Integrand size = 13, antiderivative size = 146 \[ \int \frac {\text {sech}^6(x)}{a+b \sinh (x)} \, dx=-\frac {2 b^6 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}+\frac {\text {sech}^5(x) (b+a \sinh (x))}{5 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (5 b^3+a \left (4 a^2+9 b^2\right ) \sinh (x)\right )}{15 \left (a^2+b^2\right )^2}+\frac {\text {sech}(x) \left (15 b^5+a \left (8 a^4+26 a^2 b^2+33 b^4\right ) \sinh (x)\right )}{15 \left (a^2+b^2\right )^3} \]
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Time = 0.28 (sec) , antiderivative size = 146, 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 = {2775, 2945, 12, 2739, 632, 212} \[ \int \frac {\text {sech}^6(x)}{a+b \sinh (x)} \, dx=-\frac {2 b^6 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}+\frac {\text {sech}^5(x) (a \sinh (x)+b)}{5 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (a \left (4 a^2+9 b^2\right ) \sinh (x)+5 b^3\right )}{15 \left (a^2+b^2\right )^2}+\frac {\text {sech}(x) \left (a \left (8 a^4+26 a^2 b^2+33 b^4\right ) \sinh (x)+15 b^5\right )}{15 \left (a^2+b^2\right )^3} \]
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Rule 12
Rule 212
Rule 632
Rule 2739
Rule 2775
Rule 2945
Rubi steps \begin{align*} \text {integral}& = \frac {\text {sech}^5(x) (b+a \sinh (x))}{5 \left (a^2+b^2\right )}-\frac {\int \frac {\text {sech}^4(x) \left (-4 a^2-5 b^2-4 a b \sinh (x)\right )}{a+b \sinh (x)} \, dx}{5 \left (a^2+b^2\right )} \\ & = \frac {\text {sech}^5(x) (b+a \sinh (x))}{5 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (5 b^3+a \left (4 a^2+9 b^2\right ) \sinh (x)\right )}{15 \left (a^2+b^2\right )^2}+\frac {\int \frac {\text {sech}^2(x) \left (8 a^4+18 a^2 b^2+15 b^4+2 a b \left (4 a^2+9 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)} \, dx}{15 \left (a^2+b^2\right )^2} \\ & = \frac {\text {sech}^5(x) (b+a \sinh (x))}{5 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (5 b^3+a \left (4 a^2+9 b^2\right ) \sinh (x)\right )}{15 \left (a^2+b^2\right )^2}+\frac {\text {sech}(x) \left (15 b^5+a \left (8 a^4+26 a^2 b^2+33 b^4\right ) \sinh (x)\right )}{15 \left (a^2+b^2\right )^3}-\frac {\int -\frac {15 b^6}{a+b \sinh (x)} \, dx}{15 \left (a^2+b^2\right )^3} \\ & = \frac {\text {sech}^5(x) (b+a \sinh (x))}{5 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (5 b^3+a \left (4 a^2+9 b^2\right ) \sinh (x)\right )}{15 \left (a^2+b^2\right )^2}+\frac {\text {sech}(x) \left (15 b^5+a \left (8 a^4+26 a^2 b^2+33 b^4\right ) \sinh (x)\right )}{15 \left (a^2+b^2\right )^3}+\frac {b^6 \int \frac {1}{a+b \sinh (x)} \, dx}{\left (a^2+b^2\right )^3} \\ & = \frac {\text {sech}^5(x) (b+a \sinh (x))}{5 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (5 b^3+a \left (4 a^2+9 b^2\right ) \sinh (x)\right )}{15 \left (a^2+b^2\right )^2}+\frac {\text {sech}(x) \left (15 b^5+a \left (8 a^4+26 a^2 b^2+33 b^4\right ) \sinh (x)\right )}{15 \left (a^2+b^2\right )^3}+\frac {\left (2 b^6\right ) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{\left (a^2+b^2\right )^3} \\ & = \frac {\text {sech}^5(x) (b+a \sinh (x))}{5 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (5 b^3+a \left (4 a^2+9 b^2\right ) \sinh (x)\right )}{15 \left (a^2+b^2\right )^2}+\frac {\text {sech}(x) \left (15 b^5+a \left (8 a^4+26 a^2 b^2+33 b^4\right ) \sinh (x)\right )}{15 \left (a^2+b^2\right )^3}-\frac {\left (4 b^6\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 )}{\left (a^2+b^2\right )^3} \\ & = -\frac {2 b^6 \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}+\frac {\text {sech}^5(x) (b+a \sinh (x))}{5 \left (a^2+b^2\right )}+\frac {\text {sech}^3(x) \left (5 b^3+a \left (4 a^2+9 b^2\right ) \sinh (x)\right )}{15 \left (a^2+b^2\right )^2}+\frac {\text {sech}(x) \left (15 b^5+a \left (8 a^4+26 a^2 b^2+33 b^4\right ) \sinh (x)\right )}{15 \left (a^2+b^2\right )^3} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}^6(x)}{a+b \sinh (x)} \, dx=\frac {\frac {30 b^6 \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+15 b^5 \text {sech}(x)+3 \left (a^2+b^2\right )^2 \text {sech}^5(x) (b+a \sinh (x))+\left (a^2+b^2\right ) \text {sech}^3(x) \left (5 b^3+a \left (4 a^2+9 b^2\right ) \sinh (x)\right )+a \left (8 a^4+26 a^2 b^2+33 b^4\right ) \tanh (x)}{15 \left (a^2+b^2\right )^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(349\) vs. \(2(134)=268\).
Time = 174.74 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.40
method | result | size |
default | \(\frac {2 b^{6} \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \sqrt {a^{2}+b^{2}}}-\frac {2 \left (\left (-a^{5}-3 a^{3} b^{2}-3 a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{9}+\left (-a^{4} b -3 a^{2} b^{3}-3 b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{8}+\left (-\frac {4}{3} a^{5}-\frac {16}{3} a^{3} b^{2}-8 a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{7}+\left (-2 a^{2} b^{3}-6 b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{6}+\left (-\frac {58}{15} a^{5}-\frac {166}{15} a^{3} b^{2}-\frac {66}{5} a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{5}+\left (-2 a^{4} b -\frac {16}{3} a^{2} b^{3}-\frac {28}{3} b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{4}+\left (-\frac {4}{3} a^{5}-\frac {16}{3} a^{3} b^{2}-8 a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{3}+\left (-\frac {2}{3} a^{2} b^{3}-\frac {14}{3} b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{2}+\left (-a^{5}-3 a^{3} b^{2}-3 a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )-\frac {a^{4} b}{5}-\frac {11 a^{2} b^{3}}{15}-\frac {23 b^{5}}{15}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{5}}\) | \(350\) |
risch | \(-\frac {2 \left (-15 b^{5} {\mathrm e}^{9 x}+15 a \,b^{4} {\mathrm e}^{8 x}-20 a^{2} b^{3} {\mathrm e}^{7 x}-80 b^{5} {\mathrm e}^{7 x}+30 a^{3} b^{2} {\mathrm e}^{6 x}+90 a \,b^{4} {\mathrm e}^{6 x}-48 a^{4} b \,{\mathrm e}^{5 x}-136 a^{2} b^{3} {\mathrm e}^{5 x}-178 \,{\mathrm e}^{5 x} b^{5}+80 a^{5} {\mathrm e}^{4 x}+230 a^{3} b^{2} {\mathrm e}^{4 x}+240 a \,{\mathrm e}^{4 x} b^{4}-20 \,{\mathrm e}^{3 x} a^{2} b^{3}-80 \,{\mathrm e}^{3 x} b^{5}+40 a^{5} {\mathrm e}^{2 x}+130 \,{\mathrm e}^{2 x} a^{3} b^{2}+150 \,{\mathrm e}^{2 x} a \,b^{4}-15 \,{\mathrm e}^{x} b^{5}+8 a^{5}+26 a^{3} b^{2}+33 a \,b^{4}\right )}{15 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (1+{\mathrm e}^{2 x}\right )^{5}}+\frac {b^{6} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a -a^{8}-4 a^{6} b^{2}-6 b^{4} a^{4}-4 a^{2} b^{6}-b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}-\frac {b^{6} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a +a^{8}+4 a^{6} b^{2}+6 b^{4} a^{4}+4 a^{2} b^{6}+b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\) | \(389\) |
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Leaf count of result is larger than twice the leaf count of optimal. 3175 vs. \(2 (136) = 272\).
