Integrand size = 16, antiderivative size = 77 \[ \int e^{a+b x} \tanh ^3(a+b x) \, dx=\frac {e^{a+b x}}{b}-\frac {2 e^{a+b x}}{b \left (1+e^{2 a+2 b x}\right )^2}+\frac {3 e^{a+b x}}{b \left (1+e^{2 a+2 b x}\right )}-\frac {3 \arctan \left (e^{a+b x}\right )}{b} \]
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Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2320, 398, 1172, 12, 294, 209} \[ \int e^{a+b x} \tanh ^3(a+b x) \, dx=-\frac {3 \arctan \left (e^{a+b x}\right )}{b}+\frac {e^{a+b x}}{b}+\frac {3 e^{a+b x}}{b \left (e^{2 a+2 b x}+1\right )}-\frac {2 e^{a+b x}}{b \left (e^{2 a+2 b x}+1\right )^2} \]
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Rule 12
Rule 209
Rule 294
Rule 398
Rule 1172
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (-1+x^2\right )^3}{\left (1+x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (1-\frac {2 \left (1+3 x^4\right )}{\left (1+x^2\right )^3}\right ) \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {e^{a+b x}}{b}-\frac {2 \text {Subst}\left (\int \frac {1+3 x^4}{\left (1+x^2\right )^3} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {e^{a+b x}}{b}-\frac {2 e^{a+b x}}{b \left (1+e^{2 a+2 b x}\right )^2}+\frac {\text {Subst}\left (\int -\frac {12 x^2}{\left (1+x^2\right )^2} \, dx,x,e^{a+b x}\right )}{2 b} \\ & = \frac {e^{a+b x}}{b}-\frac {2 e^{a+b x}}{b \left (1+e^{2 a+2 b x}\right )^2}-\frac {6 \text {Subst}\left (\int \frac {x^2}{\left (1+x^2\right )^2} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {e^{a+b x}}{b}-\frac {2 e^{a+b x}}{b \left (1+e^{2 a+2 b x}\right )^2}+\frac {3 e^{a+b x}}{b \left (1+e^{2 a+2 b x}\right )}-\frac {3 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {e^{a+b x}}{b}-\frac {2 e^{a+b x}}{b \left (1+e^{2 a+2 b x}\right )^2}+\frac {3 e^{a+b x}}{b \left (1+e^{2 a+2 b x}\right )}-\frac {3 \arctan \left (e^{a+b x}\right )}{b} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.78 \[ \int e^{a+b x} \tanh ^3(a+b x) \, dx=\frac {e^{a+b x} \left (2+5 e^{2 (a+b x)}+e^{4 (a+b x)}\right )}{b \left (1+e^{2 (a+b x)}\right )^2}-\frac {3 \arctan \left (e^{a+b x}\right )}{b} \]
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Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {{\mathrm e}^{b x +a}}{b}+\frac {{\mathrm e}^{b x +a} \left (3 \,{\mathrm e}^{2 b x +2 a}+1\right )}{b \left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}+\frac {3 i \ln \left ({\mathrm e}^{b x +a}-i\right )}{2 b}-\frac {3 i \ln \left ({\mathrm e}^{b x +a}+i\right )}{2 b}\) | \(80\) |
derivativedivides | \(\frac {\frac {\sinh \left (b x +a \right )^{3}}{\cosh \left (b x +a \right )^{2}}+\frac {3 \sinh \left (b x +a \right )}{\cosh \left (b x +a \right )^{2}}-\frac {3 \,\operatorname {sech}\left (b x +a \right ) \tanh \left (b x +a \right )}{2}-3 \arctan \left ({\mathrm e}^{b x +a}\right )+\frac {\sinh \left (b x +a \right )^{2}}{\cosh \left (b x +a \right )}+\frac {2}{\cosh \left (b x +a \right )}}{b}\) | \(89\) |
default | \(\frac {\frac {\sinh \left (b x +a \right )^{3}}{\cosh \left (b x +a \right )^{2}}+\frac {3 \sinh \left (b x +a \right )}{\cosh \left (b x +a \right )^{2}}-\frac {3 \,\operatorname {sech}\left (b x +a \right ) \tanh \left (b x +a \right )}{2}-3 \arctan \left ({\mathrm e}^{b x +a}\right )+\frac {\sinh \left (b x +a \right )^{2}}{\cosh \left (b x +a \right )}+\frac {2}{\cosh \left (b x +a \right )}}{b}\) | \(89\) |
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Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (71) = 142\).
Time = 0.27 (sec) , antiderivative size = 339, normalized size of antiderivative = 4.40 \[ \int e^{a+b x} \tanh ^3(a+b x) \, dx=\frac {\cosh \left (b x + a\right )^{5} + 5 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + \sinh \left (b x + a\right )^{5} + 5 \, {\left (2 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{3} + 5 \, \cosh \left (b x + a\right )^{3} + 5 \, {\left (2 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 3 \, {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + {\left (5 \, \cosh \left (b x + a\right )^{4} + 15 \, \cosh \left (b x + a\right )^{2} + 2\right )} \sinh \left (b x + a\right ) + 2 \, \cosh \left (b x + a\right )}{b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} + 2 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b} \]
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\[ \int e^{a+b x} \tanh ^3(a+b x) \, dx=e^{a} \int e^{b x} \tanh ^{3}{\left (a + b x \right )}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.90 \[ \int e^{a+b x} \tanh ^3(a+b x) \, dx=-\frac {3 \, \arctan \left (e^{\left (b x + a\right )}\right )}{b} + \frac {e^{\left (b x + a\right )}}{b} + \frac {3 \, e^{\left (3 \, b x + 3 \, a\right )} + e^{\left (b x + a\right )}}{b {\left (e^{\left (4 \, b x + 4 \, a\right )} + 2 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}} \]
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Time = 0.30 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.68 \[ \int e^{a+b x} \tanh ^3(a+b x) \, dx=\frac {\frac {3 \, e^{\left (3 \, b x + 3 \, a\right )} + e^{\left (b x + a\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{2}} - 3 \, \arctan \left (e^{\left (b x + a\right )}\right ) + e^{\left (b x + a\right )}}{b} \]
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Time = 1.72 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.21 \[ \int e^{a+b x} \tanh ^3(a+b x) \, dx=\frac {{\mathrm {e}}^{a+b\,x}}{b}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {b^2}}{b}\right )}{\sqrt {b^2}}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left (2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1\right )}+\frac {3\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]
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