Optimal. Leaf size=43 \[ -\frac {b (2 a+b) \coth (c+d x)}{d}+x (a+b)^2-\frac {b^2 \coth ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3661, 390, 206} \[ -\frac {b (2 a+b) \coth (c+d x)}{d}+x (a+b)^2-\frac {b^2 \coth ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 390
Rule 3661
Rubi steps
\begin {align*} \int \left (a+b \coth ^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^2}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-b (2 a+b)-b^2 x^2+\frac {(a+b)^2}{1-x^2}\right ) \, dx,x,\coth (c+d x)\right )}{d}\\ &=-\frac {b (2 a+b) \coth (c+d x)}{d}-\frac {b^2 \coth ^3(c+d x)}{3 d}+\frac {(a+b)^2 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=(a+b)^2 x-\frac {b (2 a+b) \coth (c+d x)}{d}-\frac {b^2 \coth ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 65, normalized size = 1.51 \[ -\frac {\coth (c+d x) \left (b \left (6 a+b \coth ^2(c+d x)+3 b\right )-3 (a+b)^2 \tanh ^{-1}\left (\sqrt {\tanh ^2(c+d x)}\right ) \sqrt {\tanh ^2(c+d x)}\right )}{3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 197, normalized size = 4.58 \[ -\frac {2 \, {\left (3 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 6 \, {\left (3 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} - {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 6 \, a b + 4 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} - 6 \, a b \cosh \left (d x + c\right ) + 3 \, {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x - {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d x + 6 \, a b + 4 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 6 \, a b + 4 \, b^{2}\right )} \sinh \left (d x + c\right )}{3 \, {\left (d \sinh \left (d x + c\right )^{3} + 3 \, {\left (d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 103, normalized size = 2.40 \[ \frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (d x + c\right )} - \frac {4 \, {\left (3 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a b + 2 \, b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 144, normalized size = 3.35 \[ -\frac {b^{2} \left (\coth ^{3}\left (d x +c \right )\right )}{3 d}-\frac {2 a b \coth \left (d x +c \right )}{d}-\frac {b^{2} \coth \left (d x +c \right )}{d}-\frac {\ln \left (\coth \left (d x +c \right )-1\right ) a^{2}}{2 d}-\frac {\ln \left (\coth \left (d x +c \right )-1\right ) a b}{d}-\frac {\ln \left (\coth \left (d x +c \right )-1\right ) b^{2}}{2 d}+\frac {\ln \left (\coth \left (d x +c \right )+1\right ) a^{2}}{2 d}+\frac {\ln \left (\coth \left (d x +c \right )+1\right ) a b}{d}+\frac {\ln \left (\coth \left (d x +c \right )+1\right ) b^{2}}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 114, normalized size = 2.65 \[ \frac {1}{3} \, b^{2} {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + 2 \, a b {\left (x + \frac {c}{d} + \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 41, normalized size = 0.95 \[ x\,{\left (a+b\right )}^2-\frac {b^2\,{\mathrm {coth}\left (c+d\,x\right )}^3}{3\,d}-\frac {b\,\mathrm {coth}\left (c+d\,x\right )\,\left (2\,a+b\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.79, size = 102, normalized size = 2.37 \[ \begin {cases} a^{2} x + \tilde {\infty } a b x + \tilde {\infty } b^{2} x & \text {for}\: c = \log {\left (- e^{- d x} \right )} \vee c = \log {\left (e^{- d x} \right )} \\x \left (a + b \coth ^{2}{\relax (c )}\right )^{2} & \text {for}\: d = 0 \\a^{2} x + 2 a b x - \frac {2 a b}{d \tanh {\left (c + d x \right )}} + b^{2} x - \frac {b^{2}}{d \tanh {\left (c + d x \right )}} - \frac {b^{2}}{3 d \tanh ^{3}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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