Optimal. Leaf size=28 \[ \frac {6}{x+1}-\frac {6}{(x+1)^2}+\frac {8}{3 (x+1)^3}+\log (x+1) \]
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Rubi [A] time = 0.03, 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 = {1680, 43} \[ \frac {6}{x+1}-\frac {6}{(x+1)^2}+\frac {8}{3 (x+1)^3}+\log (x+1) \]
Antiderivative was successfully verified.
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Rule 43
Rule 1680
Rubi steps
\begin {align*} \int \frac {-1+3 x-3 x^2+x^3}{1+4 x+6 x^2+4 x^3+x^4} \, dx &=\operatorname {Subst}\left (\int \frac {(-2+x)^3}{x^4} \, dx,x,1+x\right )\\ &=\operatorname {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 {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 0.86 \[ \frac {2 \left (9 x^2+9 x+4\right )}{3 (x+1)^3}+\log (x+1) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 46, normalized size = 1.64 \[ \frac {18 \, x^{2} + 3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (x + 1\right ) + 18 \, x + 8}{3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 23, normalized size = 0.82 \[ \frac {2 \, {\left (9 \, x^{2} + 9 \, x + 4\right )}}{3 \, {\left (x + 1\right )}^{3}} + \log \left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 27, normalized size = 0.96 \[ \ln \left (x +1\right )+\frac {8}{3 \left (x +1\right )^{3}}-\frac {6}{\left (x +1\right )^{2}}+\frac {6}{x +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 32, normalized size = 1.14 \[ \frac {2 \, {\left (9 \, x^{2} + 9 \, x + 4\right )}}{3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} + \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 21, normalized size = 0.75 \[ \ln \left (x+1\right )+\frac {6\,x^2+6\,x+\frac {8}{3}}{{\left (x+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 29, normalized size = 1.04 \[ \frac {18 x^{2} + 18 x + 8}{3 x^{3} + 9 x^{2} + 9 x + 3} + \log {\left (x + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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