Integrand size = 21, antiderivative size = 269 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=-\frac {9 a^2 b \arctan (\sinh (c+d x))}{2 d}-\frac {315 b^3 \arctan (\sinh (c+d x))}{128 d}+\frac {a^3 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {9 a^2 b \sinh (c+d x)}{2 d}+\frac {315 b^3 \sinh (c+d x)}{128 d}-\frac {3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}-\frac {105 b^3 \sinh (c+d x) \tanh ^2(c+d x)}{128 d}-\frac {21 b^3 \sinh (c+d x) \tanh ^4(c+d x)}{64 d}-\frac {3 b^3 \sinh (c+d x) \tanh ^6(c+d x)}{16 d}-\frac {b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d} \]
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Time = 0.18 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3747, 2718, 2672, 294, 327, 209, 2670, 276} \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {a^3 \cosh (c+d x)}{d}-\frac {9 a^2 b \arctan (\sinh (c+d x))}{2 d}+\frac {9 a^2 b \sinh (c+d x)}{2 d}-\frac {3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}-\frac {315 b^3 \arctan (\sinh (c+d x))}{128 d}+\frac {315 b^3 \sinh (c+d x)}{128 d}-\frac {b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d}-\frac {3 b^3 \sinh (c+d x) \tanh ^6(c+d x)}{16 d}-\frac {21 b^3 \sinh (c+d x) \tanh ^4(c+d x)}{64 d}-\frac {105 b^3 \sinh (c+d x) \tanh ^2(c+d x)}{128 d} \]
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Rule 209
Rule 276
Rule 294
Rule 327
Rule 2670
Rule 2672
Rule 2718
Rule 3747
Rubi steps \begin{align*} \text {integral}& = -\left (i \int \left (i a^3 \sinh (c+d x)+3 i a^2 b \sinh (c+d x) \tanh ^3(c+d x)+3 i a b^2 \sinh (c+d x) \tanh ^6(c+d x)+i b^3 \sinh (c+d x) \tanh ^9(c+d x)\right ) \, dx\right ) \\ & = a^3 \int \sinh (c+d x) \, dx+\left (3 a^2 b\right ) \int \sinh (c+d x) \tanh ^3(c+d x) \, dx+\left (3 a b^2\right ) \int \sinh (c+d x) \tanh ^6(c+d x) \, dx+b^3 \int \sinh (c+d x) \tanh ^9(c+d x) \, dx \\ & = \frac {a^3 \cosh (c+d x)}{d}+\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}-\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^6} \, dx,x,\cosh (c+d x)\right )}{d}+\frac {b^3 \text {Subst}\left (\int \frac {x^{10}}{\left (1+x^2\right )^5} \, dx,x,\sinh (c+d x)\right )}{d} \\ & = \frac {a^3 \cosh (c+d x)}{d}-\frac {3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}-\frac {b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d}+\frac {\left (9 a^2 b\right ) \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}-\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \left (-1+\frac {1}{x^6}-\frac {3}{x^4}+\frac {3}{x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{d}+\frac {\left (9 b^3\right ) \text {Subst}\left (\int \frac {x^8}{\left (1+x^2\right )^4} \, dx,x,\sinh (c+d x)\right )}{8 d} \\ & = \frac {a^3 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {9 a^2 b \sinh (c+d x)}{2 d}-\frac {3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}-\frac {3 b^3 \sinh (c+d x) \tanh ^6(c+d x)}{16 d}-\frac {b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d}-\frac {\left (9 a^2 b\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}+\frac {\left (21 b^3\right ) \text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{16 d} \\ & = -\frac {9 a^2 b \arctan (\sinh (c+d x))}{2 d}+\frac {a^3 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {9 a^2 b \sinh (c+d x)}{2 d}-\frac {3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}-\frac {21 b^3 \sinh (c+d x) \tanh ^4(c+d x)}{64 d}-\frac {3 b^3 \sinh (c+d x) \tanh ^6(c+d x)}{16 d}-\frac {b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d}+\frac {\left (105 b^3\right ) \text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{64 d} \\ & = -\frac {9 a^2 b \arctan (\sinh (c+d x))}{2 d}+\frac {a^3 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {9 a^2 b \sinh (c+d x)}{2 d}-\frac {3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}-\frac {105 b^3 \sinh (c+d x) \tanh ^2(c+d x)}{128 d}-\frac {21 b^3 \sinh (c+d x) \tanh ^4(c+d x)}{64 d}-\frac {3 b^3 \sinh (c+d x) \tanh ^6(c+d x)}{16 d}-\frac {b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d}+\frac {\left (315 b^3\right ) \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{128 d} \\ & = -\frac {9 a^2 b \arctan (\sinh (c+d x))}{2 d}+\frac {a^3 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {9 a^2 b \sinh (c+d x)}{2 d}+\frac {315 b^3 \sinh (c+d x)}{128 d}-\frac {3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}-\frac {105 b^3 \sinh (c+d x) \tanh ^2(c+d x)}{128 d}-\frac {21 b^3 \sinh (c+d x) \tanh ^4(c+d x)}{64 d}-\frac {3 b^3 \sinh (c+d x) \tanh ^6(c+d x)}{16 d}-\frac {b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d}-\frac {\left (315 b^3\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{128 d} \\ & = -\frac {9 a^2 b \arctan (\sinh (c+d x))}{2 d}-\frac {315 b^3 \arctan (\sinh (c+d x))}{128 d}+\frac {a^3 \cosh (c+d x)}{d}+\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {9 a^2 b \sinh (c+d x)}{2 d}+\frac {315 b^3 \sinh (c+d x)}{128 d}-\frac {3 a^2 b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}-\frac {105 b^3 \sinh (c+d x) \tanh ^2(c+d x)}{128 d}-\frac {21 b^3 \sinh (c+d x) \tanh ^4(c+d x)}{64 d}-\frac {3 b^3 \sinh (c+d x) \tanh ^6(c+d x)}{16 d}-\frac {b^3 \sinh (c+d x) \tanh ^8(c+d x)}{8 d} \\ \end{align*}
Time = 7.69 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.87 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=-\frac {9 \left (64 a^2 b+35 b^3\right ) \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}+\frac {a \left (a^2+3 b^2\right ) \cosh (c+d x)}{d}+\frac {9 a b^2 \text {sech}(c+d x)}{d}-\frac {3 a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {b \left (3 a^2+b^2\right ) \sinh (c+d x)}{d}+\frac {\text {sech}^2(c+d x) \left (192 a^2 b \sinh (c+d x)+325 b^3 \sinh (c+d x)\right )}{128 d}-\frac {105 b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{64 d}+\frac {11 b^3 \text {sech}^5(c+d x) \tanh (c+d x)}{16 d}-\frac {b^3 \text {sech}^7(c+d x) \tanh (c+d x)}{8 d} \]
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Time = 5.72 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.10
method | result | size |
derivativedivides | \(\frac {a^{3} \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{2}}+\frac {3 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}-3 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right )^{6}}{\cosh \left (d x +c \right )^{5}}+\frac {6 \sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{5}}+\frac {8 \sinh \left (d x +c \right )^{2}}{\cosh \left (d x +c \right )^{5}}+\frac {16}{5 \cosh \left (d x +c \right )^{5}}\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{9}}{\cosh \left (d x +c \right )^{8}}+\frac {9 \sinh \left (d x +c \right )^{7}}{\cosh \left (d x +c \right )^{8}}+\frac {21 \sinh \left (d x +c \right )^{5}}{\cosh \left (d x +c \right )^{8}}+\frac {21 \sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{8}}+\frac {9 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{8}}-9 \left (\frac {\operatorname {sech}\left (d x +c \right )^{7}}{8}+\frac {7 \operatorname {sech}\left (d x +c \right )^{5}}{48}+\frac {35 \operatorname {sech}\left (d x +c \right )^{3}}{192}+\frac {35 \,\operatorname {sech}\left (d x +c \right )}{128}\right ) \tanh \left (d x +c \right )-\frac {315 \arctan \left ({\mathrm e}^{d x +c}\right )}{64}\right )}{d}\) | \(297\) |
default | \(\frac {a^{3} \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {\sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{2}}+\frac {3 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}-3 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right )^{6}}{\cosh \left (d x +c \right )^{5}}+\frac {6 \sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{5}}+\frac {8 \sinh \left (d x +c \right )^{2}}{\cosh \left (d x +c \right )^{5}}+\frac {16}{5 \cosh \left (d x +c \right )^{5}}\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{9}}{\cosh \left (d x +c \right )^{8}}+\frac {9 \sinh \left (d x +c \right )^{7}}{\cosh \left (d x +c \right )^{8}}+\frac {21 \sinh \left (d x +c \right )^{5}}{\cosh \left (d x +c \right )^{8}}+\frac {21 \sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{8}}+\frac {9 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{8}}-9 \left (\frac {\operatorname {sech}\left (d x +c \right )^{7}}{8}+\frac {7 \operatorname {sech}\left (d x +c \right )^{5}}{48}+\frac {35 \operatorname {sech}\left (d x +c \right )^{3}}{192}+\frac {35 \,\operatorname {sech}\left (d x +c \right )}{128}\right ) \tanh \left (d x +c \right )-\frac {315 \arctan \left ({\mathrm e}^{d x +c}\right )}{64}\right )}{d}\) | \(297\) |
