Integrand size = 13, antiderivative size = 198 \[ \int \frac {\text {csch}^4(x)}{(a+b \sinh (x))^2} \, dx=-\frac {b \left (a^2-4 b^2\right ) \text {arctanh}(\cosh (x))}{a^5}-\frac {2 b^4 \left (5 a^2+4 b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^5 \left (a^2+b^2\right )^{3/2}}+\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{3 a^4 \left (a^2+b^2\right )}+\frac {b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 \left (a^2+b^2\right )}-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))} \]
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Time = 0.60 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2881, 3134, 3080, 3855, 2739, 632, 212} \[ \int \frac {\text {csch}^4(x)}{(a+b \sinh (x))^2} \, dx=-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {b \left (a^2-4 b^2\right ) \text {arctanh}(\cosh (x))}{a^5}-\frac {2 b^4 \left (5 a^2+4 b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^5 \left (a^2+b^2\right )^{3/2}}+\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{3 a^4 \left (a^2+b^2\right )}+\frac {b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 \left (a^2+b^2\right )} \]
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Rule 212
Rule 632
Rule 2739
Rule 2881
Rule 3080
Rule 3134
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\int \frac {\text {csch}^4(x) \left (a^2+4 b^2-a b \sinh (x)+3 b^2 \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{a \left (a^2+b^2\right )} \\ & = -\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {i \int \frac {\text {csch}^3(x) \left (6 i b \left (a^2+2 b^2\right )+i a \left (2 a^2-b^2\right ) \sinh (x)+2 i b \left (a^2+4 b^2\right ) \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{3 a^2 \left (a^2+b^2\right )} \\ & = \frac {b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 \left (a^2+b^2\right )}-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {\int \frac {\text {csch}^2(x) \left (2 \left (2 a^4-7 a^2 b^2-12 b^4\right )-2 a b \left (a^2-2 b^2\right ) \sinh (x)-6 b^2 \left (a^2+2 b^2\right ) \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{6 a^3 \left (a^2+b^2\right )} \\ & = \frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{3 a^4 \left (a^2+b^2\right )}+\frac {b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 \left (a^2+b^2\right )}-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {i \int \frac {\text {csch}(x) \left (6 i b \left (a^4-3 a^2 b^2-4 b^4\right )+6 i a b^2 \left (a^2+2 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)} \, dx}{6 a^4 \left (a^2+b^2\right )} \\ & = \frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{3 a^4 \left (a^2+b^2\right )}+\frac {b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 \left (a^2+b^2\right )}-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\left (b \left (a^2-4 b^2\right )\right ) \int \text {csch}(x) \, dx}{a^5}+\frac {\left (b^4 \left (5 a^2+4 b^2\right )\right ) \int \frac {1}{a+b \sinh (x)} \, dx}{a^5 \left (a^2+b^2\right )} \\ & = -\frac {b \left (a^2-4 b^2\right ) \text {arctanh}(\cosh (x))}{a^5}+\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{3 a^4 \left (a^2+b^2\right )}+\frac {b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 \left (a^2+b^2\right )}-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\left (2 b^4 \left (5 a^2+4 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^5 \left (a^2+b^2\right )} \\ & = -\frac {b \left (a^2-4 b^2\right ) \text {arctanh}(\cosh (x))}{a^5}+\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{3 a^4 \left (a^2+b^2\right )}+\frac {b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 \left (a^2+b^2\right )}-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {\left (4 b^4 \left (5 a^2+4 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a^5 \left (a^2+b^2\right )} \\ & = -\frac {b \left (a^2-4 b^2\right ) \text {arctanh}(\cosh (x))}{a^5}-\frac {2 b^4 \left (5 a^2+4 b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^5 \left (a^2+b^2\right )^{3/2}}+\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{3 a^4 \left (a^2+b^2\right )}+\frac {b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 \left (a^2+b^2\right )}-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))} \\ \end{align*}
