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 = 1.74, 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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\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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {Subst}\left (\int \frac {e^x \log ^2\left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )+16 \operatorname {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 \operatorname {Subst}\left (\int \frac {e^x \log \left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )+400 \operatorname {Subst}\left (\int \frac {e^x}{1+e^x} \, dx,x,\frac {x}{4}\right )+400 \operatorname {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 \operatorname {Subst}\left (\int \frac {e^x \log ^2\left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )+16 \operatorname {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 \operatorname {Subst}\left (\int \frac {e^x \log \left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )+400 \operatorname {Subst}\left (\int \frac {1}{1+e^x} \, dx,x,\frac {x}{4}\right )+400 \operatorname {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 \operatorname {Subst}\left (\int \frac {e^x \log ^2\left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )+16 \operatorname {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 \operatorname {Subst}\left (\int \frac {e^x \log \left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )+400 \operatorname {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 \operatorname {Subst}\left (\int \frac {e^x \log ^2\left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )+16 \operatorname {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 \operatorname {Subst}\left (\int \frac {e^x \log \left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )+400 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^{x/4}\right )-400 \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^{x/4}\right )\\ &=100 x-\frac {100 x}{1+e^{x/4}}+16 \operatorname {Subst}\left (\int \frac {e^x \log ^2\left (256 x^4\right )}{1+e^x} \, dx,x,\frac {x}{4}\right )+16 \operatorname {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 \operatorname {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 = 0.43, 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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fricas [A] time = 0.96, 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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giac [A] time = 0.23, 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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maple [A] time = 0.36, size = 31, normalized size = 1.19
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 | \(\frac {\left (4 \left (\ln \left (x^{4}\right )-4 \ln \relax (x )\right )^{2}+100\right ) x \,{\mathrm e}^{\frac {x}{4}}+64 \,{\mathrm e}^{\frac {x}{4}} \ln \relax (x )^{2} x +32 \,{\mathrm e}^{\frac {x}{4}} \ln \relax (x ) x \left (\ln \left (x^{4}\right )-4 \ln \relax (x )\right )}{{\mathrm e}^{\frac {x}{4}}+1}\) | \(61\) |
risch | \(\frac {64 x \,{\mathrm e}^{\frac {x}{4}} \ln \relax (x )^{2}}{{\mathrm e}^{\frac {x}{4}}+1}-\frac {16 i \pi x \,{\mathrm e}^{\frac {x}{4}} \left (\mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )-\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{2}+\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )-\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+\mathrm {csgn}\left (i x^{2}\right )^{3}-\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2}+\mathrm {csgn}\left (i x^{3}\right )^{3}-\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+\mathrm {csgn}\left (i x^{4}\right )^{3}\right ) \ln \relax (x )}{{\mathrm e}^{\frac {x}{4}}+1}-\frac {x \left (-100-2 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{5}-2 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{4}\right )^{5}-2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{4} \mathrm {csgn}\left (i x^{3}\right )^{2}+2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x^{3}\right )^{3}+2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x^{4}\right )^{3}+\pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x^{3}\right )^{4}-2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{5}-2 \pi ^{2} \mathrm {csgn}\left (i x^{3}\right )^{4} \mathrm {csgn}\left (i x^{4}\right )^{2}+2 \pi ^{2} \mathrm {csgn}\left (i x^{3}\right )^{3} \mathrm {csgn}\left (i x^{4}\right )^{3}+\pi ^{2} \mathrm {csgn}\left (i x^{3}\right )^{2} \mathrm {csgn}\left (i x^{4}\right )^{4}-2 \pi ^{2} \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{5}+\pi ^{2} \mathrm {csgn}\left (i x^{3}\right )^{6}+\pi ^{2} \mathrm {csgn}\left (i x^{4}\right )^{6}+2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{3} \mathrm {csgn}\left (i x^{4}\right )^{2}-2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2} \mathrm {csgn}\left (i x^{4}\right )^{3}+3 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{3}\right )^{2} \mathrm {csgn}\left (i x^{4}\right )^{2}-2 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{3}+2 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{4} \mathrm {csgn}\left (i x^{3}\right )+2 \pi ^{2} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x^{3}\right )-2 \pi ^{2} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2}-2 \pi ^{2} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2}-4 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x^{3}\right )+2 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x^{3}\right )^{2}-2 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x^{4}\right )^{2}-6 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x^{3}\right )^{3}-4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x^{4}\right )^{3}+4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{4}+2 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{4} \mathrm {csgn}\left (i x^{4}\right )-4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{2} \mathrm {csgn}\left (i x^{4}\right )^{3}+4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{4}+3 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x^{3}\right )^{2}+4 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x^{4}\right )^{2}+2 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{4}\right )^{3}-2 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{3}\right )^{3} \mathrm {csgn}\left (i x^{4}\right )-2 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+\pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{6}+\pi ^{2} \mathrm {csgn}\left (i x \right )^{4} \mathrm {csgn}\left (i x^{2}\right )^{2}-4 \pi ^{2} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{3}-4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{5}-4 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2}-2 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{3} \mathrm {csgn}\left (i x^{4}\right )+2 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{3}+\pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{3}\right )^{4}+\pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{4}\right )^{4}+6 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{4}+2 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )+4 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2}+2 \pi ^{2} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )-4 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )+2 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2} \mathrm {csgn}\left (i x^{4}\right )\right ) {\mathrm e}^{\frac {x}{4}}}{{\mathrm e}^{\frac {x}{4}}+1}\) | \(1515\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 35, normalized size = 1.35 \begin {gather*} \frac {64 \, x e^{\left (\frac {1}{4} \, x\right )} \log \relax (x)^{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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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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sympy [A] time = 0.27, 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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