Integrand size = 13, antiderivative size = 194 \[ \int \frac {\tanh ^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {b^5 \arctan (\sinh (x))}{\left (a^2+b^2\right )^3}-\frac {b^3 \arctan (\sinh (x))}{2 \left (a^2+b^2\right )^2}-\frac {3 b \arctan (\sinh (x))}{8 \left (a^2+b^2\right )}+\frac {b^6 \log (a+b \text {csch}(x))}{a \left (a^2+b^2\right )^3}+\frac {\log (\sinh (x))}{a}-\frac {a \left (a^4+3 a^2 b^2+3 b^4\right ) \log (\tanh (x))}{\left (a^2+b^2\right )^3}+\frac {3 b \text {sech}(x) \tanh (x)}{8 \left (a^2+b^2\right )}-\frac {\left (a \left (a^2+2 b^2\right )-b^3 \text {csch}(x)\right ) \tanh ^2(x)}{2 \left (a^2+b^2\right )^2}-\frac {(a-b \text {csch}(x)) \tanh ^4(x)}{4 \left (a^2+b^2\right )} \]
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Time = 0.20 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3970, 908, 653, 205, 209, 649, 266} \[ \int \frac {\tanh ^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {3 b \arctan (\sinh (x))}{8 \left (a^2+b^2\right )}-\frac {b^5 \arctan (\sinh (x))}{\left (a^2+b^2\right )^3}-\frac {b^3 \arctan (\sinh (x))}{2 \left (a^2+b^2\right )^2}-\frac {\tanh ^4(x) (a-b \text {csch}(x))}{4 \left (a^2+b^2\right )}+\frac {3 b \tanh (x) \text {sech}(x)}{8 \left (a^2+b^2\right )}+\frac {b^6 \log (a+b \text {csch}(x))}{a \left (a^2+b^2\right )^3}-\frac {\tanh ^2(x) \left (a \left (a^2+2 b^2\right )-b^3 \text {csch}(x)\right )}{2 \left (a^2+b^2\right )^2}-\frac {a \left (a^4+3 a^2 b^2+3 b^4\right ) \log (\tanh (x))}{\left (a^2+b^2\right )^3}+\frac {\log (\sinh (x))}{a} \]
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Rule 205
Rule 209
Rule 266
Rule 649
Rule 653
Rule 908
Rule 3970
Rubi steps \begin{align*} \text {integral}& = b^6 \text {Subst}\left (\int \frac {1}{x (a+x) \left (-b^2-x^2\right )^3} \, dx,x,b \text {csch}(x)\right ) \\ & = b^6 \text {Subst}\left (\int \left (-\frac {1}{a b^6 x}+\frac {1}{a \left (a^2+b^2\right )^3 (a+x)}+\frac {b^2+a x}{b^2 \left (a^2+b^2\right ) \left (b^2+x^2\right )^3}+\frac {b^4+a \left (a^2+2 b^2\right ) x}{b^4 \left (a^2+b^2\right )^2 \left (b^2+x^2\right )^2}+\frac {b^6+a \left (a^4+3 a^2 b^2+3 b^4\right ) x}{b^6 \left (a^2+b^2\right )^3 \left (b^2+x^2\right )}\right ) \, dx,x,b \text {csch}(x)\right ) \\ & = \frac {b^6 \log (a+b \text {csch}(x))}{a \left (a^2+b^2\right )^3}+\frac {\log (\sinh (x))}{a}+\frac {\text {Subst}\left (\int \frac {b^6+a \left (a^4+3 a^2 b^2+3 b^4\right ) x}{b^2+x^2} \, dx,x,b \text {csch}(x)\right )}{\left (a^2+b^2\right )^3}+\frac {b^2 \text {Subst}\left (\int \frac {b^4+a \left (a^2+2 b^2\right ) x}{\left (b^2+x^2\right )^2} \, dx,x,b \text {csch}(x)\right )}{\left (a^2+b^2\right )^2}+\frac {b^4 \text {Subst}\left (\int \frac {b^2+a x}{\left (b^2+x^2\right )^3} \, dx,x,b \text {csch}(x)\right )}{a^2+b^2} \\ & = \frac {b^6 \log (a+b \text {csch}(x))}{a \left (a^2+b^2\right )^3}+\frac {\log (\sinh (x))}{a}-\frac {\left (a \left (a^2+2 b^2\right )-b^3 \text {csch}(x)\right ) \tanh ^2(x)}{2 \left (a^2+b^2\right )^2}-\frac {(a-b \text {csch}(x)) \tanh ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b^6 \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \text {csch}(x)\right )}{\left (a^2+b^2\right )^3}+\frac {b^4 \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \text {csch}(x)\right )}{2 \left (a^2+b^2\right )^2}+\frac {\left (3 b^4\right ) \text {Subst}\left (\int \frac {1}{\left (b^2+x^2\right )^2} \, dx,x,b \text {csch}(x)\right )}{4 \left (a^2+b^2\right )}+\frac {\left (a \left (a^4+3 a^2 b^2+3 b^4\right )\right ) \text {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \text {csch}(x)\right )}{\left (a^2+b^2\right )^3} \\ & = -\frac {b^5 \arctan (\sinh (x))}{\left (a^2+b^2\right )^3}-\frac {b^3 \arctan (\sinh (x))}{2 \left (a^2+b^2\right )^2}+\frac {b^6 \log (a+b \text {csch}(x))}{a \left (a^2+b^2\right )^3}+\frac {\log (\sinh (x))}{a}-\frac {a \left (a^4+3 a^2 b^2+3 b^4\right ) \log (\tanh (x))}{\left (a^2+b^2\right )^3}+\frac {3 b \text {sech}(x) \tanh (x)}{8 \left (a^2+b^2\right )}-\frac {\left (a \left (a^2+2 b^2\right )-b^3 \text {csch}(x)\right ) \tanh ^2(x)}{2 \left (a^2+b^2\right )^2}-\frac {(a-b \text {csch}(x)) \tanh ^4(x)}{4 \left (a^2+b^2\right )}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \text {csch}(x)\right )}{8 \left (a^2+b^2\right )} \\ & = -\frac {b^5 \arctan (\sinh (x))}{\left (a^2+b^2\right )^3}-\frac {b^3 \arctan (\sinh (x))}{2 \left (a^2+b^2\right )^2}-\frac {3 b \arctan (\sinh (x))}{8 \left (a^2+b^2\right )}+\frac {b^6 \log (a+b \text {csch}(x))}{a \left (a^2+b^2\right )^3}+\frac {\log (\sinh (x))}{a}-\frac {a \left (a^4+3 a^2 b^2+3 b^4\right ) \log (\tanh (x))}{\left (a^2+b^2\right )^3}+\frac {3 b \text {sech}(x) \tanh (x)}{8 \left (a^2+b^2\right )}-\frac {\left (a \left (a^2+2 b^2\right )-b^3 \text {csch}(x)\right ) \tanh ^2(x)}{2 \left (a^2+b^2\right )^2}-\frac {(a-b \text {csch}(x)) \tanh ^4(x)}{4 \left (a^2+b^2\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.30 \[ \int \frac {\tanh ^5(x)}{a+b \text {csch}(x)} \, dx=\frac {a b \left (5 a^4+14 a^2 b^2+9 b^4\right ) \arctan (\sinh (x))+4 a \left (a^5+i a^4 b+3 a^3 b^2+3 i a^2 b^3+3 a b^4+3 i b^5\right ) \log (i-\sinh (x))+4 a \left (a^5-i a^4 b+3 a^3 b^2-3 i a^2 b^3+3 a b^4-3 i b^5\right ) \log (i+\sinh (x))+8 b^6 \log (b+a \sinh (x))+4 a^2 \left (2 a^4+5 a^2 b^2+3 b^4\right ) \text {sech}^2(x)-2 a^2 \left (a^2+b^2\right )^2 \text {sech}^4(x)+a b \left (5 a^4+14 a^2 b^2+9 b^4\right ) \text {sech}(x) \tanh (x)-2 a b \left (a^2+b^2\right )^2 \text {sech}^3(x) \tanh (x)}{8 a \left (a^2+b^2\right )^3} \]
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Time = 0.81 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.88
method | result | size |
default | \(\frac {\frac {2 \left (\left (-\frac {3}{8} a^{4} b -\frac {5}{4} a^{2} b^{3}-\frac {7}{8} b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{7}+\left (-a^{5}-3 a^{3} b^{2}-2 a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{6}+\left (-\frac {13}{4} a^{2} b^{3}-\frac {15}{8} b^{5}-\frac {11}{8} a^{4} b \right ) \tanh \left (\frac {x}{2}\right )^{5}+\left (-4 a^{5}-10 a^{3} b^{2}-6 a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{4}+\left (\frac {13}{4} a^{2} b^{3}+\frac {15}{8} b^{5}+\frac {11}{8} a^{4} b \right ) \tanh \left (\frac {x}{2}\right )^{3}+\left (-a^{5}-3 a^{3} b^{2}-2 a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{2}+\left (\frac {3}{8} a^{4} b +\frac {5}{4} a^{2} b^{3}+\frac {7}{8} b^{5}\right ) \tanh \left (\frac {x}{2}\right )\right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{4}}+\frac {\left (8 a^{5}+24 a^{3} b^{2}+24 a \,b^{4}\right ) \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )}{8}+\frac {\left (-3 a^{4} b -10 a^{2} b^{3}-15 b^{5}\right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{4}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}+\frac {b^{6} \ln \left (-\tanh \left (\frac {x}{2}\right )^{2} b +2 a \tanh \left (\frac {x}{2}\right )+b \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right ) a}\) | \(364\) |
