Integrand size = 13, antiderivative size = 145 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=-\frac {a \left (8 a^4+20 a^2 b^2+15 b^4\right ) x}{8 b^6}-\frac {2 \left (a^2+b^2\right )^{5/2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^6}+\frac {\cosh ^5(x)}{5 b}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5} \]
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Time = 0.29 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2774, 2944, 2814, 2739, 632, 212} \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=-\frac {2 \left (a^2+b^2\right )^{5/2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^6}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}-\frac {a x \left (8 a^4+20 a^2 b^2+15 b^4\right )}{8 b^6}+\frac {\cosh ^5(x)}{5 b} \]
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Rule 212
Rule 632
Rule 2739
Rule 2774
Rule 2814
Rule 2944
Rubi steps \begin{align*} \text {integral}& = \frac {\cosh ^5(x)}{5 b}+\frac {i \int \frac {\cosh ^4(x) (-i b+i a \sinh (x))}{a+b \sinh (x)} \, dx}{b} \\ & = \frac {\cosh ^5(x)}{5 b}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}-\frac {i \int \frac {\cosh ^2(x) \left (i b \left (a^2+4 b^2\right )-i a \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)} \, dx}{4 b^3} \\ & = \frac {\cosh ^5(x)}{5 b}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5}+\frac {i \int \frac {-i b \left (4 a^4+9 a^2 b^2+8 b^4\right )+i a \left (8 a^4+20 a^2 b^2+15 b^4\right ) \sinh (x)}{a+b \sinh (x)} \, dx}{8 b^5} \\ & = -\frac {a \left (8 a^4+20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac {\cosh ^5(x)}{5 b}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5}+\frac {\left (a^2+b^2\right )^3 \int \frac {1}{a+b \sinh (x)} \, dx}{b^6} \\ & = -\frac {a \left (8 a^4+20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac {\cosh ^5(x)}{5 b}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5}+\frac {\left (2 \left (a^2+b^2\right )^3\right ) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b^6} \\ & = -\frac {a \left (8 a^4+20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac {\cosh ^5(x)}{5 b}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5}-\frac {\left (4 \left (a^2+b^2\right )^3\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 )}{b^6} \\ & = -\frac {a \left (8 a^4+20 a^2 b^2+15 b^4\right ) x}{8 b^6}-\frac {2 \left (a^2+b^2\right )^{5/2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^6}+\frac {\cosh ^5(x)}{5 b}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5} \\ \end{align*}
Result contains complex when optimal does not.
Time = 4.68 (sec) , antiderivative size = 463, normalized size of antiderivative = 3.19 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=\frac {\cosh (x) \left (8 \left (15 a^4+35 a^2 b^2+23 b^4\right )-15 a b \left (4 a^2+9 b^2\right ) \sinh (x)+8 b^2 \left (5 a^2+11 b^2\right ) \sinh ^2(x)-30 a b^3 \sinh ^3(x)+24 b^4 \sinh ^4(x)-\frac {30 (-1)^{3/4} \sqrt {b} \left (8 a^4-4 i a^3 b+16 a^2 b^2-7 i a b^3+8 b^4\right ) \arcsin \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {a-i b} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}{\sqrt {b}}\right )}{\sqrt {a-i b} \sqrt {1+i \sinh (x)} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}-\frac {240 \left (a^2+b^2\right )^2 \text {arctanh}\left (\frac {\sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}{\sqrt {-\frac {b (-i+\sinh (x))}{a+i b}}}\right )}{\sqrt {-\frac {b (-i+\sinh (x))}{a+i b}} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}+\frac {240 (a-i b)^{5/2} (a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a-i b} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}{\sqrt {a+i b} \sqrt {-\frac {b (-i+\sinh (x))}{a+i b}}}\right )}{\sqrt {-\frac {b (-i+\sinh (x))}{a+i b}} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}\right )}{120 b^5} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(343\) vs. \(2(132)=264\).
