Optimal. Leaf size=67 \[ \frac {e^{c (a+b x)}}{b c}-\frac {2 e^{c (a+b x)} \, _2F_1\left (1,\frac {b c}{2 e};\frac {b c}{2 e}+1;-e^{2 (d+e x)}\right )}{b c} \]
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Rubi [A] time = 0.07, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5484, 2194, 2251} \[ \frac {e^{c (a+b x)}}{b c}-\frac {2 e^{c (a+b x)} \, _2F_1\left (1,\frac {b c}{2 e};\frac {b c}{2 e}+1;-e^{2 (d+e x)}\right )}{b c} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2251
Rule 5484
Rubi steps
\begin {align*} \int e^{c (a+b x)} \tanh (d+e x) \, dx &=\int \left (e^{c (a+b x)}-\frac {2 e^{c (a+b x)}}{1+e^{2 (d+e x)}}\right ) \, dx\\ &=-\left (2 \int \frac {e^{c (a+b x)}}{1+e^{2 (d+e x)}} \, dx\right )+\int e^{c (a+b x)} \, dx\\ &=\frac {e^{c (a+b x)}}{b c}-\frac {2 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}\\ \end {align*}
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Mathematica [B] time = 1.96, size = 141, normalized size = 2.10 \[ \frac {e^{c (a+b x)} \left (2 b c e^{2 (d+e x)} \, _2F_1\left (1,\frac {b c}{2 e}+1;\frac {b c}{2 e}+2;-e^{2 (d+e x)}\right )-(b c+2 e) \left (2 e^{2 d} \, _2F_1\left (1,\frac {b c}{2 e};\frac {b c}{2 e}+1;-e^{2 (d+e x)}\right )-e^{2 d}+1\right )\right )}{b c \left (e^{2 d}+1\right ) (b c+2 e)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (e^{\left (b c x + a c\right )} \tanh \left (e x + d\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{\left ({\left (b x + a\right )} c\right )} \tanh \left (e x + d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.27, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{c \left (b x +a \right )} \tanh \left (e x +d \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ 4 \, e \int \frac {e^{\left (b c x + a c\right )}}{b c + {\left (b c e^{\left (4 \, d\right )} - 2 \, e e^{\left (4 \, d\right )}\right )} e^{\left (4 \, e x\right )} + 2 \, {\left (b c e^{\left (2 \, d\right )} - 2 \, e e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )} - 2 \, e}\,{d x} - \frac {{\left (b c e^{\left (a c\right )} + 2 \, e e^{\left (a c\right )} - {\left (b c e^{\left (a c + 2 \, d\right )} - 2 \, e e^{\left (a c + 2 \, d\right )}\right )} e^{\left (2 \, e x\right )}\right )} e^{\left (b c x\right )}}{b^{2} c^{2} - 2 \, b c e + {\left (b^{2} c^{2} e^{\left (2 \, d\right )} - 2 \, b c e e^{\left (2 \, d\right )}\right )} e^{\left (2 \, e x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{c\,\left (a+b\,x\right )}\,\mathrm {tanh}\left (d+e\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{a c} \int e^{b c x} \tanh {\left (d + e x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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