Integrand size = 13, antiderivative size = 149 \[ \int \frac {\text {sech}^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {a (3 i a+b) \log (i-\sinh (x))}{16 (a-i b)^3}+\frac {a (3 a+i b) \log (i+\sinh (x))}{16 (i a-b)^3}-\frac {a^4 b \log (b+a \sinh (x))}{\left (a^2+b^2\right )^3}-\frac {\text {sech}^4(x) (b-a \sinh (x))}{4 \left (a^2+b^2\right )}-\frac {\text {sech}^2(x) \left (4 a^2 b-a \left (3 a^2-b^2\right ) \sinh (x)\right )}{8 \left (a^2+b^2\right )^2} \]
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Time = 0.25 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3957, 2916, 12, 837, 815} \[ \int \frac {\text {sech}^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {\text {sech}^4(x) (b-a \sinh (x))}{4 \left (a^2+b^2\right )}-\frac {\text {sech}^2(x) \left (4 a^2 b-a \left (3 a^2-b^2\right ) \sinh (x)\right )}{8 \left (a^2+b^2\right )^2}-\frac {a^4 b \log (a \sinh (x)+b)}{\left (a^2+b^2\right )^3}-\frac {a (b+3 i a) \log (-\sinh (x)+i)}{16 (a-i b)^3}+\frac {a (3 a+i b) \log (\sinh (x)+i)}{16 (-b+i a)^3} \]
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Rule 12
Rule 815
Rule 837
Rule 2916
Rule 3957
Rubi steps \begin{align*} \text {integral}& = i \int \frac {\text {sech}^4(x) \tanh (x)}{i b+i a \sinh (x)} \, dx \\ & = -\left (\left (i a^5\right ) \text {Subst}\left (\int \frac {x}{a (i b+x) \left (a^2-x^2\right )^3} \, dx,x,i a \sinh (x)\right )\right ) \\ & = -\left (\left (i a^4\right ) \text {Subst}\left (\int \frac {x}{(i b+x) \left (a^2-x^2\right )^3} \, dx,x,i a \sinh (x)\right )\right ) \\ & = -\frac {\text {sech}^4(x) (b-a \sinh (x))}{4 \left (a^2+b^2\right )}-\frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {-i a^2 b+3 a^2 x}{(i b+x) \left (a^2-x^2\right )^2} \, dx,x,i a \sinh (x)\right )}{4 \left (a^2+b^2\right )} \\ & = -\frac {\text {sech}^4(x) (b-a \sinh (x))}{4 \left (a^2+b^2\right )}-\frac {\text {sech}^2(x) \left (4 a^2 b-a \left (3 a^2-b^2\right ) \sinh (x)\right )}{8 \left (a^2+b^2\right )^2}-\frac {i \text {Subst}\left (\int \frac {-i a^2 b \left (5 a^2+b^2\right )+a^2 \left (3 a^2-b^2\right ) x}{(i b+x) \left (a^2-x^2\right )} \, dx,x,i a \sinh (x)\right )}{8 \left (a^2+b^2\right )^2} \\ & = -\frac {\text {sech}^4(x) (b-a \sinh (x))}{4 \left (a^2+b^2\right )}-\frac {\text {sech}^2(x) \left (4 a^2 b-a \left (3 a^2-b^2\right ) \sinh (x)\right )}{8 \left (a^2+b^2\right )^2}-\frac {i \text {Subst}\left (\int \left (\frac {a (a-i b)^2 (3 a+i b)}{2 (a+i b) (a-x)}-\frac {8 a^4 b}{\left (a^2+b^2\right ) (b-i x)}+\frac {a (3 a-i b) (a+i b)^2}{2 (a-i b) (a+x)}\right ) \, dx,x,i a \sinh (x)\right )}{8 \left (a^2+b^2\right )^2} \\ & = -\frac {a (3 i a+b) \log (i-\sinh (x))}{16 (a-i b)^3}+\frac {a (3 a+i b) \log (i+\sinh (x))}{16 (i a-b)^3}-\frac {a^4 b \log (b+a \sinh (x))}{\left (a^2+b^2\right )^3}-\frac {\text {sech}^4(x) (b-a \sinh (x))}{4 \left (a^2+b^2\right )}-\frac {\text {sech}^2(x) \left (4 a^2 b-a \left (3 a^2-b^2\right ) \sinh (x)\right )}{8 \left (a^2+b^2\right )^2} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.09 \[ \int \frac {\text {sech}^5(x)}{a+b \text {csch}(x)} \, dx=\frac {a (i a-b)^3 (3 a-i b) \log (i-\sinh (x))-a (3 a+i b) (i a+b)^3 \log (i+\sinh (x))-16 a^4 b \log (b+a \sinh (x))-8 a^2 b \left (a^2+b^2\right ) \text {sech}^2(x)-4 b \left (a^2+b^2\right )^2 \text {sech}^4(x)+2 a \left (3 a^4+2 a^2 b^2-b^4\right ) \text {sech}(x) \tanh (x)+4 a \left (a^2+b^2\right )^2 \text {sech}^3(x) \tanh (x)}{16 \left (a^2+b^2\right )^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(315\) vs. \(2(135)=270\).
