Optimal. Leaf size=26 \[ \frac {\frac {5}{4}+x}{\frac {2 x^2}{3}+\log \left (-1+\frac {x}{1+x}\right )} \]
________________________________________________________________________________________
Rubi [F] time = 0.59, 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 = {} \begin {gather*} \int \frac {45-24 x-84 x^2-24 x^3+(36+36 x) \log \left (-\frac {1}{1+x}\right )}{16 x^4+16 x^5+\left (48 x^2+48 x^3\right ) \log \left (-\frac {1}{1+x}\right )+(36+36 x) \log ^2\left (-\frac {1}{1+x}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3 \left (-15+8 x+28 x^2+8 x^3\right )+36 (1+x) \log \left (-\frac {1}{1+x}\right )}{4 (1+x) \left (2 x^2+3 \log \left (-\frac {1}{1+x}\right )\right )^2} \, dx\\ &=\frac {1}{4} \int \frac {-3 \left (-15+8 x+28 x^2+8 x^3\right )+36 (1+x) \log \left (-\frac {1}{1+x}\right )}{(1+x) \left (2 x^2+3 \log \left (-\frac {1}{1+x}\right )\right )^2} \, dx\\ &=\frac {1}{4} \int \left (-\frac {3 \left (-15+8 x+36 x^2+16 x^3\right )}{(1+x) \left (2 x^2+3 \log \left (-\frac {1}{1+x}\right )\right )^2}+\frac {12}{2 x^2+3 \log \left (-\frac {1}{1+x}\right )}\right ) \, dx\\ &=-\left (\frac {3}{4} \int \frac {-15+8 x+36 x^2+16 x^3}{(1+x) \left (2 x^2+3 \log \left (-\frac {1}{1+x}\right )\right )^2} \, dx\right )+3 \int \frac {1}{2 x^2+3 \log \left (-\frac {1}{1+x}\right )} \, dx\\ &=-\left (\frac {3}{4} \int \left (-\frac {12}{\left (2 x^2+3 \log \left (-\frac {1}{1+x}\right )\right )^2}+\frac {20 x}{\left (2 x^2+3 \log \left (-\frac {1}{1+x}\right )\right )^2}+\frac {16 x^2}{\left (2 x^2+3 \log \left (-\frac {1}{1+x}\right )\right )^2}-\frac {3}{(1+x) \left (2 x^2+3 \log \left (-\frac {1}{1+x}\right )\right )^2}\right ) \, dx\right )+3 \int \frac {1}{2 x^2+3 \log \left (-\frac {1}{1+x}\right )} \, dx\\ &=\frac {9}{4} \int \frac {1}{(1+x) \left (2 x^2+3 \log \left (-\frac {1}{1+x}\right )\right )^2} \, dx+3 \int \frac {1}{2 x^2+3 \log \left (-\frac {1}{1+x}\right )} \, dx+9 \int \frac {1}{\left (2 x^2+3 \log \left (-\frac {1}{1+x}\right )\right )^2} \, dx-12 \int \frac {x^2}{\left (2 x^2+3 \log \left (-\frac {1}{1+x}\right )\right )^2} \, dx-15 \int \frac {x}{\left (2 x^2+3 \log \left (-\frac {1}{1+x}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.86, size = 27, normalized size = 1.04 \begin {gather*} \frac {3 (5+4 x)}{4 \left (2 x^2+3 \log \left (-\frac {1}{1+x}\right )\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.60, size = 25, normalized size = 0.96 \begin {gather*} \frac {3 \, {\left (4 \, x + 5\right )}}{4 \, {\left (2 \, x^{2} + 3 \, \log \left (-\frac {1}{x + 1}\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 48, normalized size = 1.85 \begin {gather*} -\frac {3 \, {\left (\frac {4}{x + 1} + \frac {1}{{\left (x + 1\right )}^{2}}\right )}}{4 \, {\left (\frac {4}{x + 1} - \frac {3 \, \log \left (-\frac {1}{x + 1}\right )}{{\left (x + 1\right )}^{2}} - \frac {2}{{\left (x + 1\right )}^{2}} - 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 25, normalized size = 0.96
method | result | size |
norman | \(\frac {\frac {15}{4}+3 x}{2 x^{2}+3 \ln \left (-\frac {1}{x +1}\right )}\) | \(25\) |
risch | \(\frac {\frac {15}{4}+3 x}{2 x^{2}+3 \ln \left (-\frac {1}{x +1}\right )}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.39, size = 23, normalized size = 0.88 \begin {gather*} \frac {3 \, {\left (4 \, x + 5\right )}}{4 \, {\left (2 \, x^{2} - 3 \, \log \left (-x - 1\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 7.63, size = 25, normalized size = 0.96 \begin {gather*} \frac {3\,\left (4\,x+5\right )}{4\,\left (3\,\ln \left (-\frac {1}{x+1}\right )+2\,x^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.13, size = 19, normalized size = 0.73 \begin {gather*} \frac {12 x + 15}{8 x^{2} + 12 \log {\left (- \frac {1}{x + 1} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________