(1/2*a) + sqrt(2)*x*e^(1/2*a)*log(x^2*e^(4*a)
+ sqrt(2)*x*e^(7/2*a) + e^(3*a)) - sqrt(2)*x*e^(1/2*a)*log(x^2*e^(4*a) - sqrt(2)*x*e^(7/2*a) + e^(3*a)) - 4)/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(tanh(a + 2*log(x))/x**2, x)

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Giac [A]  time = 1.18468, size = 163, normalized size = 1.11 \begin{align*} \frac{1}{2} \, \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{1}{2} \, a\right )} + \frac{1}{2} \, \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{1}{2} \, a\right )} - \frac{1}{4} \, \sqrt{2} e^{\left (\frac{1}{2} \, a\right )} \log \left (\sqrt{2} x e^{\left (-\frac{1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) + \frac{1}{4} \, \sqrt{2} e^{\left (\frac{1}{2} \, a\right )} \log \left (-\sqrt{2} x e^{\left (-\frac{1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) + \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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