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]

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

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Rubi [F]  time = 0.02, 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 = {} \[ \int \frac {\tanh (a+2 \log (x))}{x^2} \, dx \]

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*}

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Mathematica [C]  time = 0.17, size = 59, normalized size = 0.40 \[ \frac {2-x (\sinh (a)+\cosh (a))^2 \text {RootSum}\left [-\text {$\#$1}^4 \sinh (a)+\text {$\#$1}^4 \cosh (a)+\sinh (a)+\cosh (a)\& ,\frac {\log \left (\frac {1}{x}-\text {$\#$1}\right )+\log (x)}{\text {$\#$1}^3}\& \right ]}{2 x} \]

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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fricas [B]  time = 0.61, size = 204, normalized size = 1.39 \[ -\frac {4 \, \sqrt {2} x \arctan \left (-{\left (\sqrt {2} x e^{\left (\frac {5}{2} \, a\right )} - \sqrt {2} \sqrt {x^{2} e^{\left (4 \, a\right )} + \sqrt {2} x e^{\left (\frac {7}{2} \, a\right )} + e^{\left (3 \, a\right )}} e^{\left (\frac {1}{2} \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (-2 \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} + 4 \, \sqrt {2} x \arctan \left (-{\left (\sqrt {2} x e^{\left (\frac {5}{2} \, a\right )} - \sqrt {2} \sqrt {x^{2} e^{\left (4 \, a\right )} - \sqrt {2} x e^{\left (\frac {7}{2} \, a\right )} + e^{\left (3 \, a\right )}} e^{\left (\frac {1}{2} \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (-2 \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} + \sqrt {2} x e^{\left (\frac {1}{2} \, a\right )} \log \left (x^{2} e^{\left (4 \, a\right )} + \sqrt {2} x e^{\left (\frac {7}{2} \, a\right )} + e^{\left (3 \, a\right )}\right ) - \sqrt {2} x e^{\left (\frac {1}{2} \, a\right )} \log \left (x^{2} e^{\left (4 \, a\right )} - \sqrt {2} x e^{\left (\frac {7}{2} \, a\right )} + e^{\left (3 \, a\right )}\right ) - 4}{4 \, x} \]

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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giac [A]  time = 0.13, size = 121, normalized size = 0.82 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} e^{\left (-\frac {1}{2} \, a\right )} + 2 \, x\right )} e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} + \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} e^{\left (-\frac {1}{2} \, a\right )} - 2 \, x\right )} e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} - \frac {1}{4} \, \sqrt {2} e^{\left (\frac {1}{2} \, a\right )} \log \left (\sqrt {2} x e^{\left (-\frac {1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) + \frac {1}{4} \, \sqrt {2} e^{\left (\frac {1}{2} \, a\right )} \log \left (-\sqrt {2} x e^{\left (-\frac {1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) + \frac {1}{x} \]

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

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maple [C]  time = 0.11, size = 42, normalized size = 0.29 \[ \frac {1}{x}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+{\mathrm e}^{2 a}\right )}{\sum }\textit {\_R} \ln \left (\left (5 \textit {\_R}^{4}+4 \,{\mathrm e}^{2 a}\right ) x -\textit {\_R}^{3}\right )\right )}{2} \]

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 = 0.45, size = 125, normalized size = 0.85 \[ -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} e^{\left (\frac {1}{2} \, a\right )} + \frac {2}{x}\right )} e^{\left (-\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} - \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} e^{\left (\frac {1}{2} \, a\right )} - \frac {2}{x}\right )} e^{\left (-\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} - \frac {1}{4} \, \sqrt {2} e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {\sqrt {2} e^{\left (\frac {1}{2} \, a\right )}}{x} + \frac {1}{x^{2}} + e^{a}\right ) + \frac {1}{4} \, \sqrt {2} e^{\left (\frac {1}{2} \, a\right )} \log \left (-\frac {\sqrt {2} e^{\left (\frac {1}{2} \, a\right )}}{x} + \frac {1}{x^{2}} + e^{a}\right ) + \frac {1}{x} \]

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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mupad [B]  time = 1.09, size = 45, normalized size = 0.31 \[ \mathrm {atan}\left (x\,{\left (-{\mathrm {e}}^{2\,a}\right )}^{1/4}\right )\,{\left (-{\mathrm {e}}^{2\,a}\right )}^{1/4}-\mathrm {atanh}\left (x\,{\left (-{\mathrm {e}}^{2\,a}\right )}^{1/4}\right )\,{\left (-{\mathrm {e}}^{2\,a}\right )}^{1/4}+\frac {1}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a + 2*log(x))/x^2,x)

[Out]

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

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

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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