Optimal. Leaf size=117 \[ -\frac {e^{-a-b x} \, _2F_1\left (1,-\frac {b}{2 d};1-\frac {b}{2 d};e^{2 (c+d x)}\right )}{b}-\frac {e^{a+b x} \, _2F_1\left (1,\frac {b}{2 d};\frac {b}{2 d}+1;e^{2 (c+d x)}\right )}{b}+\frac {e^{-a-b x}}{2 b}+\frac {e^{a+b x}}{2 b} \]
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Rubi [A] time = 0.11, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5603, 2194, 2251} \[ -\frac {e^{-a-b x} \, _2F_1\left (1,-\frac {b}{2 d};1-\frac {b}{2 d};e^{2 (c+d x)}\right )}{b}-\frac {e^{a+b x} \, _2F_1\left (1,\frac {b}{2 d};\frac {b}{2 d}+1;e^{2 (c+d x)}\right )}{b}+\frac {e^{-a-b x}}{2 b}+\frac {e^{a+b x}}{2 b} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2251
Rule 5603
Rubi steps
\begin {align*} \int \coth (c+d x) \sinh (a+b x) \, dx &=\int \left (-\frac {1}{2} e^{-a-b x}+\frac {1}{2} e^{a+b x}+\frac {e^{-a-b x}}{1-e^{2 (c+d x)}}-\frac {e^{a+b x}}{1-e^{2 (c+d x)}}\right ) \, dx\\ &=-\left (\frac {1}{2} \int e^{-a-b x} \, dx\right )+\frac {1}{2} \int e^{a+b x} \, dx+\int \frac {e^{-a-b x}}{1-e^{2 (c+d x)}} \, dx-\int \frac {e^{a+b x}}{1-e^{2 (c+d x)}} \, dx\\ &=\frac {e^{-a-b x}}{2 b}+\frac {e^{a+b x}}{2 b}-\frac {e^{-a-b x} \, _2F_1\left (1,-\frac {b}{2 d};1-\frac {b}{2 d};e^{2 (c+d x)}\right )}{b}-\frac {e^{a+b x} \, _2F_1\left (1,\frac {b}{2 d};1+\frac {b}{2 d};e^{2 (c+d x)}\right )}{b}\\ \end {align*}
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Mathematica [B] time = 6.92, size = 240, normalized size = 2.05 \[ \frac {e^{-a-b x+2 c} \left (b e^{2 d x} \, _2F_1\left (1,1-\frac {b}{2 d};2-\frac {b}{2 d};e^{2 (c+d x)}\right )-(b-2 d) \, _2F_1\left (1,-\frac {b}{2 d};1-\frac {b}{2 d};e^{2 (c+d x)}\right )\right )}{b \left (e^{2 c}-1\right ) (b-2 d)}-\frac {e^{a+2 c} \left (\frac {e^{b x} \, _2F_1\left (1,\frac {b}{2 d};\frac {b}{2 d}+1;e^{2 (c+d x)}\right )}{b}-\frac {e^{x (b+2 d)} \, _2F_1\left (1,\frac {b}{2 d}+1;\frac {b}{2 d}+2;e^{2 (c+d x)}\right )}{b+2 d}\right )}{e^{2 c}-1}+\frac {\cosh (a) \coth (c) \cosh (b x)}{b}+\frac {\sinh (a) \coth (c) \sinh (b x)}{b} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\coth \left (d x + c\right ) \sinh \left (b x + a\right ), x\right ) \]
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 \[ \int \coth \left (d x + c\right ) \sinh \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.26, size = 0, normalized size = 0.00 \[ \int \coth \left (d x +c \right ) \sinh \left (b x +a \right )\, dx \]
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 \[ \frac {{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-b x - a\right )}}{2 \, b} - \frac {1}{2} \, \int \frac {e^{\left (2 \, b x + 2 \, a\right )} - 1}{e^{\left (b x + d x + a + c\right )} + e^{\left (b x + a\right )}}\,{d x} + \frac {1}{2} \, \int \frac {e^{\left (2 \, b x + 2 \, a\right )} - 1}{e^{\left (b x + d x + a + c\right )} - e^{\left (b x + a\right )}}\,{d x} \]
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 \[ \int \mathrm {coth}\left (c+d\,x\right )\,\mathrm {sinh}\left (a+b\,x\right ) \,d x \]
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 \[ \int \sinh {\left (a + b x \right )} \coth {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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