Optimal. Leaf size=28 \[ \frac{6}{x+1}-\frac{6}{(x+1)^2}+\frac{8}{3 (x+1)^3}+\log (x+1) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0283567, antiderivative size = 28, normalized size of antiderivative = 1., 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.
[In]
[Out]
Rule 1680
Rule 43
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*}
Mathematica [A] time = 0.0123057, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 27, normalized size = 1. \begin{align*}{\frac{8}{3\, \left ( 1+x \right ) ^{3}}}-6\, \left ( 1+x \right ) ^{-2}+6\, \left ( 1+x \right ) ^{-1}+\ln \left ( 1+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00807, size = 43, normalized size = 1.54 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.49448, size = 119, normalized size = 4.25 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.104068, size = 29, normalized size = 1.04 \begin{align*} \frac{18 x^{2} + 18 x + 8}{3 x^{3} + 9 x^{2} + 9 x + 3} + \log{\left (x + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.28322, size = 31, normalized size = 1.11 \begin{align*} \frac{2 \,{\left (9 \, x^{2} + 9 \, x + 4\right )}}{3 \,{\left (x + 1\right )}^{3}} + \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]