Optimal. Leaf size=103 \[ -\frac {(b-d) e^{a+b x+c+d x} \, _2F_1\left (1,\frac {b+d}{2 d};\frac {1}{2} \left (\frac {b}{d}+3\right );-e^{2 (c+d x)}\right )}{d^2}+\frac {b e^{a+b x} \text {sech}(c+d x)}{2 d^2}+\frac {e^{a+b x} \tanh (c+d x) \text {sech}(c+d x)}{2 d} \]
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Rubi [A] time = 0.05, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5490, 5492} \[ -\frac {(b-d) e^{a+b x+c+d x} \, _2F_1\left (1,\frac {b+d}{2 d};\frac {1}{2} \left (\frac {b}{d}+3\right );-e^{2 (c+d x)}\right )}{d^2}+\frac {b e^{a+b x} \text {sech}(c+d x)}{2 d^2}+\frac {e^{a+b x} \tanh (c+d x) \text {sech}(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 5490
Rule 5492
Rubi steps
\begin {align*} \int e^{a+b x} \text {sech}^3(c+d x) \, dx &=\frac {b e^{a+b x} \text {sech}(c+d x)}{2 d^2}+\frac {e^{a+b x} \text {sech}(c+d x) \tanh (c+d x)}{2 d}+\frac {1}{2} \left (1-\frac {b^2}{d^2}\right ) \int e^{a+b x} \text {sech}(c+d x) \, dx\\ &=-\frac {(b-d) e^{a+c+b x+d x} \, _2F_1\left (1,\frac {b+d}{2 d};\frac {1}{2} \left (3+\frac {b}{d}\right );-e^{2 (c+d x)}\right )}{d^2}+\frac {b e^{a+b x} \text {sech}(c+d x)}{2 d^2}+\frac {e^{a+b x} \text {sech}(c+d x) \tanh (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 80, normalized size = 0.78 \[ \frac {e^{a+b x} \left (\text {sech}(c+d x) (b+d \tanh (c+d x))-2 (b-d) e^{c+d x} \, _2F_1\left (1,\frac {b+d}{2 d};\frac {1}{2} \left (\frac {b}{d}+3\right );-e^{2 (c+d x)}\right )\right )}{2 d^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )^{3}, x\right ) \]
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 \[ \int e^{\left (b x + a\right )} \operatorname {sech}\left (d x + c\right )^{3}\,{d x} \]
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 \[ \int {\mathrm e}^{b x +a} \mathrm {sech}\left (d x +c \right )^{3}\, dx \]
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 \[ -8 \, {\left (b^{2} e^{c} - d^{2} e^{c}\right )} \int \frac {e^{\left (b x + d x + a\right )}}{8 \, {\left (d^{2} e^{\left (2 \, d x + 2 \, c\right )} + d^{2}\right )}}\,{d x} + \frac {{\left (b e^{\left (3 \, c\right )} + d e^{\left (3 \, c\right )}\right )} e^{\left (b x + 3 \, d x + a\right )} + {\left (b e^{c} - d e^{c}\right )} e^{\left (b x + d x + a\right )}}{d^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, d^{2} e^{\left (2 \, d x + 2 \, c\right )} + d^{2}} \]
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 \[ \int \frac {{\mathrm {e}}^{a+b\,x}}{{\mathrm {cosh}\left (c+d\,x\right )}^3} \,d x \]
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 \[ e^{a} \int e^{b x} \operatorname {sech}^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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