Integrand size = 11, antiderivative size = 112 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^5} \, dx=\frac {3 \arctan \left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{8 \left (a^2-b^2\right )^{5/2}}+\frac {b \cosh (x)+a \sinh (x)}{4 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^4}+\frac {3 (b \cosh (x)+a \sinh (x))}{8 \left (a^2-b^2\right )^2 (a \cosh (x)+b \sinh (x))^2} \]
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Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3155, 3153, 212} \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^5} \, dx=\frac {3 \arctan \left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{8 \left (a^2-b^2\right )^{5/2}}+\frac {3 (a \sinh (x)+b \cosh (x))}{8 \left (a^2-b^2\right )^2 (a \cosh (x)+b \sinh (x))^2}+\frac {a \sinh (x)+b \cosh (x)}{4 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^4} \]
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Rule 212
Rule 3153
Rule 3155
Rubi steps \begin{align*} \text {integral}& = \frac {b \cosh (x)+a \sinh (x)}{4 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^4}+\frac {3 \int \frac {1}{(a \cosh (x)+b \sinh (x))^3} \, dx}{4 \left (a^2-b^2\right )} \\ & = \frac {b \cosh (x)+a \sinh (x)}{4 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^4}+\frac {3 (b \cosh (x)+a \sinh (x))}{8 \left (a^2-b^2\right )^2 (a \cosh (x)+b \sinh (x))^2}+\frac {3 \int \frac {1}{a \cosh (x)+b \sinh (x)} \, dx}{8 \left (a^2-b^2\right )^2} \\ & = \frac {b \cosh (x)+a \sinh (x)}{4 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^4}+\frac {3 (b \cosh (x)+a \sinh (x))}{8 \left (a^2-b^2\right )^2 (a \cosh (x)+b \sinh (x))^2}+\frac {(3 i) \text {Subst}\left (\int \frac {1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{8 \left (a^2-b^2\right )^2} \\ & = \frac {3 \arctan \left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{8 \left (a^2-b^2\right )^{5/2}}+\frac {b \cosh (x)+a \sinh (x)}{4 \left (a^2-b^2\right ) (a \cosh (x)+b \sinh (x))^4}+\frac {3 (b \cosh (x)+a \sinh (x))}{8 \left (a^2-b^2\right )^2 (a \cosh (x)+b \sinh (x))^2} \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.31 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^5} \, dx=\frac {1}{8} \left (\frac {6 \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )}{(a-b)^{5/2} (a+b)^{5/2}}+\frac {b \left (2 (a-b) (a+b)+3 (a \cosh (x)+b \sinh (x))^2\right )}{a (a-b)^2 (a+b)^2 (a \cosh (x)+b \sinh (x))^3}+\frac {\sinh (x) \left (2+\frac {3 (a \cosh (x)+b \sinh (x))^2}{(a-b) (a+b)}\right )}{a (a \cosh (x)+b \sinh (x))^4}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(461\) vs. \(2(102)=204\).
Time = 0.29 (sec) , antiderivative size = 462, normalized size of antiderivative = 4.12
\[\frac {-\frac {\left (5 a^{4}-16 a^{2} b^{2}+8 b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{7}}{4 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {3 b \left (a^{4}-16 a^{2} b^{2}+8 b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{6}}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a^{2}}+\frac {\left (3 a^{6}+36 a^{4} b^{2}+56 a^{2} b^{4}-32 b^{6}\right ) \tanh \left (\frac {x}{2}\right )^{5}}{4 a^{3} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {b \left (15 a^{6}+114 a^{4} b^{2}-8 a^{2} b^{4}-16 b^{6}\right ) \tanh \left (\frac {x}{2}\right )^{4}}{4 a^{4} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (3 a^{6}-84 a^{4} b^{2}-56 a^{2} b^{4}+32 b^{6}\right ) \tanh \left (\frac {x}{2}\right )^{3}}{4 a^{3} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {b \left (23 a^{4}+64 a^{2} b^{2}-24 b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{2}}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a^{2}}+\frac {\left (5 a^{4}+24 a^{2} b^{2}-8 b^{4}\right ) \tanh \left (\frac {x}{2}\right )}{4 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {2 \left (5 a^{2}-2 b^{2}\right ) b}{8 a^{4}-16 a^{2} b^{2}+8 b^{4}}}{\left (\tanh \left (\frac {x}{2}\right )^{2} a +2 b \tanh \left (\frac {x}{2}\right )+a \right )^{4}}+\frac {3 \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\]
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Leaf count of result is larger than twice the leaf count of optimal. 3408 vs. \(2 (102) = 204\).
