Integrand size = 21, antiderivative size = 98 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {5 b \arctan (\sinh (c+d x))}{2 d}-\frac {a \cosh (c+d x)}{d}+\frac {a \cosh ^3(c+d x)}{3 d}-\frac {5 b \sinh (c+d x)}{2 d}+\frac {5 b \sinh ^3(c+d x)}{6 d}-\frac {b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d} \]
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Time = 0.09 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3747, 2713, 2672, 294, 308, 209} \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {a \cosh ^3(c+d x)}{3 d}-\frac {a \cosh (c+d x)}{d}+\frac {5 b \arctan (\sinh (c+d x))}{2 d}+\frac {5 b \sinh ^3(c+d x)}{6 d}-\frac {5 b \sinh (c+d x)}{2 d}-\frac {b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d} \]
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Rule 209
Rule 294
Rule 308
Rule 2672
Rule 2713
Rule 3747
Rubi steps \begin{align*} \text {integral}& = i \int \left (-i a \sinh ^3(c+d x)-i b \sinh ^3(c+d x) \tanh ^3(c+d x)\right ) \, dx \\ & = a \int \sinh ^3(c+d x) \, dx+b \int \sinh ^3(c+d x) \tanh ^3(c+d x) \, dx \\ & = -\frac {a \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}+\frac {b \text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d} \\ & = -\frac {a \cosh (c+d x)}{d}+\frac {a \cosh ^3(c+d x)}{3 d}-\frac {b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}+\frac {(5 b) \text {Subst}\left (\int \frac {x^4}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d} \\ & = -\frac {a \cosh (c+d x)}{d}+\frac {a \cosh ^3(c+d x)}{3 d}-\frac {b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}+\frac {(5 b) \text {Subst}\left (\int \left (-1+x^2+\frac {1}{1+x^2}\right ) \, dx,x,\sinh (c+d x)\right )}{2 d} \\ & = -\frac {a \cosh (c+d x)}{d}+\frac {a \cosh ^3(c+d x)}{3 d}-\frac {5 b \sinh (c+d x)}{2 d}+\frac {5 b \sinh ^3(c+d x)}{6 d}-\frac {b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}+\frac {(5 b) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d} \\ & = \frac {5 b \arctan (\sinh (c+d x))}{2 d}-\frac {a \cosh (c+d x)}{d}+\frac {a \cosh ^3(c+d x)}{3 d}-\frac {5 b \sinh (c+d x)}{2 d}+\frac {5 b \sinh ^3(c+d x)}{6 d}-\frac {b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {5 b \arctan (\sinh (c+d x))}{2 d}-\frac {3 a \cosh (c+d x)}{4 d}+\frac {a \cosh (3 (c+d x))}{12 d}-\frac {5 b \text {sech}(c+d x) \tanh (c+d x)}{2 d}-\frac {5 b \sinh (c+d x) \tanh ^2(c+d x)}{3 d}+\frac {b \sinh ^3(c+d x) \tanh ^2(c+d x)}{3 d} \]
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Time = 0.98 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+b \left (\frac {\sinh \left (d x +c \right )^{5}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}+\frac {5 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+5 \arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(104\) |
default | \(\frac {a \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+b \left (\frac {\sinh \left (d x +c \right )^{5}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}+\frac {5 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+5 \arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(104\) |
risch | \(\frac {{\mathrm e}^{3 d x +3 c} a}{24 d}+\frac {{\mathrm e}^{3 d x +3 c} b}{24 d}-\frac {3 \,{\mathrm e}^{d x +c} a}{8 d}-\frac {9 \,{\mathrm e}^{d x +c} b}{8 d}-\frac {3 \,{\mathrm e}^{-d x -c} a}{8 d}+\frac {9 \,{\mathrm e}^{-d x -c} b}{8 d}+\frac {{\mathrm e}^{-3 d x -3 c} a}{24 d}-\frac {{\mathrm e}^{-3 d x -3 c} b}{24 d}-\frac {b \,{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}+\frac {5 i b \ln \left ({\mathrm e}^{d x +c}+i\right )}{2 d}-\frac {5 i b \ln \left ({\mathrm e}^{d x +c}-i\right )}{2 d}\) | \(186\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1070 vs. \(2 (88) = 176\).
Time = 0.28 (sec) , antiderivative size = 1070, normalized size of antiderivative = 10.92 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\text {Too large to display} \]
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\[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right ) \sinh ^{3}{\left (c + d x \right )}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.78 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {1}{24} \, b {\left (\frac {27 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} - \frac {120 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {25 \, e^{\left (-2 \, d x - 2 \, c\right )} + 77 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, e^{\left (-6 \, d x - 6 \, c\right )} - 1}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + 2 \, e^{\left (-5 \, d x - 5 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )}\right )}}\right )} + \frac {1}{24} \, a {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]
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Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.35 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {120 \, b \arctan \left (e^{\left (d x + c\right )}\right ) + a e^{\left (3 \, d x + 3 \, c\right )} + b e^{\left (3 \, d x + 3 \, c\right )} - 9 \, a e^{\left (d x + c\right )} - 27 \, b e^{\left (d x + c\right )} - {\left (9 \, a e^{\left (2 \, d x + 2 \, c\right )} - 27 \, b e^{\left (2 \, d x + 2 \, c\right )} - a + b\right )} e^{\left (-3 \, d x - 3 \, c\right )} - \frac {24 \, {\left (b e^{\left (3 \, d x + 3 \, c\right )} - b e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2}}}{24 \, d} \]
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Time = 0.25 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.74 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (a+b\right )}{24\,d}+\frac {5\,\mathrm {atan}\left (\frac {b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {d^2}}{d\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {d^2}}+\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (a-b\right )}{24\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a+9\,b\right )}{8\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (3\,a-9\,b\right )}{8\,d}-\frac {b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]
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