Time = 23.74 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.12
method | result | size |
default | \(\frac {\frac {2 \left (\left (-\frac {5}{8} a^{5}-\frac {3}{4} a^{3} b^{2}-\frac {1}{8} a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{7}+\left (2 a^{4} b +3 a^{2} b^{3}+b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{6}+\left (\frac {3}{8} a^{5}+\frac {5}{4} a^{3} b^{2}+\frac {7}{8} a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{5}+\left (2 a^{4} b +2 a^{2} b^{3}\right ) \tanh \left (\frac {x}{2}\right )^{4}+\left (-\frac {3}{8} a^{5}-\frac {5}{4} a^{3} b^{2}-\frac {7}{8} a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{3}+\left (2 a^{4} b +3 a^{2} b^{3}+b^{5}\right ) \tanh \left (\frac {x}{2}\right )^{2}+\left (\frac {5}{8} a^{5}+\frac {3}{4} a^{3} b^{2}+\frac {1}{8} a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )\right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{4}}+\frac {a \left (4 a^{3} b \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )+\left (3 a^{4}-6 a^{2} b^{2}-b^{4}\right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )\right )}{4}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {a^{4} b \ln \left (-\tanh \left (\frac {x}{2}\right )^{2} b +2 a \tanh \left (\frac {x}{2}\right )+b \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}\) | \(316\) |
risch | \(\frac {\left (3 a^{3} {\mathrm e}^{6 x}-a \,b^{2} {\mathrm e}^{6 x}-8 a^{2} b \,{\mathrm e}^{5 x}+11 a^{3} {\mathrm e}^{4 x}+7 a \,b^{2} {\mathrm e}^{4 x}-32 a^{2} b \,{\mathrm e}^{3 x}-16 b^{3} {\mathrm e}^{3 x}-11 a^{3} {\mathrm e}^{2 x}-7 a \,b^{2} {\mathrm e}^{2 x}-8 a^{2} b \,{\mathrm e}^{x}-3 a^{3}+a \,b^{2}\right ) {\mathrm e}^{x}}{4 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+{\mathrm e}^{2 x}\right )^{4}}-\frac {3 i a^{5} \ln \left ({\mathrm e}^{x}-i\right )}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {3 i a^{3} \ln \left ({\mathrm e}^{x}-i\right ) b^{2}}{4 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {i a \ln \left ({\mathrm e}^{x}-i\right ) b^{4}}{8 a^{6}+24 a^{4} b^{2}+24 a^{2} b^{4}+8 b^{6}}+\frac {a^{4} \ln \left ({\mathrm e}^{x}-i\right ) b}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {3 i a^{5} \ln \left ({\mathrm e}^{x}+i\right )}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {3 i a^{3} \ln \left ({\mathrm e}^{x}+i\right ) b^{2}}{4 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {i a \ln \left ({\mathrm e}^{x}+i\right ) b^{4}}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {a^{4} \ln \left ({\mathrm e}^{x}+i\right ) b}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {a^{4} b \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\) | \(484\) |
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Leaf count of result is larger than twice the leaf count of optimal. 2778 vs. \(2 (130) = 260\).
Time = 0.31 (sec) , antiderivative size = 2778, normalized size of antiderivative = 18.64 \[ \int \frac {\text {sech}^5(x)}{a+b \text {csch}(x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\text {sech}^5(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\operatorname {sech}^{5}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (130) = 260\).