risch | \(\frac {{\mathrm e}^{d x +c} a^{3}}{2 d}+\frac {3 \,{\mathrm e}^{d x +c} a^{2} b}{2 d}+\frac {3 \,{\mathrm e}^{d x +c} a \,b^{2}}{2 d}+\frac {b^{3} {\mathrm e}^{d x +c}}{2 d}+\frac {{\mathrm e}^{-d x -c} a^{3}}{2 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{2} b}{2 d}+\frac {3 \,{\mathrm e}^{-d x -c} a \,b^{2}}{2 d}-\frac {{\mathrm e}^{-d x -c} b^{3}}{2 d}+\frac {b \,{\mathrm e}^{d x +c} \left (960 a^{2} {\mathrm e}^{14 d x +14 c}+5760 a b \,{\mathrm e}^{14 d x +14 c}+1625 b^{2} {\mathrm e}^{14 d x +14 c}+4800 a^{2} {\mathrm e}^{12 d x +12 c}+32640 a b \,{\mathrm e}^{12 d x +12 c}+3925 b^{2} {\mathrm e}^{12 d x +12 c}+8640 a^{2} {\mathrm e}^{10 d x +10 c}+88704 a b \,{\mathrm e}^{10 d x +10 c}+9065 b^{2} {\mathrm e}^{10 d x +10 c}+4800 a^{2} {\mathrm e}^{8 d x +8 c}+143232 a b \,{\mathrm e}^{8 d x +8 c}+1645 b^{2} {\mathrm e}^{8 d x +8 c}-4800 a^{2} {\mathrm e}^{6 d x +6 c}+143232 a b \,{\mathrm e}^{6 d x +6 c}-1645 b^{2} {\mathrm e}^{6 d x +6 c}-8640 a^{2} {\mathrm e}^{4 d x +4 c}+88704 a b \,{\mathrm e}^{4 d x +4 c}-9065 \,{\mathrm e}^{4 d x +4 c} b^{2}-4800 a^{2} {\mathrm e}^{2 d x +2 c}+32640 a b \,{\mathrm e}^{2 d x +2 c}-3925 \,{\mathrm e}^{2 d x +2 c} b^{2}-960 a^{2}+5760 a b -1625 b^{2}\right )}{320 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{8}}+\frac {9 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{2 d}+\frac {315 i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{128 d}-\frac {9 i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{2 d}-\frac {315 i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{128 d}\) | \(509\) |
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Leaf count of result is larger than twice the leaf count of optimal. 6410 vs. \(2 (249) = 498\).
Time = 0.32 (sec) , antiderivative size = 6410, normalized size of antiderivative = 23.83 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\text {Too large to display} \]
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\[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right )^{3} \sinh {\left (c + d x \right )}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.80 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {1}{64} \, b^{3} {\left (\frac {315 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {32 \, e^{\left (-d x - c\right )}}{d} + \frac {581 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1681 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3605 \, e^{\left (-6 \, d x - 6 \, c\right )} + 2569 \, e^{\left (-8 \, d x - 8 \, c\right )} + 1463 \, e^{\left (-10 \, d x - 10 \, c\right )} - 917 \, e^{\left (-12 \, d x - 12 \, c\right )} - 529 \, e^{\left (-14 \, d x - 14 \, c\right )} - 293 \, e^{\left (-16 \, d x - 16 \, c\right )} + 32}{d {\left (e^{\left (-d x - c\right )} + 8 \, e^{\left (-3 \, d x - 3 \, c\right )} + 28 \, e^{\left (-5 \, d x - 5 \, c\right )} + 56 \, e^{\left (-7 \, d x - 7 \, c\right )} + 70 \, e^{\left (-9 \, d x - 9 \, c\right )} + 56 \, e^{\left (-11 \, d x - 11 \, c\right )} + 28 \, e^{\left (-13 \, d x - 13 \, c\right )} + 8 \, e^{\left (-15 \, d x - 15 \, c\right )} + e^{\left (-17 \, d x - 17 \, c\right )}\right )}}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {6 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )}}{d} + \frac {4 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + 1}{d {\left (e^{\left (-d x - c\right )} + 2 \, e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} + \frac {3}{10} \, a b^{2} {\left (\frac {5 \, e^{\left (-d x - c\right )}}{d} + \frac {85 \, e^{\left (-2 \, d x - 2 \, c\right )} + 210 \, e^{\left (-4 \, d x - 4 \, c\right )} + 314 \, e^{\left (-6 \, d x - 6 \, c\right )} + 185 \, e^{\left (-8 \, d x - 8 \, c\right )} + 65 \, e^{\left (-10 \, d x - 10 \, c\right )} + 5}{d {\left (e^{\left (-d x - c\right )} + 5 \, e^{\left (-3 \, d x - 3 \, c\right )} + 10 \, e^{\left (-5 \, d x - 5 \, c\right )} + 10 \, e^{\left (-7 \, d x - 7 \, c\right )} + 5 \, e^{\left (-9 \, d x - 9 \, c\right )} + e^{\left (-11 \, d x - 11 \, c\right )}\right )}}\right )} + \frac {a^{3} \cosh \left (d x + c\right )}{d} \]
