Optimal. Leaf size=36 \[ a^2 x-\frac {(a+b)^2 \coth (c+d x)}{d}-\frac {b^2 \tanh (c+d x)}{d} \]
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Rubi [A] time = 0.09, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4141, 1802, 207} \[ a^2 x-\frac {(a+b)^2 \coth (c+d x)}{d}-\frac {b^2 \tanh (c+d x)}{d} \]
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 )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \left (1-x^2\right )\right )^2}{x^2 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-b^2+\frac {(a+b)^2}{x^2}-\frac {a^2}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {(a+b)^2 \coth (c+d x)}{d}-\frac {b^2 \tanh (c+d x)}{d}-\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a^2 x-\frac {(a+b)^2 \coth (c+d x)}{d}-\frac {b^2 \tanh (c+d x)}{d}\\ \end {align*}
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Mathematica [B] time = 0.74, size = 82, normalized size = 2.28 \[ \frac {4 \text {sech}(c+d x) \left (a \cosh ^2(c+d x)+b\right )^2 \left (a^2 d x \cosh (c+d x)+\sinh (d x) \left ((a+b)^2 \text {csch}(c) \coth (c+d x)-b^2 \text {sech}(c)\right )\right )}{d (a \cosh (2 (c+d x))+a+2 b)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 106, normalized size = 2.94 \[ -\frac {{\left (a^{2} + 2 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} - 2 \, {\left (a^{2} d x + a^{2} + 2 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} + 2 \, a b + 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} + 2 \, a b}{2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 65, normalized size = 1.81 \[ \frac {a^{2} d x - \frac {2 \, {\left (a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b e^{\left (2 \, d x + 2 \, c\right )} + a^{2} + 2 \, a b + 2 \, b^{2}\right )}}{e^{\left (4 \, d x + 4 \, c\right )} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 64, normalized size = 1.78 \[ \frac {a^{2} \left (d x +c -\coth \left (d x +c \right )\right )-2 a b \coth \left (d x +c \right )+b^{2} \left (-\frac {1}{\sinh \left (d x +c \right ) \cosh \left (d x +c \right )}-2 \tanh \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 71, normalized size = 1.97 \[ a^{2} {\left (x + \frac {c}{d} + \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + \frac {4 \, a b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} + \frac {4 \, b^{2}}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.42, size = 60, normalized size = 1.67 \[ a^2\,x-\frac {\frac {2\,\left (a^2+2\,a\,b+2\,b^2\right )}{d}+\frac {2\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+2\,b\right )}{d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-1} \]
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \coth ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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