Integrand size = 13, antiderivative size = 87 \[ \int \frac {\text {sech}^4(x)}{a+b \text {sech}(x)} \, dx=\frac {\left (2 a^2+b^2\right ) \arctan (\sinh (x))}{2 b^3}-\frac {2 a^3 \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^3 \sqrt {a+b}}-\frac {a \tanh (x)}{b^2}+\frac {\text {sech}(x) \tanh (x)}{2 b} \]
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Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3936, 4167, 4083, 3855, 3916, 2738, 211} \[ \int \frac {\text {sech}^4(x)}{a+b \text {sech}(x)} \, dx=-\frac {2 a^3 \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{b^3 \sqrt {a-b} \sqrt {a+b}}+\frac {\left (2 a^2+b^2\right ) \arctan (\sinh (x))}{2 b^3}-\frac {a \tanh (x)}{b^2}+\frac {\tanh (x) \text {sech}(x)}{2 b} \]
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Rule 211
Rule 2738
Rule 3855
Rule 3916
Rule 3936
Rule 4083
Rule 4167
Rubi steps \begin{align*} \text {integral}& = \frac {\text {sech}(x) \tanh (x)}{2 b}+\frac {\int \frac {\text {sech}(x) \left (a+b \text {sech}(x)-2 a \text {sech}^2(x)\right )}{a+b \text {sech}(x)} \, dx}{2 b} \\ & = -\frac {a \tanh (x)}{b^2}+\frac {\text {sech}(x) \tanh (x)}{2 b}+\frac {\int \frac {\text {sech}(x) \left (a b+\left (2 a^2+b^2\right ) \text {sech}(x)\right )}{a+b \text {sech}(x)} \, dx}{2 b^2} \\ & = -\frac {a \tanh (x)}{b^2}+\frac {\text {sech}(x) \tanh (x)}{2 b}-\frac {a^3 \int \frac {\text {sech}(x)}{a+b \text {sech}(x)} \, dx}{b^3}+\frac {\left (2 a^2+b^2\right ) \int \text {sech}(x) \, dx}{2 b^3} \\ & = \frac {\left (2 a^2+b^2\right ) \arctan (\sinh (x))}{2 b^3}-\frac {a \tanh (x)}{b^2}+\frac {\text {sech}(x) \tanh (x)}{2 b}-\frac {a^3 \int \frac {1}{1+\frac {a \cosh (x)}{b}} \, dx}{b^4} \\ & = \frac {\left (2 a^2+b^2\right ) \arctan (\sinh (x))}{2 b^3}-\frac {a \tanh (x)}{b^2}+\frac {\text {sech}(x) \tanh (x)}{2 b}-\frac {\left (2 a^3\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}-\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b^4} \\ & = \frac {\left (2 a^2+b^2\right ) \arctan (\sinh (x))}{2 b^3}-\frac {2 a^3 \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^3 \sqrt {a+b}}-\frac {a \tanh (x)}{b^2}+\frac {\text {sech}(x) \tanh (x)}{2 b} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94 \[ \int \frac {\text {sech}^4(x)}{a+b \text {sech}(x)} \, dx=\frac {2 \left (2 a^2+b^2\right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )+\frac {4 a^3 \arctan \left (\frac {(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+b (-2 a+b \text {sech}(x)) \tanh (x)}{2 b^3} \]
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Time = 0.57 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.25
method | result | size |
default | \(\frac {\frac {2 \left (\left (-a b -\frac {1}{2} b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{3}+\left (-a b +\frac {1}{2} b^{2}\right ) \tanh \left (\frac {x}{2}\right )\right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{2}}+\left (2 a^{2}+b^{2}\right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{b^{3}}-\frac {2 a^{3} \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{3} \sqrt {\left (a +b \right ) \left (a -b \right )}}\) | \(109\) |
risch | \(\frac {{\mathrm e}^{3 x} b +2 a \,{\mathrm e}^{2 x}-{\mathrm e}^{x} b +2 a}{\left (1+{\mathrm e}^{2 x}\right )^{2} b^{2}}+\frac {i \ln \left ({\mathrm e}^{x}+i\right ) a^{2}}{b^{3}}+\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{2 b}-\frac {i \ln \left ({\mathrm e}^{x}-i\right ) a^{2}}{b^{3}}-\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{2 b}-\frac {a^{3} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, b^{3}}+\frac {a^{3} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, b^{3}}\) | \(209\) |
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Leaf count of result is larger than twice the leaf count of optimal. 682 vs. \(2 (73) = 146\).
