Optimal. Leaf size=67 \[ \frac{a^2 b \tanh ^3(c+d x)}{d}+\frac{a^3 \tanh (c+d x)}{d}+\frac{3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac{b^3 \tanh ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.0613605, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3675, 194} \[ \frac{a^2 b \tanh ^3(c+d x)}{d}+\frac{a^3 \tanh (c+d x)}{d}+\frac{3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac{b^3 \tanh ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 3675
Rule 194
Rubi steps
\begin{align*} \int \text{sech}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b x^2\right )^3 \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^3+3 a^2 b x^2+3 a b^2 x^4+b^3 x^6\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{a^3 \tanh (c+d x)}{d}+\frac{a^2 b \tanh ^3(c+d x)}{d}+\frac{3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac{b^3 \tanh ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.167314, size = 67, normalized size = 1. \[ \frac{a^2 b \tanh ^3(c+d x)}{d}+\frac{a^3 \tanh (c+d x)}{d}+\frac{3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac{b^3 \tanh ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.065, size = 227, normalized size = 3.4 \begin{align*}{\frac{1}{d} \left ({a}^{3}\tanh \left ( dx+c \right ) +3\,{a}^{2}b \left ( -1/2\,{\frac{\sinh \left ( dx+c \right ) }{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+1/2\, \left ( 2/3+1/3\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2} \right ) \tanh \left ( dx+c \right ) \right ) +3\,a{b}^{2} \left ( -1/2\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}-3/8\,{\frac{\sinh \left ( dx+c \right ) }{ \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+3/8\, \left ({\frac{8}{15}}+1/5\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}+{\frac{4\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{15}} \right ) \tanh \left ( dx+c \right ) \right ) +{b}^{3} \left ( -{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{2\, \left ( \cosh \left ( dx+c \right ) \right ) ^{7}}}-{\frac{5\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{8\, \left ( \cosh \left ( dx+c \right ) \right ) ^{7}}}-{\frac{5\,\sinh \left ( dx+c \right ) }{16\, \left ( \cosh \left ( dx+c \right ) \right ) ^{7}}}+{\frac{5\,\tanh \left ( dx+c \right ) }{16} \left ({\frac{16}{35}}+{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{6}}{7}}+{\frac{6\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}}{35}}+{\frac{8\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{35}} \right ) } \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05402, size = 96, normalized size = 1.43 \begin{align*} \frac{b^{3} \tanh \left (d x + c\right )^{7}}{7 \, d} + \frac{3 \, a b^{2} \tanh \left (d x + c\right )^{5}}{5 \, d} + \frac{a^{2} b \tanh \left (d x + c\right )^{3}}{d} + \frac{2 \, a^{3}}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.99027, size = 2034, normalized size = 30.36 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \operatorname{sech}^{2}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.70473, size = 468, normalized size = 6.99 \begin{align*} -\frac{2 \,{\left (35 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 105 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 105 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 35 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 210 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 420 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 210 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 525 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 665 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 315 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 175 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 700 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 560 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 420 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 525 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 315 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 231 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 105 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 210 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 140 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 42 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 35 \, a^{3} + 35 \, a^{2} b + 21 \, a b^{2} + 5 \, b^{3}\right )}}{35 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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