Integrand size = 23, antiderivative size = 67 \[ \int \text {sech}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {a^3 \tanh (c+d x)}{d}+\frac {a^2 b \tanh ^3(c+d x)}{d}+\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac {b^3 \tanh ^7(c+d x)}{7 d} \]
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Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3756, 200} \[ \int \text {sech}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {a^3 \tanh (c+d x)}{d}+\frac {a^2 b \tanh ^3(c+d x)}{d}+\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac {b^3 \tanh ^7(c+d x)}{7 d} \]
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Rule 200
Rule 3756
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (a+b x^2\right )^3 \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (a^3+3 a^2 b x^2+3 a b^2 x^4+b^3 x^6\right ) \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {a^3 \tanh (c+d x)}{d}+\frac {a^2 b \tanh ^3(c+d x)}{d}+\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac {b^3 \tanh ^7(c+d x)}{7 d} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \text {sech}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {a^3 \tanh (c+d x)}{d}+\frac {a^2 b \tanh ^3(c+d x)}{d}+\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac {b^3 \tanh ^7(c+d x)}{7 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(226\) vs. \(2(63)=126\).
Time = 21.44 (sec) , antiderivative size = 227, normalized size of antiderivative = 3.39
method | result | size |
derivativedivides | \(\frac {a^{3} \tanh \left (d x +c \right )+3 a^{2} b \left (-\frac {\sinh \left (d x +c \right )}{2 \cosh \left (d x +c \right )^{3}}+\frac {\left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )}{2}\right )+3 a \,b^{2} \left (-\frac {\sinh \left (d x +c \right )^{3}}{2 \cosh \left (d x +c \right )^{5}}-\frac {3 \sinh \left (d x +c \right )}{8 \cosh \left (d x +c \right )^{5}}+\frac {3 \left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{8}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{5}}{2 \cosh \left (d x +c \right )^{7}}-\frac {5 \sinh \left (d x +c \right )^{3}}{8 \cosh \left (d x +c \right )^{7}}-\frac {5 \sinh \left (d x +c \right )}{16 \cosh \left (d x +c \right )^{7}}+\frac {5 \left (\frac {16}{35}+\frac {\operatorname {sech}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {sech}\left (d x +c \right )^{4}}{35}+\frac {8 \operatorname {sech}\left (d x +c \right )^{2}}{35}\right ) \tanh \left (d x +c \right )}{16}\right )}{d}\) | \(227\) |
default | \(\frac {a^{3} \tanh \left (d x +c \right )+3 a^{2} b \left (-\frac {\sinh \left (d x +c \right )}{2 \cosh \left (d x +c \right )^{3}}+\frac {\left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )}{2}\right )+3 a \,b^{2} \left (-\frac {\sinh \left (d x +c \right )^{3}}{2 \cosh \left (d x +c \right )^{5}}-\frac {3 \sinh \left (d x +c \right )}{8 \cosh \left (d x +c \right )^{5}}+\frac {3 \left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{8}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{5}}{2 \cosh \left (d x +c \right )^{7}}-\frac {5 \sinh \left (d x +c \right )^{3}}{8 \cosh \left (d x +c \right )^{7}}-\frac {5 \sinh \left (d x +c \right )}{16 \cosh \left (d x +c \right )^{7}}+\frac {5 \left (\frac {16}{35}+\frac {\operatorname {sech}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {sech}\left (d x +c \right )^{4}}{35}+\frac {8 \operatorname {sech}\left (d x +c \right )^{2}}{35}\right ) \tanh \left (d x +c \right )}{16}\right )}{d}\) | \(227\) |
risch | \(-\frac {2 \left (105 a^{2} b \,{\mathrm e}^{12 d x +12 c}+21 a \,b^{2}+105 a \,b^{2} {\mathrm e}^{12 d x +12 c}+35 b^{3} {\mathrm e}^{12 d x +12 c}+35 a^{3}+105 \,{\mathrm e}^{4 d x +4 c} b^{3}+35 a^{2} b +42 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}+140 a^{2} b \,{\mathrm e}^{2 d x +2 c}+231 a \,b^{2} {\mathrm e}^{4 d x +4 c}+315 a^{2} b \,{\mathrm e}^{4 d x +4 c}+560 a^{2} b \,{\mathrm e}^{6 d x +6 c}+420 a \,b^{2} {\mathrm e}^{6 d x +6 c}+5 b^{3}+35 a^{3} {\mathrm e}^{12 d x +12 c}+665 a^{2} b \,{\mathrm e}^{8 d x +8 c}+315 a \,b^{2} {\mathrm e}^{8 d x +8 c}+420 a^{2} b \,{\mathrm e}^{10 d x +10 c}+210 a^{3} {\mathrm e}^{10 d x +10 c}+700 a^{3} {\mathrm e}^{6 d x +6 c}+525 a^{3} {\mathrm e}^{4 d x +4 c}+210 a^{3} {\mathrm e}^{2 d x +2 c}+175 b^{3} {\mathrm e}^{8 d x +8 c}+525 a^{3} {\mathrm e}^{8 d x +8 c}+210 a \,b^{2} {\mathrm e}^{10 d x +10 c}\right )}{35 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{7}}\) | \(348\) |
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Leaf count of result is larger than twice the leaf count of optimal. 786 vs. \(2 (63) = 126\).