Time = 0.31 (sec) , antiderivative size = 3175, normalized size of antiderivative = 21.75 \[ \int \frac {\text {sech}^6(x)}{a+b \sinh (x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\text {sech}^6(x)}{a+b \sinh (x)} \, dx=\int \frac {\operatorname {sech}^{6}{\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (136) = 272\).
Time = 0.30 (sec) , antiderivative size = 438, normalized size of antiderivative = 3.00 \[ \int \frac {\text {sech}^6(x)}{a+b \sinh (x)} \, dx=\frac {b^{6} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (15 \, b^{5} e^{\left (-x\right )} + 15 \, a b^{4} e^{\left (-8 \, x\right )} + 15 \, b^{5} e^{\left (-9 \, x\right )} + 8 \, a^{5} + 26 \, a^{3} b^{2} + 33 \, a b^{4} + 10 \, {\left (4 \, a^{5} + 13 \, a^{3} b^{2} + 15 \, a b^{4}\right )} e^{\left (-2 \, x\right )} + 20 \, {\left (a^{2} b^{3} + 4 \, b^{5}\right )} e^{\left (-3 \, x\right )} + 10 \, {\left (8 \, a^{5} + 23 \, a^{3} b^{2} + 24 \, a b^{4}\right )} e^{\left (-4 \, x\right )} + 2 \, {\left (24 \, a^{4} b + 68 \, a^{2} b^{3} + 89 \, b^{5}\right )} e^{\left (-5 \, x\right )} + 30 \, {\left (a^{3} b^{2} + 3 \, a b^{4}\right )} e^{\left (-6 \, x\right )} + 20 \, {\left (a^{2} b^{3} + 4 \, b^{5}\right )} e^{\left (-7 \, x\right )}\right )}}{15 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6} + 5 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} e^{\left (-2 \, x\right )} + 10 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} e^{\left (-4 \, x\right )} + 10 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} e^{\left (-6 \, x\right )} + 5 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} e^{\left (-8 \, x\right )} + {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} e^{\left (-10 \, x\right )}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (136) = 272\).
Time = 0.28 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.21 \[ \int \frac {\text {sech}^6(x)}{a+b \sinh (x)} \, dx=\frac {b^{6} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (15 \, b^{5} e^{\left (9 \, x\right )} - 15 \, a b^{4} e^{\left (8 \, x\right )} + 20 \, a^{2} b^{3} e^{\left (7 \, x\right )} + 80 \, b^{5} e^{\left (7 \, x\right )} - 30 \, a^{3} b^{2} e^{\left (6 \, x\right )} - 90 \, a b^{4} e^{\left (6 \, x\right )} + 48 \, a^{4} b e^{\left (5 \, x\right )} + 136 \, a^{2} b^{3} e^{\left (5 \, x\right )} + 178 \, b^{5} e^{\left (5 \, x\right )} - 80 \, a^{5} e^{\left (4 \, x\right )} - 230 \, a^{3} b^{2} e^{\left (4 \, x\right )} - 240 \, a b^{4} e^{\left (4 \, x\right )} + 20 \, a^{2} b^{3} e^{\left (3 \, x\right )} + 80 \, b^{5} e^{\left (3 \, x\right )} - 40 \, a^{5} e^{\left (2 \, x\right )} - 130 \, a^{3} b^{2} e^{\left (2 \, x\right )} - 150 \, a b^{4} e^{\left (2 \, x\right )} + 15 \, b^{5} e^{x} - 8 \, a^{5} - 26 \, a^{3} b^{2} - 33 \, a b^{4}\right )}}{15 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (e^{\left (2 \, x\right )} + 1\right )}^{5}} \]
