Integrand size = 23, antiderivative size = 94 \[ \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=(a+b)^3 x-\frac {(a+b)^3 \tanh (c+d x)}{d}-\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^2 (3 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^3 \tanh ^7(c+d x)}{7 d} \]
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Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3751, 472, 212} \[ \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=-\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^2 (3 a+b) \tanh ^5(c+d x)}{5 d}-\frac {(a+b)^3 \tanh (c+d x)}{d}+x (a+b)^3-\frac {b^3 \tanh ^7(c+d x)}{7 d} \]
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Rule 212
Rule 472
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^2 \left (a+b x^2\right )^3}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-(a+b)^3-b \left (3 a^2+3 a b+b^2\right ) x^2-b^2 (3 a+b) x^4-b^3 x^6+\frac {a^3+3 a^2 b+3 a b^2+b^3}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d} \\ & = -\frac {(a+b)^3 \tanh (c+d x)}{d}-\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^2 (3 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^3 \tanh ^7(c+d x)}{7 d}+\frac {(a+b)^3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = (a+b)^3 x-\frac {(a+b)^3 \tanh (c+d x)}{d}-\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^2 (3 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^3 \tanh ^7(c+d x)}{7 d} \\ \end{align*}
Time = 2.00 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.15 \[ \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {\tanh (c+d x) \left (-105 (a+b)^3-35 b \left (3 a^2+3 a b+b^2\right ) \tanh ^2(c+d x)-21 b^2 (3 a+b) \tanh ^4(c+d x)-15 b^3 \tanh ^6(c+d x)+\frac {105 (a+b)^3 \text {arctanh}\left (\sqrt {\tanh ^2(c+d x)}\right )}{\sqrt {\tanh ^2(c+d x)}}\right )}{105 d} \]
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Time = 0.13 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.74
method | result | size |
parallelrisch | \(-\frac {15 \tanh \left (d x +c \right )^{7} b^{3}+63 \tanh \left (d x +c \right )^{5} a \,b^{2}+21 b^{3} \tanh \left (d x +c \right )^{5}+105 \tanh \left (d x +c \right )^{3} a^{2} b +105 a \,b^{2} \tanh \left (d x +c \right )^{3}+35 b^{3} \tanh \left (d x +c \right )^{3}-105 a^{3} d x -315 a^{2} b d x -315 a \,b^{2} d x -105 b^{3} d x +105 a^{3} \tanh \left (d x +c \right )+315 a^{2} b \tanh \left (d x +c \right )+315 a \,b^{2} \tanh \left (d x +c \right )+105 b^{3} \tanh \left (d x +c \right )}{105 d}\) | \(164\) |
derivativedivides | \(\frac {-3 a^{2} b \tanh \left (d x +c \right )-3 a \,b^{2} \tanh \left (d x +c \right )-\frac {3 \tanh \left (d x +c \right )^{5} a \,b^{2}}{5}-\tanh \left (d x +c \right )^{3} a^{2} b -a \,b^{2} \tanh \left (d x +c \right )^{3}-\frac {b^{3} \tanh \left (d x +c \right )^{5}}{5}-\frac {b^{3} \tanh \left (d x +c \right )^{3}}{3}-a^{3} \tanh \left (d x +c \right )-b^{3} \tanh \left (d x +c \right )-\frac {\tanh \left (d x +c \right )^{7} b^{3}}{7}+\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (\tanh \left (d x +c \right )+1\right )}{2}-\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}}{d}\) | \(193\) |
default | \(\frac {-3 a^{2} b \tanh \left (d x +c \right )-3 a \,b^{2} \tanh \left (d x +c \right )-\frac {3 \tanh \left (d x +c \right )^{5} a \,b^{2}}{5}-\tanh \left (d x +c \right )^{3} a^{2} b -a \,b^{2} \tanh \left (d x +c \right )^{3}-\frac {b^{3} \tanh \left (d x +c \right )^{5}}{5}-\frac {b^{3} \tanh \left (d x +c \right )^{3}}{3}-a^{3} \tanh \left (d x +c \right )-b^{3} \tanh \left (d x +c \right )-\frac {\tanh \left (d x +c \right )^{7} b^{3}}{7}+\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (\tanh \left (d x +c \right )+1\right )}{2}-\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}}{d}\) | \(193\) |
parts | \(\frac {a^{3} \left (-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {b^{3} \left (-\frac {\tanh \left (d x +c \right )^{7}}{7}-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {3 a^{2} b \left (-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {3 a \,b^{2} \left (-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}\) | \(218\) |
risch | \(a^{3} x +3 b \,a^{2} x +3 a \,b^{2} x +b^{3} x +\frac {\frac {46 a \,b^{2}}{5}+2 a^{3}+12 a^{2} b \,{\mathrm e}^{12 d x +12 c}+72 a \,b^{2} {\mathrm e}^{10 d x +10 c}+60 a^{2} b \,{\mathrm e}^{10 d x +10 c}+18 a \,b^{2} {\mathrm e}^{12 d x +12 c}+\frac {232 \,{\mathrm e}^{4 d x +4 c} b^{3}}{5}+8 a^{2} b +\frac {232 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}}{5}+44 a^{2} b \,{\mathrm e}^{2 d x +2 c}+\frac {606 a \,b^{2} {\mathrm e}^{4 d x +4 c}}{5}+108 a^{2} b \,{\mathrm e}^{4 d x +4 c}+152 a^{2} b \,{\mathrm e}^{6 d x +6 c}+176 a \,b^{2} {\mathrm e}^{6 d x +6 c}+\frac {352 b^{3}}{105}+128 a^{2} b \,{\mathrm e}^{8 d x +8 c}+146 a \,b^{2} {\mathrm e}^{8 d x +8 c}+8 b^{3} {\mathrm e}^{12 d x +12 c}+40 a^{3} {\mathrm e}^{6 d x +6 c}+\frac {176 \,{\mathrm e}^{6 d x +6 c} b^{3}}{3}+30 a^{3} {\mathrm e}^{4 d x +4 c}+12 a^{3} {\mathrm e}^{2 d x +2 c}+\frac {232 \,{\mathrm e}^{2 d x +2 c} b^{3}}{15}+24 b^{3} {\mathrm e}^{10 d x +10 c}+\frac {176 b^{3} {\mathrm e}^{8 d x +8 c}}{3}+12 a^{3} {\mathrm e}^{10 d x +10 c}+2 a^{3} {\mathrm e}^{12 d x +12 c}+30 a^{3} {\mathrm e}^{8 d x +8 c}}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{7}}\) | \(415\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1036 vs. \(2 (88) = 176\).