risch | \(\frac {x}{a}-\frac {2 x \,a^{5}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {6 x \,a^{3} b^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {6 x a \,b^{4}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 x \,b^{6}}{a \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {\left (5 a^{2} b \,{\mathrm e}^{6 x}+9 b^{3} {\mathrm e}^{6 x}+16 a^{3} {\mathrm e}^{5 x}+24 a \,b^{2} {\mathrm e}^{5 x}-3 a^{2} b \,{\mathrm e}^{4 x}+b^{3} {\mathrm e}^{4 x}+16 a^{3} {\mathrm e}^{3 x}+32 a \,b^{2} {\mathrm e}^{3 x}+3 a^{2} b \,{\mathrm e}^{2 x}-b^{3} {\mathrm e}^{2 x}+16 a^{3} {\mathrm e}^{x}+24 \,{\mathrm e}^{x} b^{2} a -5 a^{2} b -9 b^{3}\right ) {\mathrm e}^{x}}{4 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+{\mathrm e}^{2 x}\right )^{4}}-\frac {5 i \ln \left ({\mathrm e}^{x}+i\right ) a^{2} b^{3}}{4 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {15 i \ln \left ({\mathrm e}^{x}-i\right ) b^{5}}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {15 i \ln \left ({\mathrm e}^{x}+i\right ) b^{5}}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {\ln \left ({\mathrm e}^{x}-i\right ) a^{5}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {3 \ln \left ({\mathrm e}^{x}-i\right ) a^{3} b^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {3 \ln \left ({\mathrm e}^{x}-i\right ) a \,b^{4}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {3 i \ln \left ({\mathrm e}^{x}+i\right ) a^{4} b}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {3 i \ln \left ({\mathrm e}^{x}-i\right ) a^{4} b}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {5 i \ln \left ({\mathrm e}^{x}-i\right ) a^{2} b^{3}}{4 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {\ln \left ({\mathrm e}^{x}+i\right ) a^{5}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {3 \ln \left ({\mathrm e}^{x}+i\right ) a^{3} b^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {3 \ln \left ({\mathrm e}^{x}+i\right ) a \,b^{4}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {b^{6} \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{a \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\) | \(787\) |
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Leaf count of result is larger than twice the leaf count of optimal. 4025 vs. \(2 (185) = 370\).
Time = 0.37 (sec) , antiderivative size = 4025, normalized size of antiderivative = 20.75 \[ \int \frac {\tanh ^5(x)}{a+b \text {csch}(x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\tanh ^5(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\tanh ^{5}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (185) = 370\).
Time = 0.29 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.97 \[ \int \frac {\tanh ^5(x)}{a+b \text {csch}(x)} \, dx=\frac {b^{6} \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}} + \frac {{\left (3 \, a^{4} b + 10 \, a^{2} b^{3} + 15 \, b^{5}\right )} \arctan \left (e^{\left (-x\right )}\right )}{4 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {{\left (a^{5} + 3 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (5 \, a^{2} b + 9 \, b^{3}\right )} e^{\left (-x\right )} + 8 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} e^{\left (-2 \, x\right )} - {\left (3 \, a^{2} b - b^{3}\right )} e^{\left (-3 \, x\right )} + 16 \, {\left (a^{3} + 2 \, a b^{2}\right )} e^{\left (-4 \, x\right )} + {\left (3 \, a^{2} b - b^{3}\right )} e^{\left (-5 \, x\right )} + 8 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} e^{\left (-6 \, x\right )} - {\left (5 \, a^{2} b + 9 \, b^{3}\right )} e^{\left (-7 \, x\right )}}{4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4} + 4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-2 \, x\right )} + 6 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-4 \, x\right )} + 4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-6 \, x\right )} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-8 \, x\right )}\right )}} + \frac {x}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (185) = 370\).
