Optimal. Leaf size=76 \[ h x+\frac {i x^2}{2}-\frac {1}{6} (d+4 f+16 h) \tanh ^{-1}\left (\frac {x}{2}\right )+\frac {1}{3} (d+f+h) \tanh ^{-1}(x)-\frac {1}{6} (e+g+i) \log \left (1-x^2\right )+\frac {1}{6} (e+4 g+16 i) \log \left (4-x^2\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {1687, 1690,
1180, 213, 1677, 1671, 646, 31} \begin {gather*} -\frac {1}{6} \tanh ^{-1}\left (\frac {x}{2}\right ) (d+4 f+16 h)+\frac {1}{3} \tanh ^{-1}(x) (d+f+h)-\frac {1}{6} \log \left (1-x^2\right ) (e+g+i)+\frac {1}{6} \log \left (4-x^2\right ) (e+4 g+16 i)+h x+\frac {i x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 213
Rule 646
Rule 1180
Rule 1671
Rule 1677
Rule 1687
Rule 1690
Rubi steps
\begin {align*} \int \frac {d+e x+f x^2+g x^3+h x^4+14 x^5}{4-5 x^2+x^4} \, dx &=\int \frac {x \left (e+g x^2+14 x^4\right )}{4-5 x^2+x^4} \, dx+\int \frac {d+f x^2+h x^4}{4-5 x^2+x^4} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {e+g x+14 x^2}{4-5 x+x^2} \, dx,x,x^2\right )+\int \left (h+\frac {d-4 h+(f+5 h) x^2}{4-5 x^2+x^4}\right ) \, dx\\ &=h x+\frac {1}{2} \text {Subst}\left (\int \left (14-\frac {56-e-(70+g) x}{4-5 x+x^2}\right ) \, dx,x,x^2\right )+\int \frac {d-4 h+(f+5 h) x^2}{4-5 x^2+x^4} \, dx\\ &=h x+7 x^2-\frac {1}{2} \text {Subst}\left (\int \frac {56-e-(70+g) x}{4-5 x+x^2} \, dx,x,x^2\right )-\frac {1}{3} (d+f+h) \int \frac {1}{-1+x^2} \, dx+\frac {1}{3} (d+4 f+16 h) \int \frac {1}{-4+x^2} \, dx\\ &=h x+7 x^2-\frac {1}{6} (d+4 f+16 h) \tanh ^{-1}\left (\frac {x}{2}\right )+\frac {1}{3} (d+f+h) \tanh ^{-1}(x)-\frac {1}{6} (-224-e-4 g) \text {Subst}\left (\int \frac {1}{-4+x} \, dx,x,x^2\right )-\frac {1}{6} (14+e+g) \text {Subst}\left (\int \frac {1}{-1+x} \, dx,x,x^2\right )\\ &=h x+7 x^2-\frac {1}{6} (d+4 f+16 h) \tanh ^{-1}\left (\frac {x}{2}\right )+\frac {1}{3} (d+f+h) \tanh ^{-1}(x)-\frac {1}{6} (14+e+g) \log \left (1-x^2\right )+\frac {1}{6} (224+e+4 g) \log \left (4-x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 98, normalized size = 1.29 \begin {gather*} \frac {1}{12} \left (12 h x+6 i x^2-2 (d+e+f+g+h+i) \log (1-x)+(d+2 e+4 (f+2 g+4 h+8 i)) \log (2-x)+2 (d-e+f-g+h-i) \log (1+x)-(d-2 (e-2 f+4 g-8 h+16 i)) \log (2+x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 107, normalized size = 1.41
method | result | size |
default | \(\frac {i \,x^{2}}{2}+h x +\left (-\frac {d}{12}+\frac {e}{6}-\frac {f}{3}+\frac {2 g}{3}-\frac {4 h}{3}+\frac {8 i}{3}\right ) \ln \left (x +2\right )+\left (\frac {d}{12}+\frac {e}{6}+\frac {f}{3}+\frac {2 g}{3}+\frac {4 h}{3}+\frac {8 i}{3}\right ) \ln \left (x -2\right )+\left (-\frac {d}{6}-\frac {e}{6}-\frac {f}{6}-\frac {g}{6}-\frac {h}{6}-\frac {i}{6}\right ) \ln \left (-1+x \right )+\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}+\frac {h}{6}-\frac {i}{6}\right ) \ln \left (1+x \right )\) | \(107\) |
norman | \(\frac {i \,x^{2}}{2}+h x +\left (-\frac {d}{12}+\frac {e}{6}-\frac {f}{3}+\frac {2 g}{3}-\frac {4 h}{3}+\frac {8 i}{3}\right ) \ln \left (x +2\right )+\left (\frac {d}{12}+\frac {e}{6}+\frac {f}{3}+\frac {2 g}{3}+\frac {4 h}{3}+\frac {8 i}{3}\right ) \ln \left (x -2\right )+\left (-\frac {d}{6}-\frac {e}{6}-\frac {f}{6}-\frac {g}{6}-\frac {h}{6}-\frac {i}{6}\right ) \ln \left (-1+x \right )+\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}+\frac {h}{6}-\frac {i}{6}\right ) \ln \left (1+x \right )\) | \(107\) |
