Optimal. Leaf size=100 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} e^x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{a} b^{2/3}}+\frac {\log \left (\sqrt [3]{b}+\sqrt [3]{a} e^x\right )}{2 \sqrt [3]{a} b^{2/3}}-\frac {\log \left (b+a e^{3 x}\right )}{6 \sqrt [3]{a} b^{2/3}} \]
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Rubi [A]
time = 0.06, antiderivative size = 123, normalized size of antiderivative = 1.23, number of steps
used = 7, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {2281, 206, 31,
648, 631, 210, 642} \begin {gather*} -\frac {\log \left (a^{2/3} e^{2 x}-\sqrt [3]{a} \sqrt [3]{b} e^x+b^{2/3}\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac {\log \left (\sqrt [3]{a} e^x+\sqrt [3]{b}\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} e^x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{a} b^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 631
Rule 642
Rule 648
Rule 2281
Rubi steps
\begin {align*} \int \frac {e^x}{b+a e^{3 x}} \, dx &=\text {Subst}\left (\int \frac {1}{b+a x^3} \, dx,x,e^x\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx,x,e^x\right )}{3 b^{2/3}}+\frac {\text {Subst}\left (\int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,e^x\right )}{3 b^{2/3}}\\ &=\frac {\log \left (\sqrt [3]{b}+\sqrt [3]{a} e^x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,e^x\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac {\text {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,e^x\right )}{2 \sqrt [3]{b}}\\ &=\frac {\log \left (\sqrt [3]{b}+\sqrt [3]{a} e^x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} e^x+a^{2/3} e^{2 x}\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} e^x}{\sqrt [3]{b}}\right )}{\sqrt [3]{a} b^{2/3}}\\ &=-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a} e^x}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} b^{2/3}}+\frac {\log \left (\sqrt [3]{b}+\sqrt [3]{a} e^x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} e^x+a^{2/3} e^{2 x}\right )}{6 \sqrt [3]{a} b^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 97, normalized size = 0.97 \begin {gather*} -\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a} e^x}{\sqrt [3]{b}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{b}+\sqrt [3]{a} e^x\right )+\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} e^x+a^{2/3} e^{2 x}\right )}{6 \sqrt [3]{a} b^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 1.91, size = 28, normalized size = 0.28 \begin {gather*} \text {RootSum}\left [-1+27 a b^2 \text {\#1}^3\&,\text {Log}\left [3 b \text {\#1}+E^x\right ] \text {\#1}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 95, normalized size = 0.95
method | result | size |
risch | \(\munderset {\textit {\_R} =\RootOf \left (27 a \,b^{2} \textit {\_Z}^{3}-1\right )}{\sum }\textit {\_R} \ln \left (3 b \textit {\_R} +{\mathrm e}^{x}\right )\) | \(26\) |
default | \(\frac {\ln \left ({\mathrm e}^{x}+\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\ln \left ({\mathrm e}^{2 x}-\left (\frac {b}{a}\right )^{\frac {1}{3}} {\mathrm e}^{x}+\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,{\mathrm e}^{x}}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}\) | \(95\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 100, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {b}{a}\right )^{\frac {1}{3}} - 2 \, e^{x}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {\log \left (-\left (\frac {b}{a}\right )^{\frac {1}{3}} e^{x} + \left (\frac {b}{a}\right )^{\frac {2}{3}} + e^{\left (2 \, x\right )}\right )}{6 \, a \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {\log \left (\left (\frac {b}{a}\right )^{\frac {1}{3}} + e^{x}\right )}{3 \, a \left (\frac {b}{a}\right )^{\frac {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 311, normalized size = 3.11 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, a b e^{\left (3 \, x\right )} - 3 \, \left (a b^{2}\right )^{\frac {1}{3}} b e^{x} - b^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b e^{\left (2 \, x\right )} + \left (a b^{2}\right )^{\frac {2}{3}} e^{x} - \left (a b^{2}\right )^{\frac {1}{3}} b\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{a e^{\left (3 \, x\right )} + b}\right ) - \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b e^{\left (2 \, x\right )} - \left (a b^{2}\right )^{\frac {2}{3}} e^{x} + \left (a b^{2}\right )^{\frac {1}{3}} b\right ) + 2 \, \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b e^{x} + \left (a b^{2}\right )^{\frac {2}{3}}\right )}{6 \, a b^{2}}, \frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a b^{2}\right )^{\frac {2}{3}} e^{x} - \left (a b^{2}\right )^{\frac {1}{3}} b\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{b^{2}}\right ) - \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b e^{\left (2 \, x\right )} - \left (a b^{2}\right )^{\frac {2}{3}} e^{x} + \left (a b^{2}\right )^{\frac {1}{3}} b\right ) + 2 \, \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b e^{x} + \left (a b^{2}\right )^{\frac {2}{3}}\right )}{6 \, a b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 22, normalized size = 0.22 \begin {gather*} \operatorname {RootSum} {\left (27 z^{3} a b^{2} - 1, \left ( i \mapsto i \log {\left (3 i b + e^{x} \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 142, normalized size = 1.42 \begin {gather*} \frac {\frac {1}{6} \left (-a^{2} b\right )^{\frac {1}{3}} \ln \left (\left (\mathrm {e}^{x}\right )^{2}+\left (-\frac {b}{a}\right )^{\frac {1}{3}} \mathrm {e}^{x}+\left (-\frac {b}{a}\right )^{\frac {1}{3}} \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right )}{a b}+\frac {\left (-a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {2 \left (\mathrm {e}^{x}+\frac {\left (-\frac {b}{a}\right )^{\frac {1}{3}}}{2}\right )}{\sqrt {3} \left (-\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} a b}-\frac {\frac {1}{3} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \ln \left |\mathrm {e}^{x}-\left (-\frac {b}{a}\right )^{\frac {1}{3}}\right |}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.51, size = 104, normalized size = 1.04 \begin {gather*} \frac {\ln \left (\frac {b^{1/3}}{a^{7/3}}+\frac {{\mathrm {e}}^x}{a^2}\right )}{3\,a^{1/3}\,b^{2/3}}+\frac {\ln \left (\frac {{\mathrm {e}}^x}{a^2}+\frac {b^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^{7/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{1/3}\,b^{2/3}}-\frac {\ln \left (\frac {{\mathrm {e}}^x}{a^2}-\frac {b^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a^{7/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{1/3}\,b^{2/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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