Integrand size = 20, antiderivative size = 103 \[ \int e^{c (a+b x)} \cot ^{-1}(\text {sech}(a c+b c x)) \, dx=\frac {e^{a c+b c x} \cot ^{-1}(\text {sech}(c (a+b x)))}{b c}-\frac {\left (1-\sqrt {2}\right ) \log \left (3-2 \sqrt {2}+e^{2 c (a+b x)}\right )}{2 b c}-\frac {\left (1+\sqrt {2}\right ) \log \left (3+2 \sqrt {2}+e^{2 c (a+b x)}\right )}{2 b c} \]
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Time = 0.10 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2225, 5316, 2320, 12, 1261, 646, 31} \[ \int e^{c (a+b x)} \cot ^{-1}(\text {sech}(a c+b c x)) \, dx=-\frac {\left (1-\sqrt {2}\right ) \log \left (e^{2 c (a+b x)}+3-2 \sqrt {2}\right )}{2 b c}-\frac {\left (1+\sqrt {2}\right ) \log \left (e^{2 c (a+b x)}+3+2 \sqrt {2}\right )}{2 b c}+\frac {e^{a c+b c x} \cot ^{-1}(\text {sech}(c (a+b x)))}{b c} \]
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Rule 12
Rule 31
Rule 646
Rule 1261
Rule 2225
Rule 2320
Rule 5316
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^x \cot ^{-1}(\text {sech}(x)) \, dx,x,a c+b c x\right )}{b c} \\ & = \frac {e^{a c+b c x} \cot ^{-1}(\text {sech}(c (a+b x)))}{b c}-\frac {\text {Subst}\left (\int \frac {e^x \text {sech}(x) \tanh (x)}{1+\text {sech}^2(x)} \, dx,x,a c+b c x\right )}{b c} \\ & = \frac {e^{a c+b c x} \cot ^{-1}(\text {sech}(c (a+b x)))}{b c}-\frac {\text {Subst}\left (\int \frac {2 x \left (-1+x^2\right )}{1+6 x^2+x^4} \, dx,x,e^{a c+b c x}\right )}{b c} \\ & = \frac {e^{a c+b c x} \cot ^{-1}(\text {sech}(c (a+b x)))}{b c}-\frac {2 \text {Subst}\left (\int \frac {x \left (-1+x^2\right )}{1+6 x^2+x^4} \, dx,x,e^{a c+b c x}\right )}{b c} \\ & = \frac {e^{a c+b c x} \cot ^{-1}(\text {sech}(c (a+b x)))}{b c}-\frac {\text {Subst}\left (\int \frac {-1+x}{1+6 x+x^2} \, dx,x,e^{2 a c+2 b c x}\right )}{b c} \\ & = \frac {e^{a c+b c x} \cot ^{-1}(\text {sech}(c (a+b x)))}{b c}-\frac {\left (1-\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{3-2 \sqrt {2}+x} \, dx,x,e^{2 a c+2 b c x}\right )}{2 b c}-\frac {\left (1+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{3+2 \sqrt {2}+x} \, dx,x,e^{2 a c+2 b c x}\right )}{2 b c} \\ & = \frac {e^{a c+b c x} \cot ^{-1}(\text {sech}(c (a+b x)))}{b c}-\frac {\left (1-\sqrt {2}\right ) \log \left (3-2 \sqrt {2}+e^{2 a c+2 b c x}\right )}{2 b c}-\frac {\left (1+\sqrt {2}\right ) \log \left (3+2 \sqrt {2}+e^{2 a c+2 b c x}\right )}{2 b c} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.41 \[ \int e^{c (a+b x)} \cot ^{-1}(\text {sech}(a c+b c x)) \, dx=\frac {-4 c (a+b x)+2 e^{c (a+b x)} \cot ^{-1}\left (\frac {2 e^{c (a+b x)}}{1+e^{2 c (a+b x)}}\right )+\text {RootSum}\left [1+6 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {a c+b c x-\log \left (e^{c (a+b x)}-\text {$\#$1}\right )+7 a c \text {$\#$1}^2+7 b c x \text {$\#$1}^2-7 \log \left (e^{c (a+b x)}-\text {$\#$1}\right ) \text {$\#$1}^2}{1+3 \text {$\#$1}^2}\&\right ]}{2 b c} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.55 (sec) , antiderivative size = 855, normalized size of antiderivative = 8.30
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Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (86) = 172\).
