Integrand size = 23, antiderivative size = 79 \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=-\frac {1}{2} (a+b) (a+5 b) x+\frac {(a+b) (a+5 b) \tanh (c+d x)}{2 d}+\frac {(a+b)^2 \sinh ^2(c+d x) \tanh (c+d x)}{2 d}+\frac {b^2 \tanh ^3(c+d x)}{3 d} \]
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Time = 0.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3744, 474, 470, 327, 212} \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {(a+b) (a+5 b) \tanh (c+d x)}{2 d}+\frac {(a+b)^2 \sinh ^2(c+d x) \tanh (c+d x)}{2 d}-\frac {1}{2} x (a+b) (a+5 b)+\frac {b^2 \tanh ^3(c+d x)}{3 d} \]
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Rule 212
Rule 327
Rule 470
Rule 474
Rule 3744
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^2 \left (a+b x^2\right )^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {(a+b)^2 \sinh ^2(c+d x) \tanh (c+d x)}{2 d}-\frac {\text {Subst}\left (\int \frac {x^2 \left (a^2+6 a b+3 b^2+2 b^2 x^2\right )}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d} \\ & = \frac {(a+b)^2 \sinh ^2(c+d x) \tanh (c+d x)}{2 d}+\frac {b^2 \tanh ^3(c+d x)}{3 d}-\frac {((a+b) (a+5 b)) \text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d} \\ & = \frac {(a+b) (a+5 b) \tanh (c+d x)}{2 d}+\frac {(a+b)^2 \sinh ^2(c+d x) \tanh (c+d x)}{2 d}+\frac {b^2 \tanh ^3(c+d x)}{3 d}-\frac {((a+b) (a+5 b)) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d} \\ & = -\frac {1}{2} (a+b) (a+5 b) x+\frac {(a+b) (a+5 b) \tanh (c+d x)}{2 d}+\frac {(a+b)^2 \sinh ^2(c+d x) \tanh (c+d x)}{2 d}+\frac {b^2 \tanh ^3(c+d x)}{3 d} \\ \end{align*}
Time = 1.55 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {-6 \left (a^2+6 a b+5 b^2\right ) (c+d x)+3 (a+b)^2 \sinh (2 (c+d x))+4 b \left (6 a+7 b-b \text {sech}^2(c+d x)\right ) \tanh (c+d x)}{12 d} \]
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Time = 0.88 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.49
method | result | size |
derivativedivides | \(\frac {a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+2 a b \left (\frac {\sinh \left (d x +c \right )^{3}}{2 \cosh \left (d x +c \right )}-\frac {3 d x}{2}-\frac {3 c}{2}+\frac {3 \tanh \left (d x +c \right )}{2}\right )+b^{2} \left (\frac {\sinh \left (d x +c \right )^{5}}{2 \cosh \left (d x +c \right )^{3}}-\frac {5 d x}{2}-\frac {5 c}{2}+\frac {5 \tanh \left (d x +c \right )}{2}+\frac {5 \tanh \left (d x +c \right )^{3}}{6}\right )}{d}\) | \(118\) |
default | \(\frac {a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+2 a b \left (\frac {\sinh \left (d x +c \right )^{3}}{2 \cosh \left (d x +c \right )}-\frac {3 d x}{2}-\frac {3 c}{2}+\frac {3 \tanh \left (d x +c \right )}{2}\right )+b^{2} \left (\frac {\sinh \left (d x +c \right )^{5}}{2 \cosh \left (d x +c \right )^{3}}-\frac {5 d x}{2}-\frac {5 c}{2}+\frac {5 \tanh \left (d x +c \right )}{2}+\frac {5 \tanh \left (d x +c \right )^{3}}{6}\right )}{d}\) | \(118\) |
risch | \(-\frac {a^{2} x}{2}-3 a b x -\frac {5 b^{2} x}{2}+\frac {{\mathrm e}^{2 d x +2 c} a^{2}}{8 d}+\frac {{\mathrm e}^{2 d x +2 c} a b}{4 d}+\frac {{\mathrm e}^{2 d x +2 c} b^{2}}{8 d}-\frac {{\mathrm e}^{-2 d x -2 c} a^{2}}{8 d}-\frac {{\mathrm e}^{-2 d x -2 c} a b}{4 d}-\frac {{\mathrm e}^{-2 d x -2 c} b^{2}}{8 d}-\frac {2 b \left (6 a \,{\mathrm e}^{4 d x +4 c}+9 b \,{\mathrm e}^{4 d x +4 c}+12 \,{\mathrm e}^{2 d x +2 c} a +12 b \,{\mathrm e}^{2 d x +2 c}+6 a +7 b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{3}}\) | \(193\) |
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Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (71) = 142\).
Time = 0.26 (sec) , antiderivative size = 291, normalized size of antiderivative = 3.68 \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{5} - 4 \, {\left (3 \, {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} d x + 12 \, a b + 14 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - 12 \, {\left (3 \, {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} d x + 12 \, a b + 14 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (30 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 9 \, a^{2} + 66 \, a b + 65 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} - 12 \, {\left (3 \, {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} d x + 12 \, a b + 14 \, b^{2}\right )} \cosh \left (d x + c\right ) + 3 \, {\left (5 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{4} + {\left (9 \, a^{2} + 66 \, a b + 65 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} + 20 \, a b + 10 \, b^{2}\right )} \sinh \left (d x + c\right )}{24 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right )\right )}} \]
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\[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2} \sinh ^{2}{\left (c + d x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (71) = 142\).
Time = 0.22 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.75 \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=-\frac {1}{8} \, a^{2} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{24} \, b^{2} {\left (\frac {60 \, {\left (d x + c\right )}}{d} + \frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {121 \, e^{\left (-2 \, d x - 2 \, c\right )} + 201 \, e^{\left (-4 \, d x - 4 \, c\right )} + 147 \, e^{\left (-6 \, d x - 6 \, c\right )} + 3}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )}\right )}}\right )} - \frac {1}{4} \, a b {\left (\frac {12 \, {\left (d x + c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {17 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )}\right )}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (71) = 142\).
Time = 0.34 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.70 \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {3 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 6 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 12 \, {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} {\left (d x + c\right )} + 3 \, {\left (2 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 10 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - a^{2} - 2 \, a b - b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - \frac {16 \, {\left (6 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 9 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 12 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 6 \, a b + 7 \, b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{24 \, d} \]
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Time = 1.87 (sec) , antiderivative size = 248, normalized size of antiderivative = 3.14 \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a+b\right )}^2}{8\,d}-x\,\left (\frac {a^2}{2}+3\,a\,b+\frac {5\,b^2}{2}\right )-\frac {\frac {2\,\left (3\,b^2+2\,a\,b\right )}{3\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (b^2+2\,a\,b\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (3\,b^2+2\,a\,b\right )}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}-\frac {2\,\left (3\,b^2+2\,a\,b\right )}{3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,{\left (a+b\right )}^2}{8\,d}-\frac {\frac {2\,\left (b^2+2\,a\,b\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,b^2+2\,a\,b\right )}{3\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1} \]
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