Optimal. Leaf size=26 \[ 4 \left (x-\frac {x}{1+e^{x/4}}\right ) \left (25+\log ^2\left (x^4\right )\right ) \]
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Rubi [F]
time = 1.20, 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 {100 e^{x/2}+e^{x/4} (100+25 x)+\left (32 e^{x/4}+32 e^{x/2}\right ) \log \left (x^4\right )+\left (4 e^{x/2}+e^{x/4} (4+x)\right ) \log ^2\left (x^4\right )}{1+2 e^{x/4}+e^{x/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{x/4} \left (25 \left (4+4 e^{x/4}+x\right )+32 \left (1+e^{x/4}\right ) \log \left (x^4\right )+\left (4+4 e^{x/4}+x\right ) \log ^2\left (x^4\right )\right )}{\left (1+e^{x/4}\right )^2} \, dx\\ &=4 \text {Subst}\left (\int \frac {e^x \left (25 \left (4+4 e^x+4 x\right )+32 \left (1+e^x\right ) \log \left (256 x^4\right )+\left (4+4 e^x+4 x\right ) \log ^2\left (256 x^4\right )\right )}{\left (1+e^x\right )^2} \, dx,x,\frac {x}{4}\right )\\ &=4 \text {Subst}\left (\int \left (\frac {4 e^x x \left (25+\log ^2\left (256 x^4\right )\right )}{\left (1+e^x\right )^2}+\frac {4 e^x \left (25+8 \log \left (256 x^4\right )+\log ^2\left (256 x^4\right )\right )}{1+e^x}\right ) \, dx,x,\frac {x}{4}\right )\\ &=16 \text {Subst}\left (\int \frac {e^x x \left (25+\log ^2\left (256 x^4\right )\right )}{\left (1+e^x\right )^2} \, dx,x,\frac {x}{4}\right )+16 \text {Subst}\left (\int \frac {e^x \left (25+8 \log \left (256 x^4\right )+\log ^2\left (256 x^4\right )\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )\\ &=16 \text {Subst}\left (\int \left (\frac {25 e^x}{1+e^x}+\frac {8 e^x \log \left (256 x^4\right )}{1+e^x}+\frac {e^x \log ^2\left (256 x^4\right )}{1+e^x}\right ) \, dx,x,\frac {x}{4}\right )+16 \text {Subst}\left (\int \left (\frac {25 e^x x}{\left (1+e^x\right )^2}+\frac {e^x x \log ^2\left (256 x^4\right )}{\left (1+e^x\right )^2}\right ) \, dx,x,\frac {x}{4}\right )\\ &=16 \text {Subst}\left (\int \frac {e^x \log ^2\left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )+16 \text {Subst}\left (\int \frac {e^x x \log ^2\left (256 x^4\right )}{\left (1+e^x\right )^2} \, dx,x,\frac {x}{4}\right )+128 \text {Subst}\left (\int \frac {e^x \log \left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )+400 \text {Subst}\left (\int \frac {e^x}{1+e^x} \, dx,x,\frac {x}{4}\right )+400 \text {Subst}\left (\int \frac {e^x x}{\left (1+e^x\right )^2} \, dx,x,\frac {x}{4}\right )\\ &=-\frac {100 x}{1+e^{x/4}}+16 \text {Subst}\left (\int \frac {e^x \log ^2\left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )+16 \text {Subst}\left (\int \frac {e^x x \log ^2\left (256 x^4\right )}{\left (1+e^x\right )^2} \, dx,x,\frac {x}{4}\right )+128 \text {Subst}\left (\int \frac {e^x \log \left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )+400 \text {Subst}\left (\int \frac {1}{1+e^x} \, dx,x,\frac {x}{4}\right )+400 \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^{x/4}\right )\\ &=-\frac {100 x}{1+e^{x/4}}+400 \log \left (1+e^{x/4}\right )+16 \text {Subst}\left (\int \frac {e^x \log ^2\left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )+16 \text {Subst}\left (\int \frac {e^x x \log ^2\left (256 x^4\right )}{\left (1+e^x\right )^2} \, dx,x,\frac {x}{4}\right )+128 \text {Subst}\left (\int \frac {e^x \log \left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )+400 \text {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,e^{x/4}\right )\\ &=-\frac {100 x}{1+e^{x/4}}+400 \log \left (1+e^{x/4}\right )+16 \text {Subst}\left (\int \frac {e^x \log ^2\left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )+16 \text {Subst}\left (\int \frac {e^x x \log ^2\left (256 x^4\right )}{\left (1+e^x\right )^2} \, dx,x,\frac {x}{4}\right )+128 \text {Subst}\left (\int \frac {e^x \log \left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )+400 \text {Subst}\left (\int \frac {1}{x} \, dx,x,e^{x/4}\right )-400 \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^{x/4}\right )\\ &=100 x-\frac {100 x}{1+e^{x/4}}+16 \text {Subst}\left (\int \frac {e^x \log ^2\left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )+16 \text {Subst}\left (\int \frac {e^x x \log ^2\left (256 x^4\right )}{\left (1+e^x\right )^2} \, dx,x,\frac {x}{4}\right )+128 \text {Subst}\left (\int \frac {e^x \log \left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.24, size = 29, normalized size = 1.12 \begin {gather*} \frac {4 e^{x/4} x \left (25+\log ^2\left (x^4\right )\right )}{1+e^{x/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(60\) vs.
\(2(23)=46\).
time = 1.10, size = 61, normalized size = 2.35
method | result | size |
norman | \(\frac {100 x \,{\mathrm e}^{\frac {x}{4}}+4 \ln \left (x^{4}\right )^{2} x \,{\mathrm e}^{\frac {x}{4}}}{{\mathrm e}^{\frac {x}{4}}+1}\) | \(31\) |
default | \(\frac {\left (4 \left (\ln \left (x^{4}\right )-4 \ln \left (x \right )\right )^{2}+100\right ) x \,{\mathrm e}^{\frac {x}{4}}+64 \,{\mathrm e}^{\frac {x}{4}} \ln \left (x \right )^{2} x +32 \,{\mathrm e}^{\frac {x}{4}} \ln \left (x \right ) x \left (\ln \left (x^{4}\right )-4 \ln \left (x \right )\right )}{{\mathrm e}^{\frac {x}{4}}+1}\) | \(61\) |
risch | \(\text {Expression too large to display}\) | \(1515\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 35, normalized size = 1.35 \begin {gather*} \frac {64 \, x e^{\left (\frac {1}{4} \, x\right )} \log \left (x\right )^{2}}{e^{\left (\frac {1}{4} \, x\right )} + 1} + \frac {100 \, x e^{\left (\frac {1}{4} \, x\right )}}{e^{\left (\frac {1}{4} \, x\right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 30, normalized size = 1.15 \begin {gather*} \frac {4 \, {\left (x e^{\left (\frac {1}{4} \, x\right )} \log \left (x^{4}\right )^{2} + 25 \, x e^{\left (\frac {1}{4} \, x\right )}\right )}}{e^{\left (\frac {1}{4} \, x\right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 34, normalized size = 1.31 \begin {gather*} 4 x \log {\left (x^{4} \right )}^{2} + 100 x + \frac {- 4 x \log {\left (x^{4} \right )}^{2} - 100 x}{e^{\frac {x}{4}} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 30, normalized size = 1.15 \begin {gather*} \frac {4 \, {\left (x e^{\left (\frac {1}{4} \, x\right )} \log \left (x^{4}\right )^{2} + 25 \, x e^{\left (\frac {1}{4} \, x\right )}\right )}}{e^{\left (\frac {1}{4} \, x\right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.57, size = 23, normalized size = 0.88 \begin {gather*} \frac {4\,x\,{\mathrm {e}}^{x/4}\,\left ({\ln \left (x^4\right )}^2+25\right )}{{\mathrm {e}}^{x/4}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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