Optimal. Leaf size=32 \[ -2 e^x+e^{-x} \log \left (1+e^{2 x}\right )+e^x \log \left (1+e^{2 x}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2320, 2526,
2498, 327, 209, 2505} \begin {gather*} -2 e^x+e^{-x} \log \left (e^{2 x}+1\right )+e^x \log \left (e^{2 x}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 327
Rule 2320
Rule 2498
Rule 2505
Rule 2526
Rubi steps
\begin {align*} \int \left (-e^{-x}+e^x\right ) \log \left (1+e^{2 x}\right ) \, dx &=\text {Subst}\left (\int \frac {\left (-1+x^2\right ) \log \left (1+x^2\right )}{x^2} \, dx,x,e^x\right )\\ &=\text {Subst}\left (\int \left (\log \left (1+x^2\right )-\frac {\log \left (1+x^2\right )}{x^2}\right ) \, dx,x,e^x\right )\\ &=\text {Subst}\left (\int \log \left (1+x^2\right ) \, dx,x,e^x\right )-\text {Subst}\left (\int \frac {\log \left (1+x^2\right )}{x^2} \, dx,x,e^x\right )\\ &=e^{-x} \log \left (1+e^{2 x}\right )+e^x \log \left (1+e^{2 x}\right )-2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^x\right )-2 \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,e^x\right )\\ &=-2 e^x-2 \tan ^{-1}\left (e^x\right )+e^{-x} \log \left (1+e^{2 x}\right )+e^x \log \left (1+e^{2 x}\right )+2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^x\right )\\ &=-2 e^x+e^{-x} \log \left (1+e^{2 x}\right )+e^x \log \left (1+e^{2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 24, normalized size = 0.75 \begin {gather*} -2 e^x+\left (e^{-x}+e^x\right ) \log \left (1+e^{2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 24, normalized size = 0.75
method | result | size |
risch | \(\left (1+{\mathrm e}^{2 x}\right ) {\mathrm e}^{-x} \ln \left (1+{\mathrm e}^{2 x}\right )-2 \,{\mathrm e}^{x}\) | \(24\) |
norman | \(\left ({\mathrm e}^{2 x} \ln \left (1+{\mathrm e}^{2 x}\right )-2 \,{\mathrm e}^{2 x}+\ln \left (1+{\mathrm e}^{2 x}\right )\right ) {\mathrm e}^{-x}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.45, size = 20, normalized size = 0.62 \begin {gather*} {\left (e^{\left (-x\right )} + e^{x}\right )} \log \left (e^{\left (2 \, x\right )} + 1\right ) - 2 \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.50, size = 26, normalized size = 0.81 \begin {gather*} {\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{\left (2 \, x\right )} + 1\right ) - 2 \, e^{\left (2 \, x\right )}\right )} e^{\left (-x\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 = 1.98, size = 20, normalized size = 0.62 \begin {gather*} {\left (e^{\left (-x\right )} + e^{x}\right )} \log \left (e^{\left (2 \, x\right )} + 1\right ) - 2 \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.38, size = 24, normalized size = 0.75 \begin {gather*} 2\,\ln \left ({\mathrm {e}}^{2\,x}+1\right )\,\mathrm {cosh}\left (x\right )-\frac {{\mathrm {e}}^{2\,x}+1}{\mathrm {cosh}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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