Time = 0.35 (sec) , antiderivative size = 6874, normalized size of antiderivative = 61.38 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^5} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^5} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^5} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (102) = 204\).
Time = 0.27 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.11 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^5} \, dx=\frac {3 \, \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} - b^{2}}} + \frac {3 \, a^{3} e^{\left (7 \, x\right )} + 9 \, a^{2} b e^{\left (7 \, x\right )} + 9 \, a b^{2} e^{\left (7 \, x\right )} + 3 \, b^{3} e^{\left (7 \, x\right )} + 11 \, a^{3} e^{\left (5 \, x\right )} + 11 \, a^{2} b e^{\left (5 \, x\right )} - 11 \, a b^{2} e^{\left (5 \, x\right )} - 11 \, b^{3} e^{\left (5 \, x\right )} - 11 \, a^{3} e^{\left (3 \, x\right )} + 11 \, a^{2} b e^{\left (3 \, x\right )} + 11 \, a b^{2} e^{\left (3 \, x\right )} - 11 \, b^{3} e^{\left (3 \, x\right )} - 3 \, a^{3} e^{x} + 9 \, a^{2} b e^{x} - 9 \, a b^{2} e^{x} + 3 \, b^{3} e^{x}}{4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}^{4}} \]
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Time = 2.41 (sec) , antiderivative size = 354, normalized size of antiderivative = 3.16 \[ \int \frac {1}{(a \cosh (x)+b \sinh (x))^5} \, dx=\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {a^{10}-5\,a^8\,b^2+10\,a^6\,b^4-10\,a^4\,b^6+5\,a^2\,b^8-b^{10}}}{a^5-a^4\,b-2\,a^3\,b^2+2\,a^2\,b^3+a\,b^4-b^5}\right )}{4\,\sqrt {a^{10}-5\,a^8\,b^2+10\,a^6\,b^4-10\,a^4\,b^6+5\,a^2\,b^8-b^{10}}}-\frac {4\,{\mathrm {e}}^{3\,x}}{\left (a+b\right )\,\left ({\mathrm {e}}^{8\,x}\,{\left (a+b\right )}^4+{\left (a-b\right )}^4+4\,{\mathrm {e}}^{2\,x}\,\left (a+b\right )\,{\left (a-b\right )}^3+4\,{\mathrm {e}}^{6\,x}\,{\left (a+b\right )}^3\,\left (a-b\right )+6\,{\mathrm {e}}^{4\,x}\,{\left (a+b\right )}^2\,{\left (a-b\right )}^2\right )}-\frac {2\,{\mathrm {e}}^x}{{\left (a+b\right )}^2\,\left ({\mathrm {e}}^{6\,x}\,{\left (a+b\right )}^3+{\left (a-b\right )}^3+3\,{\mathrm {e}}^{2\,x}\,\left (a+b\right )\,{\left (a-b\right )}^2+3\,{\mathrm {e}}^{4\,x}\,{\left (a+b\right )}^2\,\left (a-b\right )\right )}+\frac {3\,{\mathrm {e}}^x}{4\,{\left (a+b\right )}^2\,{\left (a-b\right )}^2\,\left (a-b+{\mathrm {e}}^{2\,x}\,\left (a+b\right )\right )}+\frac {{\mathrm {e}}^x}{2\,{\left (a+b\right )}^2\,\left (a-b\right )\,\left ({\mathrm {e}}^{4\,x}\,{\left (a+b\right )}^2+{\left (a-b\right )}^2+2\,{\mathrm {e}}^{2\,x}\,\left (a+b\right )\,\left (a-b\right )\right )} \]
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