Time = 0.27 (sec) , antiderivative size = 1036, normalized size of antiderivative = 11.02 \[ \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (82) = 164\).
Time = 0.22 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.04 \[ \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\begin {cases} a^{3} x - \frac {a^{3} \tanh {\left (c + d x \right )}}{d} + 3 a^{2} b x - \frac {a^{2} b \tanh ^{3}{\left (c + d x \right )}}{d} - \frac {3 a^{2} b \tanh {\left (c + d x \right )}}{d} + 3 a b^{2} x - \frac {3 a b^{2} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {a b^{2} \tanh ^{3}{\left (c + d x \right )}}{d} - \frac {3 a b^{2} \tanh {\left (c + d x \right )}}{d} + b^{3} x - \frac {b^{3} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac {b^{3} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {b^{3} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {b^{3} \tanh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\left (c \right )}\right )^{3} \tanh ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (88) = 176\).
Time = 0.21 (sec) , antiderivative size = 400, normalized size of antiderivative = 4.26 \[ \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {1}{105} \, b^{3} {\left (105 \, x + \frac {105 \, c}{d} - \frac {8 \, {\left (203 \, e^{\left (-2 \, d x - 2 \, c\right )} + 609 \, e^{\left (-4 \, d x - 4 \, c\right )} + 770 \, e^{\left (-6 \, d x - 6 \, c\right )} + 770 \, e^{\left (-8 \, d x - 8 \, c\right )} + 315 \, e^{\left (-10 \, d x - 10 \, c\right )} + 105 \, e^{\left (-12 \, d x - 12 \, c\right )} + 44\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}}\right )} + \frac {1}{5} \, a b^{2} {\left (15 \, x + \frac {15 \, c}{d} - \frac {2 \, {\left (70 \, e^{\left (-2 \, d x - 2 \, c\right )} + 140 \, e^{\left (-4 \, d x - 4 \, c\right )} + 90 \, e^{\left (-6 \, d x - 6 \, c\right )} + 45 \, e^{\left (-8 \, d x - 8 \, c\right )} + 23\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + a^{2} b {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + a^{3} {\left (x + \frac {c}{d} - \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (88) = 176\).
Time = 0.43 (sec) , antiderivative size = 418, normalized size of antiderivative = 4.45 \[ \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {105 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (105 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 630 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 945 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 420 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 630 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 3150 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 3780 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1260 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 1575 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 6720 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 7665 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 3080 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 2100 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 7980 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 9240 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 3080 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 1575 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 5670 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 6363 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2436 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 630 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2310 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 2436 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 812 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}}}{105 \, d} \]
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Time = 1.87 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.13 \[ \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=x\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )-\frac {\mathrm {tanh}\left (c+d\,x\right )\,{\left (a+b\right )}^3}{d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^3\,\left (3\,a^2\,b+3\,a\,b^2+b^3\right )}{3\,d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^5\,\left (b^3+3\,a\,b^2\right )}{5\,d}-\frac {b^3\,{\mathrm {tanh}\left (c+d\,x\right )}^7}{7\,d} \]
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