Time = 57.61 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.37
method | result | size |
risch | \(-\frac {a^{5} x}{b^{6}}-\frac {5 a^{3} x}{2 b^{4}}-\frac {15 a x}{8 b^{2}}+\frac {{\mathrm e}^{5 x}}{160 b}-\frac {a \,{\mathrm e}^{4 x}}{64 b^{2}}+\frac {{\mathrm e}^{3 x} a^{2}}{24 b^{3}}+\frac {7 \,{\mathrm e}^{3 x}}{96 b}-\frac {a^{3} {\mathrm e}^{2 x}}{8 b^{4}}-\frac {a \,{\mathrm e}^{2 x}}{4 b^{2}}+\frac {{\mathrm e}^{x} a^{4}}{2 b^{5}}+\frac {9 \,{\mathrm e}^{x} a^{2}}{8 b^{3}}+\frac {11 \,{\mathrm e}^{x}}{16 b}+\frac {{\mathrm e}^{-x} a^{4}}{2 b^{5}}+\frac {9 \,{\mathrm e}^{-x} a^{2}}{8 b^{3}}+\frac {11 \,{\mathrm e}^{-x}}{16 b}+\frac {a^{3} {\mathrm e}^{-2 x}}{8 b^{4}}+\frac {a \,{\mathrm e}^{-2 x}}{4 b^{2}}+\frac {{\mathrm e}^{-3 x} a^{2}}{24 b^{3}}+\frac {7 \,{\mathrm e}^{-3 x}}{96 b}+\frac {a \,{\mathrm e}^{-4 x}}{64 b^{2}}+\frac {{\mathrm e}^{-5 x}}{160 b}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} \ln \left ({\mathrm e}^{x}-\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}}-a^{5}-2 a^{3} b^{2}-a \,b^{4}}{b \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\right )}{b^{6}}-\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}}+a^{5}+2 a^{3} b^{2}+a \,b^{4}}{b \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\right )}{b^{6}}\) | \(344\) |
default | \(-\frac {2 \left (-a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{6} \sqrt {a^{2}+b^{2}}}-\frac {1}{5 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{5}}-\frac {2 b +a}{4 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {4 a^{2}+6 a b +13 b^{2}}{12 b^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {4 a^{3}+4 a^{2} b +11 a \,b^{2}+9 b^{3}}{8 b^{4} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {8 a^{4}+4 a^{3} b +20 a^{2} b^{2}+9 b^{3} a +15 b^{4}}{8 b^{5} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {a \left (8 a^{4}+20 a^{2} b^{2}+15 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8 b^{6}}+\frac {1}{5 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {2 b -a}{4 b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {-4 a^{2}+6 a b -13 b^{2}}{12 b^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {-4 a^{3}+4 a^{2} b -11 a \,b^{2}+9 b^{3}}{8 b^{4} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {-8 a^{4}+4 a^{3} b -20 a^{2} b^{2}+9 b^{3} a -15 b^{4}}{8 b^{5} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {a \left (8 a^{4}+20 a^{2} b^{2}+15 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8 b^{6}}\) | \(410\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1486 vs. \(2 (133) = 266\).
Time = 0.32 (sec) , antiderivative size = 1486, normalized size of antiderivative = 10.25 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (133) = 266\).
Time = 0.30 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.95 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=-\frac {{\left (15 \, a b^{3} e^{\left (-x\right )} - 6 \, b^{4} - 10 \, {\left (4 \, a^{2} b^{2} + 7 \, b^{4}\right )} e^{\left (-2 \, x\right )} + 120 \, {\left (a^{3} b + 2 \, a b^{3}\right )} e^{\left (-3 \, x\right )} - 60 \, {\left (8 \, a^{4} + 18 \, a^{2} b^{2} + 11 \, b^{4}\right )} e^{\left (-4 \, x\right )}\right )} e^{\left (5 \, x\right )}}{960 \, b^{5}} + \frac {15 \, a b^{3} e^{\left (-4 \, x\right )} + 6 \, b^{4} e^{\left (-5 \, x\right )} + 60 \, {\left (8 \, a^{4} + 18 \, a^{2} b^{2} + 11 \, b^{4}\right )} e^{\left (-x\right )} + 120 \, {\left (a^{3} b + 2 \, a b^{3}\right )} e^{\left (-2 \, x\right )} + 10 \, {\left (4 \, a^{2} b^{2} + 7 \, b^{4}\right )} e^{\left (-3 \, x\right )}}{960 \, b^{5}} - \frac {{\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x}{8 \, b^{6}} + \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (133) = 266\).