Time = 1.11 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.17 \[ \int \frac {\text {csch}^4(x)}{(a+b \sinh (x))^2} \, dx=\frac {-\frac {48 b^4 \left (5 a^2+4 b^2\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+4 a \left (2 a^2-9 b^2\right ) \coth \left (\frac {x}{2}\right )+6 a^2 b \text {csch}^2\left (\frac {x}{2}\right )-24 b \left (a^2-4 b^2\right ) \log \left (\cosh \left (\frac {x}{2}\right )\right )+24 b \left (a^2-4 b^2\right ) \log \left (\sinh \left (\frac {x}{2}\right )\right )+6 a^2 b \text {sech}^2\left (\frac {x}{2}\right )+8 a^3 \text {csch}^3(x) \sinh ^4\left (\frac {x}{2}\right )-\frac {1}{2} a^3 \text {csch}^4\left (\frac {x}{2}\right ) \sinh (x)-\frac {24 a b^5 \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+4 a \left (2 a^2-9 b^2\right ) \tanh \left (\frac {x}{2}\right )}{24 a^5} \]
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Time = 1.02 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.11
method | result | size |
default | \(-\frac {\frac {\tanh \left (\frac {x}{2}\right )^{3} a^{2}}{3}+2 a b \tanh \left (\frac {x}{2}\right )^{2}-3 a^{2} \tanh \left (\frac {x}{2}\right )+12 b^{2} \tanh \left (\frac {x}{2}\right )}{8 a^{4}}-\frac {2 b^{4} \left (\frac {-\frac {b^{2} \tanh \left (\frac {x}{2}\right )}{a^{2}+b^{2}}-\frac {a b}{a^{2}+b^{2}}}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {\left (5 a^{2}+4 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{a^{5}}-\frac {1}{24 a^{2} \tanh \left (\frac {x}{2}\right )^{3}}-\frac {-3 a^{2}+12 b^{2}}{8 a^{4} \tanh \left (\frac {x}{2}\right )}+\frac {b}{4 a^{3} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {b \left (a^{2}-4 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{5}}\) | \(219\) |
risch | \(-\frac {2 \left (-3 a^{3} b^{2} {\mathrm e}^{7 x}-6 a \,b^{4} {\mathrm e}^{7 x}-6 a^{4} b \,{\mathrm e}^{6 x}+3 a^{2} b^{3} {\mathrm e}^{6 x}+12 b^{5} {\mathrm e}^{6 x}+21 a^{3} b^{2} {\mathrm e}^{5 x}+30 a \,b^{4} {\mathrm e}^{5 x}+6 a^{4} b \,{\mathrm e}^{4 x}-21 a^{2} b^{3} {\mathrm e}^{4 x}-36 b^{5} {\mathrm e}^{4 x}+12 a^{5} {\mathrm e}^{3 x}-21 a^{3} b^{2} {\mathrm e}^{3 x}-42 a \,b^{4} {\mathrm e}^{3 x}-2 a^{4} b \,{\mathrm e}^{2 x}+25 a^{2} b^{3} {\mathrm e}^{2 x}+36 b^{5} {\mathrm e}^{2 x}-4 a^{5} {\mathrm e}^{x}+11 a^{3} b^{2} {\mathrm e}^{x}+18 b^{4} {\mathrm e}^{x} a +2 a^{4} b -7 a^{2} b^{3}-12 b^{5}\right )}{3 a^{4} \left ({\mathrm e}^{2 x}-1\right )^{3} \left (a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}+\frac {b \ln \left ({\mathrm e}^{x}-1\right )}{a^{3}}-\frac {4 b^{3} \ln \left ({\mathrm e}^{x}-1\right )}{a^{5}}-\frac {b \ln \left ({\mathrm e}^{x}+1\right )}{a^{3}}+\frac {4 b^{3} \ln \left ({\mathrm e}^{x}+1\right )}{a^{5}}+\frac {5 b^{4} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{3}}+\frac {4 b^{6} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{5}}-\frac {5 b^{4} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{3}}-\frac {4 b^{6} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{5}}\) | \(549\) |
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Leaf count of result is larger than twice the leaf count of optimal. 6430 vs. \(2 (190) = 380\).
Time = 0.54 (sec) , antiderivative size = 6430, normalized size of antiderivative = 32.47 \[ \int \frac {\text {csch}^4(x)}{(a+b \sinh (x))^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {\text {csch}^4(x)}{(a+b \sinh (x))^2} \, dx=\int \frac {\operatorname {csch}^{4}{\left (x \right )}}{\left (a + b \sinh {\left (x \right )}\right )^{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (190) = 380\).
