Optimal. Leaf size=152 \[ \frac{3 a^2 b \cosh (c+d x)}{d}-\frac{a^3 \coth (c+d x)}{d}+\frac{3 a b^2 \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{9 a b^2 \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{9}{8} a b^2 x+\frac{b^3 \cosh ^7(c+d x)}{7 d}-\frac{3 b^3 \cosh ^5(c+d x)}{5 d}+\frac{b^3 \cosh ^3(c+d x)}{d}-\frac{b^3 \cosh (c+d x)}{d} \]
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Rubi [A] time = 0.138335, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3220, 3767, 8, 2638, 2635, 2633} \[ \frac{3 a^2 b \cosh (c+d x)}{d}-\frac{a^3 \coth (c+d x)}{d}+\frac{3 a b^2 \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{9 a b^2 \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{9}{8} a b^2 x+\frac{b^3 \cosh ^7(c+d x)}{7 d}-\frac{3 b^3 \cosh ^5(c+d x)}{5 d}+\frac{b^3 \cosh ^3(c+d x)}{d}-\frac{b^3 \cosh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3220
Rule 3767
Rule 8
Rule 2638
Rule 2635
Rule 2633
Rubi steps
\begin{align*} \int \text{csch}^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx &=-\int \left (-a^3 \text{csch}^2(c+d x)-3 a^2 b \sinh (c+d x)-3 a b^2 \sinh ^4(c+d x)-b^3 \sinh ^7(c+d x)\right ) \, dx\\ &=a^3 \int \text{csch}^2(c+d x) \, dx+\left (3 a^2 b\right ) \int \sinh (c+d x) \, dx+\left (3 a b^2\right ) \int \sinh ^4(c+d x) \, dx+b^3 \int \sinh ^7(c+d x) \, dx\\ &=\frac{3 a^2 b \cosh (c+d x)}{d}+\frac{3 a b^2 \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac{1}{4} \left (9 a b^2\right ) \int \sinh ^2(c+d x) \, dx-\frac{\left (i a^3\right ) \operatorname{Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{d}-\frac{b^3 \operatorname{Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{3 a^2 b \cosh (c+d x)}{d}-\frac{b^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^3(c+d x)}{d}-\frac{3 b^3 \cosh ^5(c+d x)}{5 d}+\frac{b^3 \cosh ^7(c+d x)}{7 d}-\frac{a^3 \coth (c+d x)}{d}-\frac{9 a b^2 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{3 a b^2 \cosh (c+d x) \sinh ^3(c+d x)}{4 d}+\frac{1}{8} \left (9 a b^2\right ) \int 1 \, dx\\ &=\frac{9}{8} a b^2 x+\frac{3 a^2 b \cosh (c+d x)}{d}-\frac{b^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^3(c+d x)}{d}-\frac{3 b^3 \cosh ^5(c+d x)}{5 d}+\frac{b^3 \cosh ^7(c+d x)}{7 d}-\frac{a^3 \coth (c+d x)}{d}-\frac{9 a b^2 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{3 a b^2 \cosh (c+d x) \sinh ^3(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 1.14364, size = 140, normalized size = 0.92 \[ \frac{35 b \left (192 a^2-35 b^2\right ) \cosh (c+d x)-1120 a^3 \tanh \left (\frac{1}{2} (c+d x)\right )-1120 a^3 \coth \left (\frac{1}{2} (c+d x)\right )-1680 a b^2 \sinh (2 (c+d x))+210 a b^2 \sinh (4 (c+d x))+2520 a b^2 c+2520 a b^2 d x+245 b^3 \cosh (3 (c+d x))-49 b^3 \cosh (5 (c+d x))+5 b^3 \cosh (7 (c+d x))}{2240 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 111, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ( -{a}^{3}{\rm coth} \left (dx+c\right )+3\,{a}^{2}b\cosh \left ( dx+c \right ) +3\,a{b}^{2} \left ( \left ( 1/4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}-3/8\,\sinh \left ( dx+c \right ) \right ) \cosh \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{b}^{3} \left ( -{\frac{16}{35}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{7}}-{\frac{6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{35}}+{\frac{8\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) \cosh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20085, size = 297, normalized size = 1.95 \begin{align*} \frac{3}{64} \, a b^{2}{\left (24 \, x + \frac{e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac{8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac{1}{4480} \, b^{3}{\left (\frac{{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac{1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac{3}{2} \, a^{2} b{\left (\frac{e^{\left (d x + c\right )}}{d} + \frac{e^{\left (-d x - c\right )}}{d}\right )} + \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 = 1.87147, size = 809, normalized size = 5.32 \begin{align*} \frac{20 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 105 \, a b^{2} \cosh \left (d x + c\right )^{5} + 525 \, a b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} - 945 \, a b^{2} \cosh \left (d x + c\right )^{3} + 2 \,{\left (70 \, b^{3} \cosh \left (d x + c\right )^{3} - 81 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 4 \,{\left (35 \, b^{3} \cosh \left (d x + c\right )^{5} - 135 \, b^{3} \cosh \left (d x + c\right )^{3} + 147 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 105 \,{\left (10 \, a b^{2} \cosh \left (d x + c\right )^{3} - 27 \, a b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 280 \,{\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cosh \left (d x + c\right ) + 2 \,{\left (10 \, b^{3} \cosh \left (d x + c\right )^{7} - 81 \, b^{3} \cosh \left (d x + c\right )^{5} + 294 \, b^{3} \cosh \left (d x + c\right )^{3} + 1260 \, a b^{2} d x + 1120 \, a^{3} + 105 \,{\left (32 \, a^{2} b - 7 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2240 \, d \sinh \left (d x + c\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 = 1.44803, size = 410, normalized size = 2.7 \begin{align*} \frac{9 \,{\left (d x + c\right )} a b^{2}}{8 \, d} - \frac{{\left (1890 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 294 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 210 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 54 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, b^{3} - 35 \,{\left (192 \, a^{2} b - 35 \, b^{3}\right )} e^{\left (8 \, d x + 8 \, c\right )} + 560 \,{\left (16 \, a^{3} - 3 \, a b^{2}\right )} e^{\left (7 \, d x + 7 \, c\right )} + 210 \,{\left (32 \, a^{2} b - 7 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{4480 \, d{\left (e^{\left (d x + c\right )} + 1\right )}{\left (e^{\left (d x + c\right )} - 1\right )}} + \frac{5 \, b^{3} d^{6} e^{\left (7 \, d x + 7 \, c\right )} - 49 \, b^{3} d^{6} e^{\left (5 \, d x + 5 \, c\right )} + 210 \, a b^{2} d^{6} e^{\left (4 \, d x + 4 \, c\right )} + 245 \, b^{3} d^{6} e^{\left (3 \, d x + 3 \, c\right )} - 1680 \, a b^{2} d^{6} e^{\left (2 \, d x + 2 \, c\right )} + 6720 \, a^{2} b d^{6} e^{\left (d x + c\right )} - 1225 \, b^{3} d^{6} e^{\left (d x + c\right )}}{4480 \, d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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