Optimal. Leaf size=117 \[ -\frac{4 e^{c (a+b x)} \, _2F_1\left (1,\frac{b c}{2 e};\frac{b c}{2 e}+1;-e^{2 (d+e x)}\right )}{b c}+\frac{4 e^{c (a+b x)} \, _2F_1\left (2,\frac{b c}{2 e};\frac{b c}{2 e}+1;-e^{2 (d+e x)}\right )}{b c}+\frac{e^{c (a+b x)}}{b c} \]
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Rubi [A] time = 0.126796, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5484, 2194, 2251} \[ -\frac{4 e^{c (a+b x)} \, _2F_1\left (1,\frac{b c}{2 e};\frac{b c}{2 e}+1;-e^{2 (d+e x)}\right )}{b c}+\frac{4 e^{c (a+b x)} \, _2F_1\left (2,\frac{b c}{2 e};\frac{b c}{2 e}+1;-e^{2 (d+e x)}\right )}{b c}+\frac{e^{c (a+b x)}}{b c} \]
Antiderivative was successfully verified.
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Rule 5484
Rule 2194
Rule 2251
Rubi steps
\begin{align*} \int e^{c (a+b x)} \tanh ^2(d+e x) \, dx &=\int \left (e^{c (a+b x)}+\frac{4 e^{c (a+b x)}}{\left (1+e^{2 (d+e x)}\right )^2}-\frac{4 e^{c (a+b x)}}{1+e^{2 (d+e x)}}\right ) \, dx\\ &=4 \int \frac{e^{c (a+b x)}}{\left (1+e^{2 (d+e x)}\right )^2} \, dx-4 \int \frac{e^{c (a+b x)}}{1+e^{2 (d+e x)}} \, dx+\int e^{c (a+b x)} \, dx\\ &=\frac{e^{c (a+b x)}}{b c}-\frac{4 e^{c (a+b x)} \, _2F_1\left (1,\frac{b c}{2 e};1+\frac{b c}{2 e};-e^{2 (d+e x)}\right )}{b c}+\frac{4 e^{c (a+b x)} \, _2F_1\left (2,\frac{b c}{2 e};1+\frac{b c}{2 e};-e^{2 (d+e x)}\right )}{b c}\\ \end{align*}
Mathematica [A] time = 3.25664, size = 169, normalized size = 1.44 \[ \frac{e^{c (a+b x)} \left (2 b^2 c^2 e^{2 (d+e x)} \, _2F_1\left (1,\frac{b c}{2 e}+1;\frac{b c}{2 e}+2;-e^{2 (d+e x)}\right )-(b c+2 e) \left (2 b c e^{2 d} \, _2F_1\left (1,\frac{b c}{2 e};\frac{b c}{2 e}+1;-e^{2 (d+e x)}\right )-\left (e^{2 d}+1\right ) (e-b c \text{sech}(d) \sinh (e x) \text{sech}(d+e x))\right )\right )}{b c \left (e^{2 d}+1\right ) e (b c+2 e)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.068, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{c \left ( bx+a \right ) }} \left ( \tanh \left ( ex+d \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -16 \, b c e \int \frac{e^{\left (b c x + a c\right )}}{b^{2} c^{2} - 6 \, b c e + 8 \, e^{2} +{\left (b^{2} c^{2} e^{\left (6 \, d\right )} - 6 \, b c e e^{\left (6 \, d\right )} + 8 \, e^{2} e^{\left (6 \, d\right )}\right )} e^{\left (6 \, e x\right )} + 3 \,{\left (b^{2} c^{2} e^{\left (4 \, d\right )} - 6 \, b c e e^{\left (4 \, d\right )} + 8 \, e^{2} e^{\left (4 \, d\right )}\right )} e^{\left (4 \, e x\right )} + 3 \,{\left (b^{2} c^{2} e^{\left (2 \, d\right )} - 6 \, b c e e^{\left (2 \, d\right )} + 8 \, e^{2} e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )}}\,{d x} + \frac{{\left (b^{2} c^{2} e^{\left (a c\right )} + 10 \, b c e e^{\left (a c\right )} + 8 \, e^{2} e^{\left (a c\right )} +{\left (b^{2} c^{2} e^{\left (a c + 4 \, d\right )} - 6 \, b c e e^{\left (a c + 4 \, d\right )} + 8 \, e^{2} e^{\left (a c + 4 \, d\right )}\right )} e^{\left (4 \, e x\right )} - 2 \,{\left (b^{2} c^{2} e^{\left (a c + 2 \, d\right )} - 2 \, b c e e^{\left (a c + 2 \, d\right )} - 8 \, e^{2} e^{\left (a c + 2 \, d\right )}\right )} e^{\left (2 \, e x\right )}\right )} e^{\left (b c x\right )}}{b^{3} c^{3} - 6 \, b^{2} c^{2} e + 8 \, b c e^{2} +{\left (b^{3} c^{3} e^{\left (4 \, d\right )} - 6 \, b^{2} c^{2} e e^{\left (4 \, d\right )} + 8 \, b c e^{2} e^{\left (4 \, d\right )}\right )} e^{\left (4 \, e x\right )} + 2 \,{\left (b^{3} c^{3} e^{\left (2 \, d\right )} - 6 \, b^{2} c^{2} e e^{\left (2 \, d\right )} + 8 \, b c e^{2} e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (e^{\left (b c x + a c\right )} \tanh \left (e x + d\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} e^{a c} \int e^{b c x} \tanh ^{2}{\left (d + e x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left ({\left (b x + a\right )} c\right )} \tanh \left (e x + d\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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