Time = 0.28 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.41 \[ \int \frac {\text {csch}^4(x)}{(a+b \sinh (x))^2} \, dx=\frac {{\left (5 \, a^{2} b^{4} + 4 \, b^{6}\right )} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{7} + a^{5} b^{2}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (2 \, a^{4} b - 7 \, a^{2} b^{3} - 12 \, b^{5} + {\left (4 \, a^{5} - 11 \, a^{3} b^{2} - 18 \, a b^{4}\right )} e^{\left (-x\right )} - {\left (2 \, a^{4} b - 25 \, a^{2} b^{3} - 36 \, b^{5}\right )} e^{\left (-2 \, x\right )} - 3 \, {\left (4 \, a^{5} - 7 \, a^{3} b^{2} - 14 \, a b^{4}\right )} e^{\left (-3 \, x\right )} + 3 \, {\left (2 \, a^{4} b - 7 \, a^{2} b^{3} - 12 \, b^{5}\right )} e^{\left (-4 \, x\right )} - 3 \, {\left (7 \, a^{3} b^{2} + 10 \, a b^{4}\right )} e^{\left (-5 \, x\right )} - 3 \, {\left (2 \, a^{4} b - a^{2} b^{3} - 4 \, b^{5}\right )} e^{\left (-6 \, x\right )} + 3 \, {\left (a^{3} b^{2} + 2 \, a b^{4}\right )} e^{\left (-7 \, x\right )}\right )}}{3 \, {\left (a^{6} b + a^{4} b^{3} + 2 \, {\left (a^{7} + a^{5} b^{2}\right )} e^{\left (-x\right )} - 4 \, {\left (a^{6} b + a^{4} b^{3}\right )} e^{\left (-2 \, x\right )} - 6 \, {\left (a^{7} + a^{5} b^{2}\right )} e^{\left (-3 \, x\right )} + 6 \, {\left (a^{6} b + a^{4} b^{3}\right )} e^{\left (-4 \, x\right )} + 6 \, {\left (a^{7} + a^{5} b^{2}\right )} e^{\left (-5 \, x\right )} - 4 \, {\left (a^{6} b + a^{4} b^{3}\right )} e^{\left (-6 \, x\right )} - 2 \, {\left (a^{7} + a^{5} b^{2}\right )} e^{\left (-7 \, x\right )} + {\left (a^{6} b + a^{4} b^{3}\right )} e^{\left (-8 \, x\right )}\right )}} - \frac {{\left (a^{2} b - 4 \, b^{3}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{a^{5}} + \frac {{\left (a^{2} b - 4 \, b^{3}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{a^{5}} \]
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Time = 0.30 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.19 \[ \int \frac {\text {csch}^4(x)}{(a+b \sinh (x))^2} \, dx=\frac {{\left (5 \, a^{2} b^{4} + 4 \, b^{6}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{7} + a^{5} b^{2}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (a b^{4} e^{x} - b^{5}\right )}}{{\left (a^{6} + a^{4} b^{2}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} - \frac {{\left (a^{2} b - 4 \, b^{3}\right )} \log \left (e^{x} + 1\right )}{a^{5}} + \frac {{\left (a^{2} b - 4 \, b^{3}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{a^{5}} + \frac {2 \, {\left (3 \, a b e^{\left (5 \, x\right )} - 9 \, b^{2} e^{\left (4 \, x\right )} - 6 \, a^{2} e^{\left (2 \, x\right )} + 18 \, b^{2} e^{\left (2 \, x\right )} - 3 \, a b e^{x} + 2 \, a^{2} - 9 \, b^{2}\right )}}{3 \, a^{4} {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \]