Time = 0.30 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.15 \[ \int e^{c (a+b x)} \cot ^{-1}(\text {sech}(a c+b c x)) \, dx=\frac {2 \, {\left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right )} \arctan \left (\cosh \left (b c x + a c\right )\right ) + \sqrt {2} \log \left (-\frac {3 \, {\left (2 \, \sqrt {2} - 3\right )} \cosh \left (b c x + a c\right )^{2} - 4 \, {\left (3 \, \sqrt {2} - 4\right )} \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + 3 \, {\left (2 \, \sqrt {2} - 3\right )} \sinh \left (b c x + a c\right )^{2} + 2 \, \sqrt {2} - 3}{\cosh \left (b c x + a c\right )^{2} + \sinh \left (b c x + a c\right )^{2} + 3}\right ) - \log \left (\frac {2 \, {\left (\cosh \left (b c x + a c\right )^{2} + \sinh \left (b c x + a c\right )^{2} + 3\right )}}{\cosh \left (b c x + a c\right )^{2} - 2 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )^{2}}\right )}{2 \, b c} \]
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Timed out. \[ \int e^{c (a+b x)} \cot ^{-1}(\text {sech}(a c+b c x)) \, dx=\text {Timed out} \]
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Time = 0.34 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.64 \[ \int e^{c (a+b x)} \cot ^{-1}(\text {sech}(a c+b c x)) \, dx=\frac {\operatorname {arccot}\left (\operatorname {sech}\left (b c x + a c\right )\right ) e^{\left ({\left (b x + a\right )} c\right )}}{b c} + \frac {3 \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (2 \, b c x + 2 \, a c\right )} - 3}{2 \, \sqrt {2} + e^{\left (2 \, b c x + 2 \, a c\right )} + 3}\right )}{8 \, b c} - \frac {\sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (-2 \, b c x - 2 \, a c\right )} - 3}{2 \, \sqrt {2} + e^{\left (-2 \, b c x - 2 \, a c\right )} + 3}\right )}{8 \, b c} - \frac {\log \left (e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}{2 \, b c} \]
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Time = 0.28 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.50 \[ \int e^{c (a+b x)} \cot ^{-1}(\text {sech}(a c+b c x)) \, dx=\frac {{\left (\sqrt {2} e^{\left (-a c\right )} \log \left (-\frac {2 \, \sqrt {2} e^{\left (2 \, a c\right )} - e^{\left (2 \, b c x + 4 \, a c\right )} - 3 \, e^{\left (2 \, a c\right )}}{2 \, \sqrt {2} e^{\left (2 \, a c\right )} + e^{\left (2 \, b c x + 4 \, a c\right )} + 3 \, e^{\left (2 \, a c\right )}}\right ) + 2 \, \arctan \left (\frac {1}{2} \, e^{\left (b c x + a c\right )} + \frac {1}{2} \, e^{\left (-b c x - a c\right )}\right ) e^{\left (b c x\right )} - e^{\left (-a c\right )} \log \left (e^{\left (4 \, b c x + 4 \, a c\right )} + 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )\right )} e^{\left (a c\right )}}{2 \, b c} \]
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Time = 0.97 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.31 \[ \int e^{c (a+b x)} \cot ^{-1}(\text {sech}(a c+b c x)) \, dx=\frac {{\mathrm {e}}^{a\,c+b\,c\,x}\,\mathrm {acot}\left (\frac {1}{\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}+\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}\right )}{b\,c}+\frac {\ln \left (-8\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}-2\,\sqrt {2}-6\,\sqrt {2}\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}\right )\,\left (\sqrt {2}-1\right )}{2\,b\,c}-\frac {\ln \left (2\,\sqrt {2}-8\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}+6\,\sqrt {2}\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}\right )\,\left (\sqrt {2}+1\right )}{2\,b\,c} \]
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