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Time = 3.02 (sec) , antiderivative size = 1010, normalized size of antiderivative = 6.92 \[ \int \frac {\text {sech}^6(x)}{a+b \sinh (x)} \, dx=\frac {\frac {2\,b^5\,{\mathrm {e}}^x}{{\left (a^2+b^2\right )}^3}-\frac {2\,a\,b^4}{{\left (a^2+b^2\right )}^3}}{{\mathrm {e}}^{2\,x}+1}-\frac {\frac {8\,\left (4\,a^3+3\,a\,b^2\right )}{3\,{\left (a^2+b^2\right )}^2}-\frac {8\,{\mathrm {e}}^x\,\left (12\,a^2\,b+7\,b^3\right )}{15\,{\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\,\left (a^3\,b^2+a\,b^4\right )}{{\left (a^2+b^2\right )}^3}-\frac {8\,{\mathrm {e}}^x\,\left (a^2\,b^3+b^5\right )}{3\,{\left (a^2+b^2\right )}^3}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}-\frac {\frac {32\,a}{5\,\left (a^2+b^2\right )}-\frac {32\,b\,{\mathrm {e}}^x}{5\,\left (a^2+b^2\right )}}{5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}+5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}+1}+\frac {\frac {16\,\left (a^3+a\,b^2\right )}{{\left (a^2+b^2\right )}^2}-\frac {64\,{\mathrm {e}}^x\,\left (a^2\,b+b^3\right )}{5\,{\left (a^2+b^2\right )}^2}}{4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {2\,\mathrm {atan}\left (\left ({\mathrm {e}}^x\,\left (\frac {2\,b^4}{\sqrt {b^{12}}\,{\left (a^2+b^2\right )}^3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {2\,a\,\left (a^7\,\sqrt {b^{12}}+3\,a^3\,b^4\,\sqrt {b^{12}}+3\,a^5\,b^2\,\sqrt {b^{12}}+a\,b^6\,\sqrt {b^{12}}\right )}{b^8\,\sqrt {-{\left (a^2+b^2\right )}^7}\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )\,\sqrt {-a^{14}-7\,a^{12}\,b^2-21\,a^{10}\,b^4-35\,a^8\,b^6-35\,a^6\,b^8-21\,a^4\,b^{10}-7\,a^2\,b^{12}-b^{14}}}\right )-\frac {2\,a\,\left (b^7\,\sqrt {b^{12}}+3\,a^2\,b^5\,\sqrt {b^{12}}+3\,a^4\,b^3\,\sqrt {b^{12}}+a^6\,b\,\sqrt {b^{12}}\right )}{b^8\,\sqrt {-{\left (a^2+b^2\right )}^7}\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )\,\sqrt {-a^{14}-7\,a^{12}\,b^2-21\,a^{10}\,b^4-35\,a^8\,b^6-35\,a^6\,b^8-21\,a^4\,b^{10}-7\,a^2\,b^{12}-b^{14}}}\right )\,\left (\frac {b^7\,\sqrt {-a^{14}-7\,a^{12}\,b^2-21\,a^{10}\,b^4-35\,a^8\,b^6-35\,a^6\,b^8-21\,a^4\,b^{10}-7\,a^2\,b^{12}-b^{14}}}{2}+\frac {3\,a^2\,b^5\,\sqrt {-a^{14}-7\,a^{12}\,b^2-21\,a^{10}\,b^4-35\,a^8\,b^6-35\,a^6\,b^8-21\,a^4\,b^{10}-7\,a^2\,b^{12}-b^{14}}}{2}+\frac {3\,a^4\,b^3\,\sqrt {-a^{14}-7\,a^{12}\,b^2-21\,a^{10}\,b^4-35\,a^8\,b^6-35\,a^6\,b^8-21\,a^4\,b^{10}-7\,a^2\,b^{12}-b^{14}}}{2}+\frac {a^6\,b\,\sqrt {-a^{14}-7\,a^{12}\,b^2-21\,a^{10}\,b^4-35\,a^8\,b^6-35\,a^6\,b^8-21\,a^4\,b^{10}-7\,a^2\,b^{12}-b^{14}}}{2}\right )\right )\,\sqrt {b^{12}}}{\sqrt {-a^{14}-7\,a^{12}\,b^2-21\,a^{10}\,b^4-35\,a^8\,b^6-35\,a^6\,b^8-21\,a^4\,b^{10}-7\,a^2\,b^{12}-b^{14}}} \]
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