3.152 \(\int \frac {\tanh (a+2 \log (x))}{x^3} \, dx\)

Optimal. Leaf size=20 \[ e^a \tan ^{-1}\left (e^a x^2\right )+\frac {1}{2 x^2} \]

[Out]

1/2/x^2+exp(a)*arctan(exp(a)*x^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 = {} \[ \int \frac {\tanh (a+2 \log (x))}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tanh[a + 2*Log[x]]/x^3,x]

[Out]

Defer[Int][Tanh[a + 2*Log[x]]/x^3, x]

Rubi steps

\begin {align*} \int \frac {\tanh (a+2 \log (x))}{x^3} \, dx &=\int \frac {\tanh (a+2 \log (x))}{x^3} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 40, normalized size = 2.00 \[ \cosh (a) \left (-\tan ^{-1}\left (\frac {\cosh (a)-\sinh (a)}{x^2}\right )\right )-\sinh (a) \tan ^{-1}\left (\frac {\cosh (a)-\sinh (a)}{x^2}\right )+\frac {1}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[a + 2*Log[x]]/x^3,x]

[Out]

1/(2*x^2) - ArcTan[(Cosh[a] - Sinh[a])/x^2]*Cosh[a] - ArcTan[(Cosh[a] - Sinh[a])/x^2]*Sinh[a]

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fricas [A]  time = 0.92, size = 21, normalized size = 1.05 \[ \frac {2 \, x^{2} \arctan \left (x^{2} e^{a}\right ) e^{a} + 1}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+2*log(x))/x^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.11, size = 16, normalized size = 0.80 \[ \arctan \left (x^{2} e^{a}\right ) e^{a} + \frac {1}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+2*log(x))/x^3,x, algorithm="giac")

[Out]

arctan(x^2*e^a)*e^a + 1/2/x^2

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maple [C]  time = 0.11, size = 44, normalized size = 2.20 \[ \frac {1}{2 x^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left ({\mathrm e}^{2 a}+\textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (4 \,{\mathrm e}^{2 a}+5 \textit {\_R}^{2}\right ) x^{2}-\textit {\_R} \right )\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a+2*ln(x))/x^3,x)

[Out]

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

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maxima [A]  time = 0.43, size = 19, normalized size = 0.95 \[ -\arctan \left (\frac {e^{\left (-a\right )}}{x^{2}}\right ) e^{a} + \frac {1}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+2*log(x))/x^3,x, algorithm="maxima")

[Out]

-arctan(e^(-a)/x^2)*e^a + 1/2/x^2

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mupad [B]  time = 1.05, size = 24, normalized size = 1.20 \[ \mathrm {atan}\left (x^2\,\sqrt {{\mathrm {e}}^{2\,a}}\right )\,\sqrt {{\mathrm {e}}^{2\,a}}+\frac {1}{2\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a + 2*log(x))/x^3,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh {\left (a + 2 \log {\relax (x )} \right )}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(a+2*ln(x))/x**3,x)

[Out]

Integral(tanh(a + 2*log(x))/x**3, x)

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