Optimal. Leaf size=122 \[ \frac {d-2 e+4 f-8 g+16 h-32 i}{12 (x+2)}-\frac {1}{18} \log (1-x) (d+e+f+g+h+i)+\frac {1}{48} \log (2-x) (d+2 e+4 f+8 g+16 h+32 i)+\frac {1}{6} \log (x+1) (d-e+f-g+h-i)-\frac {1}{144} \log (x+2) (19 d-26 e+28 f-8 g-80 h+352 i)+i x \]
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Rubi [A] time = 0.31, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {1586, 6742} \[ \frac {d-2 e+4 f-8 g+16 h-32 i}{12 (x+2)}-\frac {1}{18} \log (1-x) (d+e+f+g+h+i)+\frac {1}{48} \log (2-x) (d+2 e+4 f+8 g+16 h+32 i)+\frac {1}{6} \log (x+1) (d-e+f-g+h-i)-\frac {1}{144} \log (x+2) (19 d-26 e+28 f-8 g-80 h+352 i)+i x \]
Antiderivative was successfully verified.
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Rule 1586
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (2-x-2 x^2+x^3\right ) \left (d+e x+f x^2+g x^3+h x^4+90 x^5\right )}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac {d+e x+f x^2+g x^3+h x^4+90 x^5}{(2+x)^2 \left (2-x-2 x^2+x^3\right )} \, dx\\ &=\int \left (90+\frac {2880+d+2 e+4 f+8 g+16 h}{48 (-2+x)}+\frac {-90-d-e-f-g-h}{18 (-1+x)}+\frac {-90+d-e+f-g+h}{6 (1+x)}+\frac {2880-d+2 e-4 f+8 g-16 h}{12 (2+x)^2}+\frac {-31680-19 d+26 e-28 f+8 g+80 h}{144 (2+x)}\right ) \, dx\\ &=90 x-\frac {2880-d+2 e-4 f+8 g-16 h}{12 (2+x)}-\frac {1}{18} (90+d+e+f+g+h) \log (1-x)+\frac {1}{48} (2880+d+2 e+4 f+8 g+16 h) \log (2-x)-\frac {1}{6} (90-d+e-f+g-h) \log (1+x)-\frac {1}{144} (31680+19 d-26 e+28 f-8 g-80 h) \log (2+x)\\ \end {align*}
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Mathematica [A] time = 0.06, size = 118, normalized size = 0.97 \[ \frac {1}{144} \left (\frac {12 (d-2 (e-2 f+4 g-8 h+16 i))}{x+2}-8 \log (1-x) (d+e+f+g+h+i)+3 \log (2-x) (d+2 e+4 (f+2 g+4 h+8 i))+24 \log (x+1) (d-e+f-g+h-i)+\log (x+2) (-19 d+26 e-28 f+8 g+80 h-352 i)+144 i x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 66.48, size = 200, normalized size = 1.64 \[ \frac {144 \, i x^{2} + 288 \, i x - {\left ({\left (19 \, d - 26 \, e + 28 \, f - 8 \, g - 80 \, h + 352 \, i\right )} x + 38 \, d - 52 \, e + 56 \, f - 16 \, g - 160 \, h + 704 \, i\right )} \log \left (x + 2\right ) + 24 \, {\left ({\left (d - e + f - g + h - i\right )} x + 2 \, d - 2 \, e + 2 \, f - 2 \, g + 2 \, h - 2 \, i\right )} \log \left (x + 1\right ) - 8 \, {\left ({\left (d + e + f + g + h + i\right )} x + 2 \, d + 2 \, e + 2 \, f + 2 \, g + 2 \, h + 2 \, i\right )} \log \left (x - 1\right ) + 3 \, {\left ({\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} x + 2 \, d + 4 \, e + 8 \, f + 16 \, g + 32 \, h + 64 \, i\right )} \log \left (x - 2\right ) + 12 \, d - 24 \, e + 48 \, f - 96 \, g + 192 \, h - 384 \, i}{144 \, {\left (x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 117, normalized size = 0.96 \[ i x - \frac {1}{144} \, {\left (19 \, d + 28 \, f - 8 \, g - 80 \, h + 352 \, i - 26 \, e\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{6} \, {\left (d + f - g + h - i - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{18} \, {\left (d + f + g + h + i + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{48} \, {\left (d + 4 \, f + 8 \, g + 16 \, h + 32 \, i + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) + \frac {d + 4 \, f - 8 \, g + 16 \, h - 32 \, i - 2 \, e}{12 \, {\left (x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 221, normalized size = 1.81 \[ -\frac {22 i \ln \left (x +2\right )}{9}-\frac {i \ln \left (x -1\right )}{18}-\frac {i \ln \left (x +1\right )}{6}+\frac {2 i \ln \left (x -2\right )}{3}+\frac {5 h \ln \left (x +2\right )}{9}-\frac {h \ln \left (x -1\right )}{18}+\frac {h \ln \left (x +1\right )}{6}+\frac {h \ln \left (x -2\right )}{3}-\frac {g \ln \left (x -1\right )}{18}+\frac {g \ln \left (x +2\right )}{18}+\frac {g \ln \left (x -2\right )}{6}-\frac {g \ln \left (x +1\right )}{6}-\frac {19 d \ln \left (x +2\right )}{144}+\frac {13 e \ln \left (x +2\right )}{72}-\frac {e \ln \left (x -1\right )}{18}-\frac {d \ln \left (x -1\right )}{18}-\frac {e \ln \left (x +1\right )}{6}+\frac {d \ln \left (x +1\right )}{6}+\frac {d \ln \left (x -2\right )}{48}+\frac {e \ln \left (x -2\right )}{24}+\frac {f \ln \left (x -2\right )}{12}+\frac {f \ln \left (x +1\right )}{6}-\frac {f \ln \left (x -1\right )}{18}-\frac {7 f \ln \left (x +2\right )}{36}+i x +\frac {f}{3 x +6}+\frac {d}{12 x +24}-\frac {8 i}{3 \left (x +2\right )}+\frac {4 h}{3 \left (x +2\right )}-\frac {2 g}{3 \left (x +2\right )}-\frac {e}{6 \left (x +2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 108, normalized size = 0.89 \[ i x - \frac {1}{144} \, {\left (19 \, d - 26 \, e + 28 \, f - 8 \, g - 80 \, h + 352 \, i\right )} \log \left (x + 2\right ) + \frac {1}{6} \, {\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) - \frac {1}{18} \, {\left (d + e + f + g + h + i\right )} \log \left (x - 1\right ) + \frac {1}{48} \, {\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left (x - 2\right ) + \frac {d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, i}{12 \, {\left (x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.67, size = 127, normalized size = 1.04 \[ i\,x+\frac {\frac {d}{12}-\frac {e}{6}+\frac {f}{3}-\frac {2\,g}{3}+\frac {4\,h}{3}-\frac {8\,i}{3}}{x+2}+\ln \left (x+1\right )\,\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}+\frac {h}{6}-\frac {i}{6}\right )+\ln \left (x-2\right )\,\left (\frac {d}{48}+\frac {e}{24}+\frac {f}{12}+\frac {g}{6}+\frac {h}{3}+\frac {2\,i}{3}\right )-\ln \left (x-1\right )\,\left (\frac {d}{18}+\frac {e}{18}+\frac {f}{18}+\frac {g}{18}+\frac {h}{18}+\frac {i}{18}\right )-\ln \left (x+2\right )\,\left (\frac {19\,d}{144}-\frac {13\,e}{72}+\frac {7\,f}{36}-\frac {g}{18}-\frac {5\,h}{9}+\frac {22\,i}{9}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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