Integrand size = 23, antiderivative size = 85 \[ \int \cosh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {1}{8} \left (3 a^2-2 a b+3 b^2\right ) x+\frac {3 \left (a^2-b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {(a+b) \cosh ^3(c+d x) \sinh (c+d x) \left (a+b \tanh ^2(c+d x)\right )}{4 d} \]
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Time = 0.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3756, 424, 393, 212} \[ \int \cosh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {3 \left (a^2-b^2\right ) \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac {1}{8} x \left (3 a^2-2 a b+3 b^2\right )+\frac {(a+b) \sinh (c+d x) \cosh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )}{4 d} \]
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Rule 212
Rule 393
Rule 424
Rule 3756
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^2}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {(a+b) \cosh ^3(c+d x) \sinh (c+d x) \left (a+b \tanh ^2(c+d x)\right )}{4 d}-\frac {\text {Subst}\left (\int \frac {-a (3 a-b)-(a-3 b) b x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d} \\ & = \frac {3 \left (a^2-b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {(a+b) \cosh ^3(c+d x) \sinh (c+d x) \left (a+b \tanh ^2(c+d x)\right )}{4 d}+\frac {\left (3 a^2-2 a b+3 b^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 d} \\ & = \frac {1}{8} \left (3 a^2-2 a b+3 b^2\right ) x+\frac {3 \left (a^2-b^2\right ) \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {(a+b) \cosh ^3(c+d x) \sinh (c+d x) \left (a+b \tanh ^2(c+d x)\right )}{4 d} \\ \end{align*}
Time = 1.68 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.74 \[ \int \cosh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {4 \left (3 a^2-2 a b+3 b^2\right ) (c+d x)+8 \left (a^2-b^2\right ) \sinh (2 (c+d x))+(a+b)^2 \sinh (4 (c+d x))}{32 d} \]
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Time = 13.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.46
method | result | size |
derivativedivides | \(\frac {b^{2} \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a b \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )^{3}}{4}-\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{8}-\frac {d x}{8}-\frac {c}{8}\right )+a^{2} \left (\left (\frac {\cosh \left (d x +c \right )^{3}}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(124\) |
default | \(\frac {b^{2} \left (\left (\frac {\sinh \left (d x +c \right )^{3}}{4}-\frac {3 \sinh \left (d x +c \right )}{8}\right ) \cosh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a b \left (\frac {\sinh \left (d x +c \right ) \cosh \left (d x +c \right )^{3}}{4}-\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{8}-\frac {d x}{8}-\frac {c}{8}\right )+a^{2} \left (\left (\frac {\cosh \left (d x +c \right )^{3}}{4}+\frac {3 \cosh \left (d x +c \right )}{8}\right ) \sinh \left (d x +c \right )+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(124\) |
risch | \(\frac {3 a^{2} x}{8}-\frac {a b x}{4}+\frac {3 b^{2} x}{8}+\frac {{\mathrm e}^{4 d x +4 c} a^{2}}{64 d}+\frac {{\mathrm e}^{4 d x +4 c} a b}{32 d}+\frac {{\mathrm e}^{4 d x +4 c} b^{2}}{64 d}+\frac {{\mathrm e}^{2 d x +2 c} a^{2}}{8 d}-\frac {{\mathrm e}^{2 d x +2 c} b^{2}}{8 d}-\frac {{\mathrm e}^{-2 d x -2 c} a^{2}}{8 d}+\frac {{\mathrm e}^{-2 d x -2 c} b^{2}}{8 d}-\frac {{\mathrm e}^{-4 d x -4 c} a^{2}}{64 d}-\frac {{\mathrm e}^{-4 d x -4 c} a b}{32 d}-\frac {{\mathrm e}^{-4 d x -4 c} b^{2}}{64 d}\) | \(187\) |
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Time = 0.26 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.12 \[ \int \cosh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (3 \, a^{2} - 2 \, a b + 3 \, b^{2}\right )} d x + {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{3} + 4 \, {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{8 \, d} \]
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\[ \int \cosh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2} \cosh ^{4}{\left (c + d x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (79) = 158\).
Time = 0.20 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.01 \[ \int \cosh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {1}{64} \, a^{2} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac {1}{64} \, b^{2} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac {1}{32} \, a b {\left (\frac {8 \, {\left (d x + c\right )}}{d} - \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (79) = 158\).
Time = 0.36 (sec) , antiderivative size = 186, normalized size of antiderivative = 2.19 \[ \int \cosh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a b e^{\left (4 \, d x + 4 \, c\right )} + b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 8 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 8 \, {\left (3 \, a^{2} - 2 \, a b + 3 \, b^{2}\right )} {\left (d x + c\right )} - {\left (18 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 12 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 18 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 8 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} + 2 \, a b + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} \]
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Time = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.20 \[ \int \cosh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=x\,\left (\frac {3\,a^2}{8}-\frac {a\,b}{4}+\frac {3\,b^2}{8}\right )-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (a^2-b^2\right )}{8\,d}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^2-b^2\right )}{8\,d}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}\,{\left (a+b\right )}^2}{64\,d}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}\,{\left (a+b\right )}^2}{64\,d} \]
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