Optimal. Leaf size=107 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{a+1}}\right )}{(1-a) \sqrt {a+1} \sqrt [4]{1-a^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{a+1}}\right )}{(1-a) \sqrt {a+1} \sqrt [4]{1-a^2}}-\frac {e^x}{1-a} \]
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Rubi [A] time = 0.13, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {2282, 388, 212, 208, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{a+1}}\right )}{(1-a) \sqrt {a+1} \sqrt [4]{1-a^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{a+1}}\right )}{(1-a) \sqrt {a+1} \sqrt [4]{1-a^2}}-\frac {e^x}{1-a} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 212
Rule 388
Rule 2282
Rubi steps
\begin {align*} \int \frac {e^x}{a-\tanh (2 x)} \, dx &=\operatorname {Subst}\left (\int \frac {1+x^4}{1+a-(1-a) x^4} \, dx,x,e^x\right )\\ &=-\frac {e^x}{1-a}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{1+a+(-1+a) x^4} \, dx,x,e^x\right )}{1-a}\\ &=-\frac {e^x}{1-a}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+a}-\sqrt {1-a} x^2} \, dx,x,e^x\right )}{(1-a) \sqrt {1+a}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+a}+\sqrt {1-a} x^2} \, dx,x,e^x\right )}{(1-a) \sqrt {1+a}}\\ &=-\frac {e^x}{1-a}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{1+a}}\right )}{(1-a) \sqrt {1+a} \sqrt [4]{1-a^2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{1+a}}\right )}{(1-a) \sqrt {1+a} \sqrt [4]{1-a^2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 81, normalized size = 0.76 \[ \frac {-\sqrt [4]{1-a} (a+1)^{3/4} e^x+\tan ^{-1}\left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{a+1}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{1-a} e^x}{\sqrt [4]{a+1}}\right )}{(1-a)^{5/4} (a+1)^{3/4}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 440, normalized size = 4.11 \[ -\frac {4 \, {\left (a - 1\right )} \left (-\frac {1}{a^{8} - 2 \, a^{7} - 2 \, a^{6} + 6 \, a^{5} - 6 \, a^{3} + 2 \, a^{2} + 2 \, a - 1}\right )^{\frac {1}{4}} \arctan \left (-{\left (a^{6} - 2 \, a^{5} - a^{4} + 4 \, a^{3} - a^{2} - 2 \, a + 1\right )} \left (-\frac {1}{a^{8} - 2 \, a^{7} - 2 \, a^{6} + 6 \, a^{5} - 6 \, a^{3} + 2 \, a^{2} + 2 \, a - 1}\right )^{\frac {3}{4}} e^{x} + {\left (a^{6} - 2 \, a^{5} - a^{4} + 4 \, a^{3} - a^{2} - 2 \, a + 1\right )} \sqrt {{\left (a^{4} - 2 \, a^{2} + 1\right )} \sqrt {-\frac {1}{a^{8} - 2 \, a^{7} - 2 \, a^{6} + 6 \, a^{5} - 6 \, a^{3} + 2 \, a^{2} + 2 \, a - 1}} + e^{\left (2 \, x\right )}} \left (-\frac {1}{a^{8} - 2 \, a^{7} - 2 \, a^{6} + 6 \, a^{5} - 6 \, a^{3} + 2 \, a^{2} + 2 \, a - 1}\right )^{\frac {3}{4}}\right ) + {\left (a - 1\right )} \left (-\frac {1}{a^{8} - 2 \, a^{7} - 2 \, a^{6} + 6 \, a^{5} - 6 \, a^{3} + 2 \, a^{2} + 2 \, a - 1}\right )^{\frac {1}{4}} \log \left ({\left (a^{2} - 1\right )} \left (-\frac {1}{a^{8} - 2 \, a^{7} - 2 \, a^{6} + 6 \, a^{5} - 6 \, a^{3} + 2 \, a^{2} + 2 \, a - 1}\right )^{\frac {1}{4}} + e^{x}\right ) - {\left (a - 1\right )} \left (-\frac {1}{a^{8} - 2 \, a^{7} - 2 \, a^{6} + 6 \, a^{5} - 6 \, a^{3} + 2 \, a^{2} + 2 \, a - 1}\right )^{\frac {1}{4}} \log \left (-{\left (a^{2} - 1\right )} \left (-\frac {1}{a^{8} - 2 \, a^{7} - 2 \, a^{6} + 6 \, a^{5} - 6 \, a^{3} + 2 \, a^{2} + 2 \, a - 1}\right )^{\frac {1}{4}} + e^{x}\right ) - 2 \, e^{x}}{2 \, {\left (a - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 328, normalized size = 3.07 \[ -\frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}} + 2 \, e^{x}\right )}}{2 \, \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{3} - \sqrt {2} a^{2} - \sqrt {2} a + \sqrt {2}} - \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}} - 2 \, e^{x}\right )}}{2 \, \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{3} - \sqrt {2} a^{2} - \sqrt {2} a + \sqrt {2}} - \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \log \left (\sqrt {2} \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}} e^{x} + \sqrt {\frac {a + 1}{a - 1}} + e^{\left (2 \, x\right )}\right )}{2 \, {\left (\sqrt {2} a^{3} - \sqrt {2} a^{2} - \sqrt {2} a + \sqrt {2}\right )}} + \frac {{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac {1}{4}} \log \left (-\sqrt {2} \left (\frac {a + 1}{a - 1}\right )^{\frac {1}{4}} e^{x} + \sqrt {\frac {a + 1}{a - 1}} + e^{\left (2 \, x\right )}\right )}{2 \, {\left (\sqrt {2} a^{3} - \sqrt {2} a^{2} - \sqrt {2} a + \sqrt {2}\right )}} + \frac {e^{x}}{a - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.29, size = 87, normalized size = 0.81 \[ \frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+6 a \,\textit {\_Z}^{2}-4 \textit {\_Z} +a \right )}{\sum }\frac {\left (-\textit {\_R}^{2}+2 \textit {\_R} -1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{3} a -3 \textit {\_R}^{2}+3 \textit {\_R} a -1}}{-2+2 a}-\frac {2}{\left (-1+a \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.45, size = 163, normalized size = 1.52 \[ \frac {\ln \left (8\,a\,{\left (-a-1\right )}^{1/4}+8\,{\mathrm {e}}^x\,{\left (a-1\right )}^{5/4}-8\,{\left (-a-1\right )}^{1/4}\right )-\ln \left (8\,{\mathrm {e}}^x\,{\left (a-1\right )}^{5/4}-8\,a\,{\left (-a-1\right )}^{1/4}+8\,{\left (-a-1\right )}^{1/4}\right )+2\,{\mathrm {e}}^x\,{\left (a-1\right )}^{1/4}\,{\left (-a-1\right )}^{3/4}-\ln \left (8\,{\mathrm {e}}^x\,{\left (a-1\right )}^{5/4}-a\,{\left (-a-1\right )}^{1/4}\,8{}\mathrm {i}+{\left (-a-1\right )}^{1/4}\,8{}\mathrm {i}\right )\,1{}\mathrm {i}+\ln \left (a\,{\left (-a-1\right )}^{1/4}\,8{}\mathrm {i}+8\,{\mathrm {e}}^x\,{\left (a-1\right )}^{5/4}-{\left (-a-1\right )}^{1/4}\,8{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,{\left (a-1\right )}^{5/4}\,{\left (-a-1\right )}^{3/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{x}}{a - \tanh {\left (2 x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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