Time = 0.28 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.99 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=\frac {6 \, b^{4} e^{\left (5 \, x\right )} - 15 \, a b^{3} e^{\left (4 \, x\right )} + 40 \, a^{2} b^{2} e^{\left (3 \, x\right )} + 70 \, b^{4} e^{\left (3 \, x\right )} - 120 \, a^{3} b e^{\left (2 \, x\right )} - 240 \, a b^{3} e^{\left (2 \, x\right )} + 480 \, a^{4} e^{x} + 1080 \, a^{2} b^{2} e^{x} + 660 \, b^{4} e^{x}}{960 \, b^{5}} - \frac {{\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x}{8 \, b^{6}} + \frac {{\left (15 \, a b^{4} e^{x} + 6 \, b^{5} + 60 \, {\left (8 \, a^{4} b + 18 \, a^{2} b^{3} + 11 \, b^{5}\right )} e^{\left (4 \, x\right )} + 120 \, {\left (a^{3} b^{2} + 2 \, a b^{4}\right )} e^{\left (3 \, x\right )} + 10 \, {\left (4 \, a^{2} b^{3} + 7 \, b^{5}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-5 \, x\right )}}{960 \, b^{6}} + \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{6}} \]
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Time = 2.01 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.08 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=\frac {{\mathrm {e}}^{-5\,x}}{160\,b}+\frac {{\mathrm {e}}^{5\,x}}{160\,b}-\frac {\ln \left (-\frac {2\,{\mathrm {e}}^x\,{\left (a^2+b^2\right )}^3}{b^7}-\frac {2\,\left (b-a\,{\mathrm {e}}^x\right )\,{\left (a^2+b^2\right )}^{5/2}}{b^7}\right )\,{\left (a^2+b^2\right )}^{5/2}}{b^6}+\frac {\ln \left (\frac {2\,\left (b-a\,{\mathrm {e}}^x\right )\,{\left (a^2+b^2\right )}^{5/2}}{b^7}-\frac {2\,{\mathrm {e}}^x\,{\left (a^2+b^2\right )}^3}{b^7}\right )\,{\left (a^2+b^2\right )}^{5/2}}{b^6}-\frac {x\,\left (8\,a^5+20\,a^3\,b^2+15\,a\,b^4\right )}{8\,b^6}+\frac {{\mathrm {e}}^x\,\left (8\,a^4+18\,a^2\,b^2+11\,b^4\right )}{16\,b^5}+\frac {a\,{\mathrm {e}}^{-4\,x}}{64\,b^2}-\frac {a\,{\mathrm {e}}^{4\,x}}{64\,b^2}+\frac {{\mathrm {e}}^{-x}\,\left (8\,a^4+18\,a^2\,b^2+11\,b^4\right )}{16\,b^5}+\frac {{\mathrm {e}}^{-3\,x}\,\left (4\,a^2+7\,b^2\right )}{96\,b^3}+\frac {{\mathrm {e}}^{3\,x}\,\left (4\,a^2+7\,b^2\right )}{96\,b^3}+\frac {{\mathrm {e}}^{-2\,x}\,\left (a^3+2\,a\,b^2\right )}{8\,b^4}-\frac {{\mathrm {e}}^{2\,x}\,\left (a^3+2\,a\,b^2\right )}{8\,b^4} \]
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