3.333 \(\int \frac{\log (1+x)}{x^2} \, dx\)

Optimal. Leaf size=18 \[ \log (x)-\frac{\log (x+1)}{x}-\log (x+1) \]

[Out]

Log[x] - Log[1 + x] - Log[1 + x]/x

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Rubi [A]  time = 0.0177651, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \log (x)-\frac{\log (x+1)}{x}-\log (x+1) \]

Antiderivative was successfully verified.

[In]  Int[Log[1 + x]/x^2,x]

[Out]

Log[x] - Log[1 + x] - Log[1 + x]/x

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Rubi in Sympy [A]  time = 1.5048, size = 14, normalized size = 0.78 \[ \log{\left (x \right )} - \log{\left (x + 1 \right )} - \frac{\log{\left (x + 1 \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(ln(1+x)/x**2,x)

[Out]

log(x) - log(x + 1) - log(x + 1)/x

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Mathematica [A]  time = 0.00361293, size = 18, normalized size = 1. \[ \log (x)-\frac{\log (x+1)}{x}-\log (x+1) \]

Antiderivative was successfully verified.

[In]  Integrate[Log[1 + x]/x^2,x]

[Out]

Log[x] - Log[1 + x] - Log[1 + x]/x

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Maple [A]  time = 0.01, size = 16, normalized size = 0.9 \[ \ln \left ( x \right ) -{\frac{\ln \left ( 1+x \right ) \left ( 1+x \right ) }{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(ln(1+x)/x^2,x)

[Out]

ln(x)-ln(1+x)*(1+x)/x

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Maxima [A]  time = 1.34926, size = 24, normalized size = 1.33 \[ -\frac{\log \left (x + 1\right )}{x} - \log \left (x + 1\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(x + 1)/x^2,x, algorithm="maxima")

[Out]

-log(x + 1)/x - log(x + 1) + log(x)

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Fricas [A]  time = 0.215748, size = 26, normalized size = 1.44 \[ -\frac{{\left (x + 1\right )} \log \left (x + 1\right ) - x \log \left (x\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(x + 1)/x^2,x, algorithm="fricas")

[Out]

-((x + 1)*log(x + 1) - x*log(x))/x

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Sympy [A]  time = 0.127161, size = 14, normalized size = 0.78 \[ \log{\left (x \right )} - \log{\left (x + 1 \right )} - \frac{\log{\left (x + 1 \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(ln(1+x)/x**2,x)

[Out]

log(x) - log(x + 1) - log(x + 1)/x

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GIAC/XCAS [A]  time = 0.210111, size = 27, normalized size = 1.5 \[ -\frac{{\rm ln}\left (x + 1\right )}{x} -{\rm ln}\left ({\left | x + 1 \right |}\right ) +{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(log(x + 1)/x^2,x, algorithm="giac")

[Out]

-ln(x + 1)/x - ln(abs(x + 1)) + ln(abs(x))