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Time = 4.18 (sec) , antiderivative size = 975, normalized size of antiderivative = 4.92 \[ \int \frac {\text {csch}^4(x)}{(a+b \sinh (x))^2} \, dx=\frac {\ln \left ({\mathrm {e}}^x-1\right )\,\left (a^2\,b-4\,b^3\right )}{a^5}-\frac {8}{3\,a^2\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {\frac {4}{a^2}-\frac {4\,b\,{\mathrm {e}}^x}{a^3}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {\frac {6\,b^2}{a^4}-\frac {2\,b\,{\mathrm {e}}^x}{a^3}}{{\mathrm {e}}^{2\,x}-1}-\frac {\frac {2\,b^8}{a^4\,\left (a^2\,b^3+b^5\right )}-\frac {2\,b^7\,{\mathrm {e}}^x}{a^3\,\left (a^2\,b^3+b^5\right )}}{2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}}-\frac {\ln \left ({\mathrm {e}}^x+1\right )\,\left (a^2\,b-4\,b^3\right )}{a^5}-\frac {b^4\,\ln \left (\frac {32\,b\,\left (-5\,a^4+16\,a^2\,b^2+16\,b^4\right )\,\left (-4\,{\mathrm {e}}^x\,a^5+2\,a^4\,b+11\,{\mathrm {e}}^x\,a^3\,b^2-6\,a^2\,b^3+14\,{\mathrm {e}}^x\,a\,b^4-8\,b^5\right )}{a^{12}\,{\left (a^2+b^2\right )}^2}-\frac {32\,b\,\left (5\,a^2+4\,b^2\right )\,\left (5\,a^5\,b^9-32\,b^{11}\,\sqrt {{\left (a^2+b^2\right )}^3}-2\,a^{13}\,b+20\,a^7\,b^7+24\,a^9\,b^5+7\,a^{11}\,b^3+4\,a^{14}\,{\mathrm {e}}^x-80\,a^2\,b^9\,\sqrt {{\left (a^2+b^2\right )}^3}-50\,a^4\,b^7\,\sqrt {{\left (a^2+b^2\right )}^3}-15\,a^6\,b^8\,{\mathrm {e}}^x-50\,a^8\,b^6\,{\mathrm {e}}^x-52\,a^{10}\,b^4\,{\mathrm {e}}^x-13\,a^{12}\,b^2\,{\mathrm {e}}^x+127\,a^3\,b^8\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+79\,a^5\,b^6\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+5\,a\,b^4\,{\mathrm {e}}^x\,{\left ({\left (a^2+b^2\right )}^3\right )}^{3/2}+51\,a\,b^{10}\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}\right )}{a^{12}\,\sqrt {{\left (a^2+b^2\right )}^3}\,{\left (a^2+b^2\right )}^4}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}\,\left (5\,a^2+4\,b^2\right )}{a^{11}+3\,a^9\,b^2+3\,a^7\,b^4+a^5\,b^6}+\frac {b^4\,\ln \left (\frac {32\,b\,\left (-5\,a^4+16\,a^2\,b^2+16\,b^4\right )\,\left (-4\,{\mathrm {e}}^x\,a^5+2\,a^4\,b+11\,{\mathrm {e}}^x\,a^3\,b^2-6\,a^2\,b^3+14\,{\mathrm {e}}^x\,a\,b^4-8\,b^5\right )}{a^{12}\,{\left (a^2+b^2\right )}^2}-\frac {32\,b\,\left (5\,a^2+4\,b^2\right )\,\left (2\,a^{13}\,b-32\,b^{11}\,\sqrt {{\left (a^2+b^2\right )}^3}-5\,a^5\,b^9-20\,a^7\,b^7-24\,a^9\,b^5-7\,a^{11}\,b^3-4\,a^{14}\,{\mathrm {e}}^x-80\,a^2\,b^9\,\sqrt {{\left (a^2+b^2\right )}^3}-50\,a^4\,b^7\,\sqrt {{\left (a^2+b^2\right )}^3}+15\,a^6\,b^8\,{\mathrm {e}}^x+50\,a^8\,b^6\,{\mathrm {e}}^x+52\,a^{10}\,b^4\,{\mathrm {e}}^x+13\,a^{12}\,b^2\,{\mathrm {e}}^x+127\,a^3\,b^8\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+79\,a^5\,b^6\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+5\,a\,b^4\,{\mathrm {e}}^x\,{\left ({\left (a^2+b^2\right )}^3\right )}^{3/2}+51\,a\,b^{10}\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}\right )}{a^{12}\,\sqrt {{\left (a^2+b^2\right )}^3}\,{\left (a^2+b^2\right )}^4}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}\,\left (5\,a^2+4\,b^2\right )}{a^{11}+3\,a^9\,b^2+3\,a^7\,b^4+a^5\,b^6} \]
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