Optimal. Leaf size=74 \[ -\frac {b \left (3 a^2+3 a b+b^2\right ) \coth (c+d x)}{d}-\frac {b^2 (3 a+b) \coth ^3(c+d x)}{3 d}+x (a+b)^3-\frac {b^3 \coth ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.05, antiderivative size = 74, 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} \[ -\frac {b \left (3 a^2+3 a b+b^2\right ) \coth (c+d x)}{d}-\frac {b^2 (3 a+b) \coth ^3(c+d x)}{3 d}+x (a+b)^3-\frac {b^3 \coth ^5(c+d x)}{5 d} \]
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 )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^3}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-b \left (3 a^2+3 a b+b^2\right )-b^2 (3 a+b) x^2-b^3 x^4+\frac {(a+b)^3}{1-x^2}\right ) \, dx,x,\coth (c+d x)\right )}{d}\\ &=-\frac {b \left (3 a^2+3 a b+b^2\right ) \coth (c+d x)}{d}-\frac {b^2 (3 a+b) \coth ^3(c+d x)}{3 d}-\frac {b^3 \coth ^5(c+d x)}{5 d}+\frac {(a+b)^3 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=(a+b)^3 x-\frac {b \left (3 a^2+3 a b+b^2\right ) \coth (c+d x)}{d}-\frac {b^2 (3 a+b) \coth ^3(c+d x)}{3 d}-\frac {b^3 \coth ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 1.45, size = 100, normalized size = 1.35 \[ \frac {(a+b)^3 \tanh ^{-1}\left (\sqrt {\tanh ^2(c+d x)}\right ) \tanh (c+d x)}{d \sqrt {\tanh ^2(c+d x)}}-\frac {b \coth (c+d x) \left (15 \left (3 a^2+3 a b+b^2\right )+5 b (3 a+b) \coth ^2(c+d x)+3 b^2 \coth ^4(c+d x)\right )}{15 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 557, normalized size = 7.53 \[ -\frac {{\left (45 \, a^{2} b + 60 \, a b^{2} + 23 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 5 \, {\left (45 \, a^{2} b + 60 \, a b^{2} + 23 \, b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} - {\left (45 \, a^{2} b + 60 \, a b^{2} + 23 \, b^{3} + 15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \sinh \left (d x + c\right )^{5} - 5 \, {\left (27 \, a^{2} b + 24 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 5 \, {\left (45 \, a^{2} b + 60 \, a b^{2} + 23 \, b^{3} + 15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x - 2 \, {\left (45 \, a^{2} b + 60 \, a b^{2} + 23 \, b^{3} + 15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 5 \, {\left (2 \, {\left (45 \, a^{2} b + 60 \, a b^{2} + 23 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 3 \, {\left (27 \, a^{2} b + 24 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, {\left (9 \, a^{2} b + 6 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right ) - 5 \, {\left ({\left (45 \, a^{2} b + 60 \, a b^{2} + 23 \, b^{3} + 15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{4} + 90 \, a^{2} b + 120 \, a b^{2} + 46 \, b^{3} + 30 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x - 3 \, {\left (45 \, a^{2} b + 60 \, a b^{2} + 23 \, b^{3} + 15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{15 \, {\left (d \sinh \left (d x + c\right )^{5} + 5 \, {\left (2 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{3} + 5 \, {\left (d \cosh \left (d x + c\right )^{4} - 3 \, d \cosh \left (d x + c\right )^{2} + 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 = 0.19, size = 241, normalized size = 3.26 \[ \frac {15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (45 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 90 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 45 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 180 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 270 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 90 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 270 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 330 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 140 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 180 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 210 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 70 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 45 \, a^{2} b + 60 \, a b^{2} + 23 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{5}}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 235, normalized size = 3.18 \[ -\frac {\ln \left (\coth \left (d x +c \right )-1\right ) a^{3}}{2 d}-\frac {3 \ln \left (\coth \left (d x +c \right )-1\right ) a^{2} b}{2 d}-\frac {3 \ln \left (\coth \left (d x +c \right )-1\right ) a \,b^{2}}{2 d}-\frac {\ln \left (\coth \left (d x +c \right )-1\right ) b^{3}}{2 d}-\frac {\left (\coth ^{3}\left (d x +c \right )\right ) b^{3}}{3 d}-\frac {\coth \left (d x +c \right ) b^{3}}{d}-\frac {b^{3} \left (\coth ^{5}\left (d x +c \right )\right )}{5 d}+\frac {\ln \left (\coth \left (d x +c \right )+1\right ) a^{3}}{2 d}+\frac {3 \ln \left (\coth \left (d x +c \right )+1\right ) a^{2} b}{2 d}+\frac {3 \ln \left (\coth \left (d x +c \right )+1\right ) a \,b^{2}}{2 d}+\frac {\ln \left (\coth \left (d x +c \right )+1\right ) b^{3}}{2 d}-\frac {\left (\coth ^{3}\left (d x +c \right )\right ) a \,b^{2}}{d}-\frac {3 \coth \left (d x +c \right ) a^{2} b}{d}-\frac {3 a \,b^{2} \coth \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 239, normalized size = 3.23 \[ \frac {1}{15} \, b^{3} {\left (15 \, x + \frac {15 \, c}{d} - \frac {2 \, {\left (70 \, e^{\left (-2 \, d x - 2 \, c\right )} - 140 \, e^{\left (-4 \, d x - 4 \, c\right )} + 90 \, e^{\left (-6 \, d x - 6 \, c\right )} - 45 \, e^{\left (-8 \, d x - 8 \, c\right )} - 23\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}}\right )} + a b^{2} {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + 3 \, a^{2} b {\left (x + \frac {c}{d} + \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 72, normalized size = 0.97 \[ x\,{\left (a+b\right )}^3-\frac {{\mathrm {coth}\left (c+d\,x\right )}^3\,\left (b^3+3\,a\,b^2\right )}{3\,d}-\frac {b^3\,{\mathrm {coth}\left (c+d\,x\right )}^5}{5\,d}-\frac {b\,\mathrm {coth}\left (c+d\,x\right )\,\left (3\,a^2+3\,a\,b+b^2\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 27.59, size = 170, normalized size = 2.30 \[ \begin {cases} a^{3} x + \tilde {\infty } a^{2} b x + \tilde {\infty } a b^{2} x + \tilde {\infty } b^{3} x & \text {for}\: c = \log {\left (- e^{- d x} \right )} \vee c = \log {\left (e^{- d x} \right )} \\x \left (a + b \coth ^{2}{\relax (c )}\right )^{3} & \text {for}\: d = 0 \\a^{3} x + 3 a^{2} b x - \frac {3 a^{2} b}{d \tanh {\left (c + d x \right )}} + 3 a b^{2} x - \frac {3 a b^{2}}{d \tanh {\left (c + d x \right )}} - \frac {a b^{2}}{d \tanh ^{3}{\left (c + d x \right )}} + b^{3} x - \frac {b^{3}}{d \tanh {\left (c + d x \right )}} - \frac {b^{3}}{3 d \tanh ^{3}{\left (c + d x \right )}} - \frac {b^{3}}{5 d \tanh ^{5}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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