Time = 0.33 (sec) , antiderivative size = 1444, normalized size of antiderivative = 16.60 \[ \int \frac {\text {sech}^4(x)}{a+b \text {sech}(x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\text {sech}^4(x)}{a+b \text {sech}(x)} \, dx=\int \frac {\operatorname {sech}^{4}{\left (x \right )}}{a + b \operatorname {sech}{\left (x \right )}}\, dx \]
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Exception generated. \[ \int \frac {\text {sech}^4(x)}{a+b \text {sech}(x)} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02 \[ \int \frac {\text {sech}^4(x)}{a+b \text {sech}(x)} \, dx=-\frac {2 \, a^{3} \arctan \left (\frac {a e^{x} + b}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} b^{3}} + \frac {{\left (2 \, a^{2} + b^{2}\right )} \arctan \left (e^{x}\right )}{b^{3}} + \frac {b e^{\left (3 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - b e^{x} + 2 \, a}{b^{2} {\left (e^{\left (2 \, x\right )} + 1\right )}^{2}} \]
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Time = 5.40 (sec) , antiderivative size = 476, normalized size of antiderivative = 5.47 \[ \int \frac {\text {sech}^4(x)}{a+b \text {sech}(x)} \, dx=\frac {{\mathrm {e}}^x}{b+b\,{\mathrm {e}}^{2\,x}}-\frac {2\,{\mathrm {e}}^x}{b+2\,b\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{4\,x}}+\frac {2\,a}{b^2\,{\mathrm {e}}^{2\,x}+b^2}-\frac {\ln \left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,b}-\frac {a^2\,\left (\ln \left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}-\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,1{}\mathrm {i}\right )}{b^3}-\frac {a^3\,\ln \left (16\,a\,b^5-48\,a^5\,b-24\,a^5\,\sqrt {b^2-a^2}+32\,a^3\,b^3+24\,a^6\,{\mathrm {e}}^x+32\,b^6\,{\mathrm {e}}^x+16\,a\,b^4\,\sqrt {b^2-a^2}+40\,a^3\,b^2\,\sqrt {b^2-a^2}+32\,b^5\,{\mathrm {e}}^x\,\sqrt {b^2-a^2}+56\,a^2\,b^4\,{\mathrm {e}}^x-112\,a^4\,b^2\,{\mathrm {e}}^x+72\,a^2\,b^3\,{\mathrm {e}}^x\,\sqrt {b^2-a^2}-72\,a^4\,b\,{\mathrm {e}}^x\,\sqrt {b^2-a^2}\right )}{b^3\,\sqrt {b^2-a^2}}+\frac {a^3\,\ln \left (16\,a\,b^5-48\,a^5\,b+24\,a^5\,\sqrt {b^2-a^2}+32\,a^3\,b^3+24\,a^6\,{\mathrm {e}}^x+32\,b^6\,{\mathrm {e}}^x-16\,a\,b^4\,\sqrt {b^2-a^2}-40\,a^3\,b^2\,\sqrt {b^2-a^2}-32\,b^5\,{\mathrm {e}}^x\,\sqrt {b^2-a^2}+56\,a^2\,b^4\,{\mathrm {e}}^x-112\,a^4\,b^2\,{\mathrm {e}}^x-72\,a^2\,b^3\,{\mathrm {e}}^x\,\sqrt {b^2-a^2}+72\,a^4\,b\,{\mathrm {e}}^x\,\sqrt {b^2-a^2}\right )}{b^3\,\sqrt {b^2-a^2}} \]
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