Optimal. Leaf size=28 \[ \frac {8}{3 (1+x)^3}-\frac {6}{(1+x)^2}+\frac {6}{1+x}+\log (1+x) \]
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Rubi [A]
time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1694, 45}
\begin {gather*} \frac {6}{x+1}-\frac {6}{(x+1)^2}+\frac {8}{3 (x+1)^3}+\log (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 1694
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\text {Subst}\left (\int \frac {(-2+x)^3}{x^4} \, dx,x,1+x\right )\\ &=\text {Subst}\left (\int \left (-\frac {8}{x^4}+\frac {12}{x^3}-\frac {6}{x^2}+\frac {1}{x}\right ) \, dx,x,1+x\right )\\ &=\frac {8}{3 (1+x)^3}-\frac {6}{(1+x)^2}+\frac {6}{1+x}+\log (1+x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 24, normalized size = 0.86 \begin {gather*} \frac {2 \left (4+9 x+9 x^2\right )}{3 (1+x)^3}+\log (1+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 27, normalized size = 0.96
method | result | size |
norman | \(\frac {6 x +6 x^{2}+\frac {8}{3}}{\left (x +1\right )^{3}}+\ln \left (x +1\right )\) | \(22\) |
default | \(\frac {8}{3 \left (x +1\right )^{3}}-\frac {6}{\left (x +1\right )^{2}}+\frac {6}{x +1}+\ln \left (x +1\right )\) | \(27\) |
risch | \(\frac {6 x +6 x^{2}+\frac {8}{3}}{x^{3}+3 x^{2}+3 x +1}+\ln \left (x +1\right )\) | \(32\) |
parallelrisch | \(\frac {3 \ln \left (x +1\right ) x^{3}+8+9 \ln \left (x +1\right ) x^{2}+9 \ln \left (x +1\right ) x +18 x^{2}+3 \ln \left (x +1\right )+18 x}{3 x^{3}+9 x^{2}+9 x +3}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.38, size = 32, normalized size = 1.14 \begin {gather*} \frac {2 \, {\left (9 \, x^{2} + 9 \, x + 4\right )}}{3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} + \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.61, size = 46, normalized size = 1.64 \begin {gather*} \frac {18 \, x^{2} + 3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (x + 1\right ) + 18 \, x + 8}{3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 29, normalized size = 1.04 \begin {gather*} \frac {18 x^{2} + 18 x + 8}{3 x^{3} + 9 x^{2} + 9 x + 3} + \log {\left (x + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 23, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (9 \, x^{2} + 9 \, x + 4\right )}}{3 \, {\left (x + 1\right )}^{3}} + \log \left ({\left | x + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 21, normalized size = 0.75 \begin {gather*} \ln \left (x+1\right )+\frac {6\,x^2+6\,x+\frac {8}{3}}{{\left (x+1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F] Failed to verify
time = 1.00, size = 29, normalized size = 1.04 \begin {gather*} \frac {x^{2}}{2}-x +2 \ln \left (x +1\right )-\frac {1}{x +1}+\frac {1}{2 \left (x +1\right )^{2}} \end {gather*}
Warning: Unable to verify antiderivative.
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