Time = 0.27 (sec) , antiderivative size = 786, normalized size of antiderivative = 11.73 \[ \int \text {sech}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=-\frac {4 \, {\left ({\left (35 \, a^{3} + 70 \, a^{2} b + 63 \, a b^{2} + 20 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} + 6 \, {\left (35 \, a^{2} b + 42 \, a b^{2} + 15 \, b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + {\left (35 \, a^{3} + 70 \, a^{2} b + 63 \, a b^{2} + 20 \, b^{3}\right )} \sinh \left (d x + c\right )^{6} + 14 \, {\left (15 \, a^{3} + 20 \, a^{2} b + 9 \, a b^{2}\right )} \cosh \left (d x + c\right )^{4} + {\left (210 \, a^{3} + 280 \, a^{2} b + 126 \, a b^{2} + 15 \, {\left (35 \, a^{3} + 70 \, a^{2} b + 63 \, a b^{2} + 20 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, {\left (35 \, a^{2} b + 42 \, a b^{2} + 15 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 28 \, {\left (5 \, a^{2} b + 3 \, a b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 350 \, a^{3} + 280 \, a^{2} b + 210 \, a b^{2} + 7 \, {\left (75 \, a^{3} + 70 \, a^{2} b + 39 \, a b^{2} + 20 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + {\left (15 \, {\left (35 \, a^{3} + 70 \, a^{2} b + 63 \, a b^{2} + 20 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 525 \, a^{3} + 490 \, a^{2} b + 273 \, a b^{2} + 140 \, b^{3} + 84 \, {\left (15 \, a^{3} + 20 \, a^{2} b + 9 \, a b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (35 \, a^{2} b + 42 \, a b^{2} + 15 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 56 \, {\left (5 \, a^{2} b + 3 \, a b^{2}\right )} \cosh \left (d x + c\right )^{3} + 7 \, {\left (25 \, a^{2} b + 6 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}}{35 \, {\left (d \cosh \left (d x + c\right )^{8} + 8 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + d \sinh \left (d x + c\right )^{8} + 8 \, d \cosh \left (d x + c\right )^{6} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{2} + 2 \, d\right )} \sinh \left (d x + c\right )^{6} + 4 \, {\left (14 \, d \cosh \left (d x + c\right )^{3} + 9 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 28 \, d \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, d \cosh \left (d x + c\right )^{4} + 60 \, d \cosh \left (d x + c\right )^{2} + 14 \, d\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (7 \, d \cosh \left (d x + c\right )^{5} + 15 \, d \cosh \left (d x + c\right )^{3} + 7 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 56 \, d \cosh \left (d x + c\right )^{2} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{6} + 30 \, d \cosh \left (d x + c\right )^{4} + 42 \, d \cosh \left (d x + c\right )^{2} + 14 \, d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (2 \, d \cosh \left (d x + c\right )^{7} + 9 \, d \cosh \left (d x + c\right )^{5} + 14 \, d \cosh \left (d x + c\right )^{3} + 7 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 35 \, d\right )}} \]
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\[ \int \text {sech}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \operatorname {sech}^{2}{\left (c + d x \right )}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.06 \[ \int \text {sech}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {b^{3} \tanh \left (d x + c\right )^{7}}{7 \, d} + \frac {3 \, a b^{2} \tanh \left (d x + c\right )^{5}}{5 \, d} + \frac {a^{2} b \tanh \left (d x + c\right )^{3}}{d} + \frac {2 \, a^{3}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (63) = 126\).
Time = 0.43 (sec) , antiderivative size = 347, normalized size of antiderivative = 5.18 \[ \int \text {sech}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=-\frac {2 \, {\left (35 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 105 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 105 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 35 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 210 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 420 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 210 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 525 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 665 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 315 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 175 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 700 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 560 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 420 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 525 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 315 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 231 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 105 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 210 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 140 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 42 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 35 \, a^{3} + 35 \, a^{2} b + 21 \, a b^{2} + 5 \, b^{3}\right )}}{35 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}} \]
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Time = 2.04 (sec) , antiderivative size = 1050, normalized size of antiderivative = 15.67 \[ \int \text {sech}^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \]
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