Time = 0.30 (sec) , antiderivative size = 432, normalized size of antiderivative = 2.23 \[ \int \frac {\tanh ^5(x)}{a+b \text {csch}(x)} \, dx=\frac {b^{6} \log \left ({\left | -a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}} - \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} {\left (3 \, a^{4} b + 10 \, a^{2} b^{3} + 15 \, b^{5}\right )}}{16 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {{\left (a^{5} + 3 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac {3 \, a^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 9 \, a^{3} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 9 \, a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 5 \, a^{4} b {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 14 \, a^{2} b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 9 \, b^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 8 \, a^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 32 \, a^{3} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 48 \, a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 12 \, a^{4} b {\left (e^{\left (-x\right )} - e^{x}\right )} + 40 \, a^{2} b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )} + 28 \, b^{5} {\left (e^{\left (-x\right )} - e^{x}\right )} + 16 \, a^{3} b^{2} + 64 \, a b^{4}}{4 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}^{2}} \]
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Time = 7.33 (sec) , antiderivative size = 611, normalized size of antiderivative = 3.15 \[ \int \frac {\tanh ^5(x)}{a+b \text {csch}(x)} \, dx=\frac {\frac {6\,{\mathrm {e}}^x\,\left (a^2\,b+b^3\right )}{{\left (a^2+b^2\right )}^2}+\frac {8\,\left (a^4+a^2\,b^2\right )}{a\,{\left (a^2+b^2\right )}^2}}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}-\frac {\frac {4\,a}{a^2+b^2}+\frac {4\,b\,{\mathrm {e}}^x}{a^2+b^2}}{4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {x}{a}-\frac {\frac {{\mathrm {e}}^x\,\left (9\,a^2\,b+13\,b^3\right )}{2\,{\left (a^2+b^2\right )}^2}+\frac {2\,\left (4\,a^4+5\,a^2\,b^2\right )}{a\,{\left (a^2+b^2\right )}^2}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}+\frac {\frac {{\mathrm {e}}^x\,\left (5\,a^4\,b+14\,a^2\,b^3+9\,b^5\right )}{4\,{\left (a^2+b^2\right )}^3}+\frac {2\,\left (2\,a^6+5\,a^4\,b^2+3\,a^2\,b^4\right )}{a\,{\left (a^2+b^2\right )}^3}}{{\mathrm {e}}^{2\,x}+1}+\frac {\ln \left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )\,\left (-8\,a^2+a\,b\,21{}\mathrm {i}+15\,b^2\right )}{8\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )}+\frac {b^6\,\ln \left (64\,a^{13}\,{\mathrm {e}}^{2\,x}-64\,a\,b^{12}-64\,a^{13}+159\,a^3\,b^{10}-492\,a^5\,b^8-1214\,a^7\,b^6-1020\,a^9\,b^4-393\,a^{11}\,b^2+128\,b^{13}\,{\mathrm {e}}^x-159\,a^3\,b^{10}\,{\mathrm {e}}^{2\,x}+492\,a^5\,b^8\,{\mathrm {e}}^{2\,x}+1214\,a^7\,b^6\,{\mathrm {e}}^{2\,x}+1020\,a^9\,b^4\,{\mathrm {e}}^{2\,x}+393\,a^{11}\,b^2\,{\mathrm {e}}^{2\,x}+128\,a^{12}\,b\,{\mathrm {e}}^x+64\,a\,b^{12}\,{\mathrm {e}}^{2\,x}-318\,a^2\,b^{11}\,{\mathrm {e}}^x+984\,a^4\,b^9\,{\mathrm {e}}^x+2428\,a^6\,b^7\,{\mathrm {e}}^x+2040\,a^8\,b^5\,{\mathrm {e}}^x+786\,a^{10}\,b^3\,{\mathrm {e}}^x\right )}{a^7+3\,a^5\,b^2+3\,a^3\,b^4+a\,b^6}+\frac {\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,\left (-a^2\,8{}\mathrm {i}+21\,a\,b+b^2\,15{}\mathrm {i}\right )}{8\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )} \]
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