risch | \(h x +\frac {2 \ln \left (x +2\right ) g}{3}+\frac {2 \ln \left (2-x \right ) g}{3}-\frac {\ln \left (1+x \right ) g}{6}-\frac {\ln \left (1-x \right ) g}{6}-\frac {\ln \left (1-x \right ) f}{6}+\frac {\ln \left (2-x \right ) f}{3}-\frac {\ln \left (x +2\right ) f}{3}+\frac {\ln \left (1+x \right ) f}{6}+\frac {\ln \left (2-x \right ) e}{6}+\frac {\ln \left (2-x \right ) d}{12}-\frac {\ln \left (x +2\right ) d}{12}+\frac {\ln \left (x +2\right ) e}{6}+\frac {\ln \left (1+x \right ) d}{6}-\frac {\ln \left (1+x \right ) e}{6}-\frac {\ln \left (1-x \right ) d}{6}-\frac {\ln \left (1-x \right ) e}{6}-\frac {\ln \left (1-x \right ) h}{6}+\frac {\ln \left (1+x \right ) h}{6}+\frac {4 \ln \left (2-x \right ) h}{3}-\frac {4 \ln \left (x +2\right ) h}{3}+\frac {8 \ln \left (2-x \right ) i}{3}+\frac {i \,x^{2}}{2}-\frac {\ln \left (1+x \right ) i}{6}-\frac {\ln \left (1-x \right ) i}{6}+\frac {8 \ln \left (x +2\right ) i}{3}\) | \(203\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 85, normalized size = 1.12 \begin {gather*} h x + \frac {1}{2} i \, x^{2} - \frac {1}{12} \, {\left (d + 4 \, f - 8 \, g + 16 \, h - 2 \, e - 32 i\right )} \log \left (x + 2\right ) + \frac {1}{6} \, {\left (d + f - g + h - e - i\right )} \log \left (x + 1\right ) - \frac {1}{6} \, {\left (d + f + g + h + e + i\right )} \log \left (x - 1\right ) + \frac {1}{12} \, {\left (d + 4 \, f + 8 \, g + 16 \, h + 2 \, e + 32 i\right )} \log \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 6.52, size = 88, normalized size = 1.16 \begin {gather*} \frac {1}{2} \, i x^{2} + h x - \frac {1}{12} \, {\left (d - 2 \, e + 4 \, f - 8 \, g + 16 \, h - 32 \, 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}{6} \, {\left (d + e + f + g + h + i\right )} \log \left (x - 1\right ) + \frac {1}{12} \, {\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.50, size = 89, normalized size = 1.17 \begin {gather*} h x + \frac {1}{2} i \, x^{2} - \frac {1}{12} \, {\left (d + 4 \, f - 8 \, g + 16 \, h - 2 \, e - 32 i\right )} \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{6} \, {\left (d + f - g + h - e - i\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{6} \, {\left (d + f + g + h + e + i\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{12} \, {\left (d + 4 \, f + 8 \, g + 16 \, h + 2 \, e + 32 i\right )} \log \left ({\left | x - 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.19, size = 108, normalized size = 1.42 \begin {gather*} h\,x+\frac {i\,x^2}{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+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}{12}+\frac {e}{6}+\frac {f}{3}+\frac {2\,g}{3}+\frac {4\,h}{3}+\frac {8\,i}{3}\right )-\ln \left (x+2\right )\,\left (\frac {d}{12}-\frac {e}{6}+\frac {f}{3}-\frac {2\,g}{3}+\frac {4\,h}{3}-\frac {8\,i}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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