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

Optimal. Leaf size=145 \[ \frac{e^{-a/2} \log \left (e^a x^2-\sqrt{2} e^{a/2} x+1\right )}{2 \sqrt{2}}-\frac{e^{-a/2} \log \left (e^a x^2+\sqrt{2} e^{a/2} x+1\right )}{2 \sqrt{2}}+\frac{e^{-a/2} \tan ^{-1}\left (1-\sqrt{2} e^{a/2} x\right )}{\sqrt{2}}-\frac{e^{-a/2} \tan ^{-1}\left (\sqrt{2} e^{a/2} x+1\right )}{\sqrt{2}}+x \]

[Out]

x + 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]*E^(a/2))

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Rubi [F]  time = 0.0073467, 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 = {} \[ \int \tanh (a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tanh[a + 2*Log[x]],x]

[Out]

Defer[Int][Tanh[a + 2*Log[x]], x]

Rubi steps

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

Mathematica [C]  time = 0.165275, size = 58, normalized size = 0.4 \[ \frac{1}{2} (\cosh (2 a)-\sinh (2 a)) \text{RootSum}\left [\text{$\#$1}^4 \sinh (a)+\text{$\#$1}^4 \cosh (a)-\sinh (a)+\cosh (a)\& ,\frac{\log (x)-\log (x-\text{$\#$1})}{\text{$\#$1}^3}\& \right ]+x \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[a + 2*Log[x]],x]

[Out]

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

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Maple [C]  time = 0.012, size = 33, normalized size = 0.2 \begin{align*} x-{\frac{{{\rm e}^{-2\,a}}}{2}\sum _{{\it \_R}={\it RootOf} \left ({{\rm e}^{2\,a}}{{\it \_Z}}^{4}+1 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.80798, size = 167, normalized size = 1.15 \begin{align*} -\frac{1}{2} \, \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{1}{2} \, a\right )} - \frac{1}{2} \, \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{1}{2} \, a\right )} - \frac{1}{4} \, \sqrt{2} e^{\left (-\frac{1}{2} \, a\right )} \log \left (x^{2} e^{a} + \sqrt{2} x e^{\left (\frac{1}{2} \, a\right )} + 1\right ) + \frac{1}{4} \, \sqrt{2} e^{\left (-\frac{1}{2} \, a\right )} \log \left (x^{2} e^{a} - \sqrt{2} x e^{\left (\frac{1}{2} \, a\right )} + 1\right ) + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.14815, size = 498, normalized size = 3.43 \begin{align*} \sqrt{2} \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{1}{2} \, a\right )} + \sqrt{2} \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{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 ) + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*arctan(-sqrt(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^(-1/2
*a) + sqrt(2)*arctan(-sqrt(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^(-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(-sqr
t(2)*x*e^(-1/2*a) + x^2 + e^(-a)) + x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(tanh(a + 2*log(x)), x)

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Giac [A]  time = 1.1658, size = 161, 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 ) + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+2*log(x)),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*sqr
t(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)) + x