Optimal. Leaf size=105 \[ \frac{b^2 (3 a+4 b) \text{sech}^3(c+d x)}{3 d}+\frac{(a+b)^3 \cosh ^3(c+d x)}{3 d}-\frac{(a+b)^2 (a+4 b) \cosh (c+d x)}{d}-\frac{3 b (a+b) (a+2 b) \text{sech}(c+d x)}{d}-\frac{b^3 \text{sech}^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.12258, antiderivative size = 105, 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 = {3664, 448} \[ \frac{b^2 (3 a+4 b) \text{sech}^3(c+d x)}{3 d}+\frac{(a+b)^3 \cosh ^3(c+d x)}{3 d}-\frac{(a+b)^2 (a+4 b) \cosh (c+d x)}{d}-\frac{3 b (a+b) (a+2 b) \text{sech}(c+d x)}{d}-\frac{b^3 \text{sech}^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3664
Rule 448
Rubi steps
\begin{align*} \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right ) \left (a+b-b x^2\right )^3}{x^4} \, dx,x,\text{sech}(c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (3 (-a-2 b) b (a+b)-\frac{(a+b)^3}{x^4}+\frac{(a+b)^2 (a+4 b)}{x^2}+b^2 (3 a+4 b) x^2-b^3 x^4\right ) \, dx,x,\text{sech}(c+d x)\right )}{d}\\ &=-\frac{(a+b)^2 (a+4 b) \cosh (c+d x)}{d}+\frac{(a+b)^3 \cosh ^3(c+d x)}{3 d}-\frac{3 b (a+b) (a+2 b) \text{sech}(c+d x)}{d}+\frac{b^2 (3 a+4 b) \text{sech}^3(c+d x)}{3 d}-\frac{b^3 \text{sech}^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.290715, size = 91, normalized size = 0.87 \[ \frac{20 b^2 (3 a+4 b) \text{sech}^3(c+d x)-45 (a+b)^2 (a+5 b) \cosh (c+d x)+5 (a+b)^3 \cosh (3 (c+d x))-180 b (a+b) (a+2 b) \text{sech}(c+d x)-12 b^3 \text{sech}^5(c+d x)}{60 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 287, normalized size = 2.7 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( dx+c \right ) +3\,{a}^{2}b \left ( 1/3\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{\cosh \left ( dx+c \right ) }}+4/3\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{\cosh \left ( dx+c \right ) }}-8/3\,\cosh \left ( dx+c \right ) \right ) +3\,a{b}^{2} \left ( 1/3\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}-2\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}-8/3\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{ \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+16/3\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{\cosh \left ( dx+c \right ) }}-16/3\,\cosh \left ( dx+c \right ) \right ) +{b}^{3} \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{8}}{3\, \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}-{\frac{8\, \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{3\, \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}-16\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{ \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}-{\frac{64\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{5\, \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+{\frac{128\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{15\, \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}}+{\frac{128\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{15\,\cosh \left ( dx+c \right ) }}-{\frac{128\,\cosh \left ( dx+c \right ) }{15}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.10065, size = 593, normalized size = 5.65 \begin{align*} -\frac{1}{120} \, b^{3}{\left (\frac{5 \,{\left (45 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d} + \frac{200 \, e^{\left (-2 \, d x - 2 \, c\right )} + 2515 \, e^{\left (-4 \, d x - 4 \, c\right )} + 6680 \, e^{\left (-6 \, d x - 6 \, c\right )} + 9073 \, e^{\left (-8 \, d x - 8 \, c\right )} + 5600 \, e^{\left (-10 \, d x - 10 \, c\right )} + 1665 \, e^{\left (-12 \, d x - 12 \, c\right )} - 5}{d{\left (e^{\left (-3 \, d x - 3 \, c\right )} + 5 \, e^{\left (-5 \, d x - 5 \, c\right )} + 10 \, e^{\left (-7 \, d x - 7 \, c\right )} + 10 \, e^{\left (-9 \, d x - 9 \, c\right )} + 5 \, e^{\left (-11 \, d x - 11 \, c\right )} + e^{\left (-13 \, d x - 13 \, c\right )}\right )}}\right )} - \frac{1}{8} \, a b^{2}{\left (\frac{33 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac{30 \, e^{\left (-2 \, d x - 2 \, c\right )} + 240 \, e^{\left (-4 \, d x - 4 \, c\right )} + 322 \, e^{\left (-6 \, d x - 6 \, c\right )} + 177 \, e^{\left (-8 \, d x - 8 \, c\right )} - 1}{d{\left (e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )} + 3 \, e^{\left (-7 \, d x - 7 \, c\right )} + e^{\left (-9 \, d x - 9 \, c\right )}\right )}}\right )} - \frac{1}{8} \, a^{2} b{\left (\frac{21 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac{20 \, e^{\left (-2 \, d x - 2 \, c\right )} + 69 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1}{d{\left (e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} + \frac{1}{24} \, a^{3}{\left (\frac{e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac{9 \, e^{\left (d x + c\right )}}{d} - \frac{9 \, e^{\left (-d x - c\right )}}{d} + \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.96694, size = 1388, normalized size = 13.22 \begin{align*} \frac{5 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{8} + 5 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sinh \left (d x + c\right )^{8} - 20 \,{\left (a^{3} + 12 \, a^{2} b + 21 \, a b^{2} + 10 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} - 20 \,{\left (a^{3} + 12 \, a^{2} b + 21 \, a b^{2} + 10 \, b^{3} - 7 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{6} - 20 \,{\left (11 \, a^{3} + 123 \, a^{2} b + 249 \, a b^{2} + 137 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 10 \,{\left (35 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{4} - 22 \, a^{3} - 246 \, a^{2} b - 498 \, a b^{2} - 274 \, b^{3} - 30 \,{\left (a^{3} + 12 \, a^{2} b + 21 \, a b^{2} + 10 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{4} - 425 \, a^{3} - 5235 \, a^{2} b - 10395 \, a b^{2} - 5649 \, b^{3} - 20 \,{\left (31 \, a^{3} + 372 \, a^{2} b + 747 \, a b^{2} + 390 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 20 \,{\left (7 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{6} - 15 \,{\left (a^{3} + 12 \, a^{2} b + 21 \, a b^{2} + 10 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} - 31 \, a^{3} - 372 \, a^{2} b - 747 \, a b^{2} - 390 \, b^{3} - 6 \,{\left (11 \, a^{3} + 123 \, a^{2} b + 249 \, a b^{2} + 137 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2}}{120 \,{\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 5 \, d \cosh \left (d x + c\right )^{3} + 5 \,{\left (2 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, d \cosh \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.43728, size = 597, normalized size = 5.69 \begin{align*} -\frac{5 \,{\left (9 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 63 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 99 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 45 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )} - 5 \,{\left (a^{3} e^{\left (3 \, d x + 48 \, c\right )} + 3 \, a^{2} b e^{\left (3 \, d x + 48 \, c\right )} + 3 \, a b^{2} e^{\left (3 \, d x + 48 \, c\right )} + b^{3} e^{\left (3 \, d x + 48 \, c\right )} - 9 \, a^{3} e^{\left (d x + 46 \, c\right )} - 63 \, a^{2} b e^{\left (d x + 46 \, c\right )} - 99 \, a b^{2} e^{\left (d x + 46 \, c\right )} - 45 \, b^{3} e^{\left (d x + 46 \, c\right )}\right )} e^{\left (-45 \, c\right )} + \frac{16 \,{\left (45 \, a^{2} b e^{\left (9 \, d x + 9 \, c\right )} + 135 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} + 90 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} + 180 \, a^{2} b e^{\left (7 \, d x + 7 \, c\right )} + 480 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 280 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} + 270 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} + 690 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 428 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 180 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 480 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 280 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 45 \, a^{2} b e^{\left (d x + c\right )} + 135 \, a b^{2} e^{\left (d x + c\right )} + 90 \, b^{3} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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