Optimal. Leaf size=18 \[ a x-\frac {(a+b) \coth (c+d x)}{d} \]
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Rubi [A] time = 0.06, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4141, 1802, 207} \[ a x-\frac {(a+b) \coth (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 207
Rule 1802
Rule 4141
Rubi steps
\begin {align*} \int \coth ^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \left (1-x^2\right )}{x^2 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a+b}{x^2}-\frac {a}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {(a+b) \coth (c+d x)}{d}-\frac {a \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a x-\frac {(a+b) \coth (c+d x)}{d}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 41, normalized size = 2.28 \[ -\frac {a \coth (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\tanh ^2(c+d x)\right )}{d}-\frac {b \coth (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 39, normalized size = 2.17 \[ -\frac {{\left (a + b\right )} \cosh \left (d x + c\right ) - {\left (a d x + a + b\right )} \sinh \left (d x + c\right )}{d \sinh \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 27, normalized size = 1.50 \[ \frac {a d x - \frac {2 \, {\left (a + b\right )}}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 30, normalized size = 1.67 \[ \frac {a \left (d x +c -\coth \left (d x +c \right )\right )-b \coth \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 47, normalized size = 2.61 \[ a {\left (x + \frac {c}{d} + \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + \frac {2 \, b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 25, normalized size = 1.39 \[ a\,x-\frac {2\,\left (a+b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \coth ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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