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Time = 0.59 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.72 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {160 \, a^{3} e^{\left (d x + c\right )} + 480 \, a^{2} b e^{\left (d x + c\right )} + 480 \, a b^{2} e^{\left (d x + c\right )} + 160 \, b^{3} e^{\left (d x + c\right )} - 45 \, {\left (64 \, a^{2} b + 35 \, b^{3}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) + 160 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} e^{\left (-d x - c\right )} + \frac {960 \, a^{2} b e^{\left (15 \, d x + 15 \, c\right )} + 5760 \, a b^{2} e^{\left (15 \, d x + 15 \, c\right )} + 1625 \, b^{3} e^{\left (15 \, d x + 15 \, c\right )} + 4800 \, a^{2} b e^{\left (13 \, d x + 13 \, c\right )} + 32640 \, a b^{2} e^{\left (13 \, d x + 13 \, c\right )} + 3925 \, b^{3} e^{\left (13 \, d x + 13 \, c\right )} + 8640 \, a^{2} b e^{\left (11 \, d x + 11 \, c\right )} + 88704 \, a b^{2} e^{\left (11 \, d x + 11 \, c\right )} + 9065 \, b^{3} e^{\left (11 \, d x + 11 \, c\right )} + 4800 \, a^{2} b e^{\left (9 \, d x + 9 \, c\right )} + 143232 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} + 1645 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} - 4800 \, a^{2} b e^{\left (7 \, d x + 7 \, c\right )} + 143232 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 1645 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} - 8640 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} + 88704 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 9065 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} - 4800 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 32640 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 3925 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 960 \, a^{2} b e^{\left (d x + c\right )} + 5760 \, a b^{2} e^{\left (d x + c\right )} - 1625 \, b^{3} e^{\left (d x + c\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{320 \, d} \]
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Time = 2.21 (sec) , antiderivative size = 707, normalized size of antiderivative = 2.63 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {{\mathrm {e}}^{c+d\,x}\,{\left (a+b\right )}^3}{2\,d}+\frac {{\mathrm {e}}^{-c-d\,x}\,{\left (a-b\right )}^3}{2\,d}-\frac {9\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (35\,b^3\,\sqrt {d^2}+64\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {4096\,a^4\,b^2+4480\,a^2\,b^4+1225\,b^6}}\right )\,\sqrt {4096\,a^4\,b^2+4480\,a^2\,b^4+1225\,b^6}}{64\,\sqrt {d^2}}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (2455\,b^3+1728\,a\,b^2\right )}{40\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (2605\,b^3+768\,a\,b^2\right )}{20\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {188\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (192\,a^2\,b+1152\,a\,b^2+325\,b^3\right )}{64\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (475\,b^3+48\,a\,b^2\right )}{5\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}+\frac {112\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (7\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}+35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}+7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (192\,a^2\,b+768\,a\,b^2+745\,b^3\right )}{32\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {32\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (8\,{\mathrm {e}}^{2\,c+2\,d\,x}+28\,{\mathrm {e}}^{4\,c+4\,d\,x}+56\,{\mathrm {e}}^{6\,c+6\,d\,x}+70\,{\mathrm {e}}^{8\,c+8\,d\,x}+56\,{\mathrm {e}}^{10\,c+10\,d\,x}+28\,{\mathrm {e}}^{12\,c+12\,d\,x}+8\,{\mathrm {e}}^{14\,c+14\,d\,x}+{\mathrm {e}}^{16\,c+16\,d\,x}+1\right )} \]
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