Time = 0.28 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.34 \[ \int \frac {\text {sech}^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {a^{4} b \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {a^{4} b \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (3 \, a^{5} - 6 \, a^{3} b^{2} - a b^{4}\right )} \arctan \left (e^{\left (-x\right )}\right )}{4 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac {8 \, a^{2} b e^{\left (-2 \, x\right )} + 8 \, a^{2} b e^{\left (-6 \, x\right )} - {\left (3 \, a^{3} - a b^{2}\right )} e^{\left (-x\right )} - {\left (11 \, a^{3} + 7 \, a b^{2}\right )} e^{\left (-3 \, x\right )} + 16 \, {\left (2 \, a^{2} b + b^{3}\right )} e^{\left (-4 \, x\right )} + {\left (11 \, a^{3} + 7 \, a b^{2}\right )} e^{\left (-5 \, x\right )} + {\left (3 \, a^{3} - a b^{2}\right )} e^{\left (-7 \, x\right )}}{4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4} + 4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-2 \, x\right )} + 6 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-4 \, x\right )} + 4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-6 \, x\right )} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-8 \, x\right )}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (130) = 260\).
Time = 0.29 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.51 \[ \int \frac {\text {sech}^5(x)}{a+b \text {csch}(x)} \, dx=-\frac {a^{5} b \log \left ({\left | -a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}} + \frac {a^{4} b \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} {\left (3 \, a^{5} - 6 \, a^{3} b^{2} - a b^{4}\right )}}{16 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac {3 \, a^{4} b {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 3 \, a^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 2 \, a^{3} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} - a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 32 \, a^{4} b {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 8 \, a^{2} b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 20 \, a^{5} {\left (e^{\left (-x\right )} - e^{x}\right )} + 24 \, a^{3} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )} + 4 \, a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )} + 96 \, a^{4} b + 64 \, a^{2} b^{3} + 16 \, b^{5}}{4 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}^{2}} \]
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Time = 6.02 (sec) , antiderivative size = 513, normalized size of antiderivative = 3.44 \[ \int \frac {\text {sech}^5(x)}{a+b \text {csch}(x)} \, dx=\frac {\frac {8\,\left (a^2\,b+b^3\right )}{{\left (a^2+b^2\right )}^2}-\frac {6\,{\mathrm {e}}^x\,\left (a^3+a\,b^2\right )}{{\left (a^2+b^2\right )}^2}}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}-\frac {\frac {2\,\left (a^2\,b+2\,b^3\right )}{{\left (a^2+b^2\right )}^2}-\frac {{\mathrm {e}}^x\,\left (a^3+5\,a\,b^2\right )}{2\,{\left (a^2+b^2\right )}^2}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}-\frac {\frac {4\,b}{a^2+b^2}-\frac {4\,a\,{\mathrm {e}}^x}{a^2+b^2}}{4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {\frac {2\,\left (a^4\,b+a^2\,b^3\right )}{{\left (a^2+b^2\right )}^3}-\frac {{\mathrm {e}}^x\,\left (3\,a^5+2\,a^3\,b^2-a\,b^4\right )}{4\,{\left (a^2+b^2\right )}^3}}{{\mathrm {e}}^{2\,x}+1}+\frac {\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,\left (a\,b-a^2\,3{}\mathrm {i}\right )}{8\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )}+\frac {\ln \left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )\,\left (-3\,a^2+a\,b\,1{}\mathrm {i}\right )}{8\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {a^4\,b\,\ln \left (9\,a^{10}\,{\mathrm {e}}^{2\,x}-9\,a^{10}-a^2\,b^8-12\,a^4\,b^6-30\,a^6\,b^4-220\,a^8\,b^2+a^2\,b^8\,{\mathrm {e}}^{2\,x}+12\,a^4\,b^6\,{\mathrm {e}}^{2\,x}+30\,a^6\,b^4\,{\mathrm {e}}^{2\,x}+220\,a^8\,b^2\,{\mathrm {e}}^{2\,x}+2\,a\,b^9\,{\mathrm {e}}^x+18\,a^9\,b\,{\mathrm {e}}^x+24\,a^3\,b^7\,{\mathrm {e}}^x+60\,a^5\,b^5\,{\mathrm {e}}^x+440\,a^7\,b^3\,{\mathrm {e}}^x\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6} \]
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