Optimal. Leaf size=77 \[ -\frac {a^2 \tanh ^3(c+d x)}{3 d}-\frac {a^2 \tanh (c+d x)}{d}+a^2 x+\frac {b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^2 \tanh ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.10, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4141, 1802, 206} \[ -\frac {a^2 \tanh ^3(c+d x)}{3 d}-\frac {a^2 \tanh (c+d x)}{d}+a^2 x+\frac {b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^2 \tanh ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 1802
Rule 4141
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right )^2 \tanh ^4(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (a+b \left (1-x^2\right )\right )^2}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a^2-a^2 x^2+b (2 a+b) x^4-b^2 x^6+\frac {a^2}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a^2 \tanh (c+d x)}{d}-\frac {a^2 \tanh ^3(c+d x)}{3 d}+\frac {b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^2 \tanh ^7(c+d x)}{7 d}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a^2 x-\frac {a^2 \tanh (c+d x)}{d}-\frac {a^2 \tanh ^3(c+d x)}{3 d}+\frac {b (2 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^2 \tanh ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [B] time = 1.18, size = 395, normalized size = 5.13 \[ \frac {\text {sech}(c) \text {sech}^7(c+d x) \left (4480 a^2 \sinh (2 c+d x)-3780 a^2 \sinh (2 c+3 d x)+2100 a^2 \sinh (4 c+3 d x)-1540 a^2 \sinh (4 c+5 d x)+420 a^2 \sinh (6 c+5 d x)-280 a^2 \sinh (6 c+7 d x)+3675 a^2 d x \cosh (2 c+d x)+2205 a^2 d x \cosh (2 c+3 d x)+2205 a^2 d x \cosh (4 c+3 d x)+735 a^2 d x \cosh (4 c+5 d x)+735 a^2 d x \cosh (6 c+5 d x)+105 a^2 d x \cosh (6 c+7 d x)+105 a^2 d x \cosh (8 c+7 d x)-5320 a^2 \sinh (d x)+3675 a^2 d x \cosh (d x)-1260 a b \sinh (2 c+d x)+924 a b \sinh (2 c+3 d x)-840 a b \sinh (4 c+3 d x)+168 a b \sinh (4 c+5 d x)-420 a b \sinh (6 c+5 d x)+84 a b \sinh (6 c+7 d x)+1680 a b \sinh (d x)+420 b^2 \sinh (2 c+d x)-168 b^2 \sinh (2 c+3 d x)-420 b^2 \sinh (4 c+3 d x)+84 b^2 \sinh (4 c+5 d x)+12 b^2 \sinh (6 c+7 d x)+840 b^2 \sinh (d x)\right )}{13440 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 721, normalized size = 9.36 \[ \frac {{\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right )^{7} + 7 \, {\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} - 2 \, {\left (70 \, a^{2} - 21 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{7} + 7 \, {\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} - 14 \, {\left (3 \, {\left (70 \, a^{2} - 21 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 40 \, a^{2} + 9 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{5} + 35 \, {\left ({\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 21 \, {\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - 14 \, {\left (5 \, {\left (70 \, a^{2} - 21 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 10 \, {\left (40 \, a^{2} + 9 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 60 \, a^{2} - 3 \, a b + 21 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + 7 \, {\left (3 \, {\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 10 \, {\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 9 \, {\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 35 \, {\left (105 \, a^{2} d x + 140 \, a^{2} - 42 \, a b - 6 \, b^{2}\right )} \cosh \left (d x + c\right ) - 14 \, {\left ({\left (70 \, a^{2} - 21 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{6} + 5 \, {\left (40 \, a^{2} + 9 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 9 \, {\left (20 \, a^{2} - a b + 7 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 30 \, a^{2} - 15 \, a b - 45 \, b^{2}\right )} \sinh \left (d x + c\right )}{105 \, {\left (d \cosh \left (d x + c\right )^{7} + 7 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + 7 \, d \cosh \left (d x + c\right )^{5} + 35 \, {\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 21 \, d \cosh \left (d x + c\right )^{3} + 7 \, {\left (3 \, d \cosh \left (d x + c\right )^{5} + 10 \, d \cosh \left (d x + c\right )^{3} + 9 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 35 \, d \cosh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 275, normalized size = 3.57 \[ \frac {105 \, a^{2} d x + \frac {4 \, {\left (105 \, a^{2} e^{\left (12 \, d x + 12 \, c\right )} - 105 \, a b e^{\left (12 \, d x + 12 \, c\right )} + 525 \, a^{2} e^{\left (10 \, d x + 10 \, c\right )} - 210 \, a b e^{\left (10 \, d x + 10 \, c\right )} - 105 \, b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1120 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} - 315 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 105 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 1330 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} - 420 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 210 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 945 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 231 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 42 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 385 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 42 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 21 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 70 \, a^{2} - 21 \, a b - 3 \, b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.44, size = 181, normalized size = 2.35 \[ \frac {a^{2} \left (d x +c -\tanh \left (d x +c \right )-\frac {\left (\tanh ^{3}\left (d x +c \right )\right )}{3}\right )+2 a b \left (-\frac {\sinh ^{3}\left (d x +c \right )}{2 \cosh \left (d x +c \right )^{5}}-\frac {3 \sinh \left (d x +c \right )}{8 \cosh \left (d x +c \right )^{5}}+\frac {3 \left (\frac {8}{15}+\frac {\mathrm {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \mathrm {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{8}\right )+b^{2} \left (-\frac {\sinh ^{3}\left (d x +c \right )}{4 \cosh \left (d x +c \right )^{7}}-\frac {\sinh \left (d x +c \right )}{8 \cosh \left (d x +c \right )^{7}}+\frac {\left (\frac {16}{35}+\frac {\mathrm {sech}\left (d x +c \right )^{6}}{7}+\frac {6 \mathrm {sech}\left (d x +c \right )^{4}}{35}+\frac {8 \mathrm {sech}\left (d x +c \right )^{2}}{35}\right ) \tanh \left (d x +c \right )}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.55, size = 649, normalized size = 8.43 \[ \frac {2 \, a b \tanh \left (d x + c\right )^{5}}{5 \, d} + \frac {1}{3} \, a^{2} {\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 )} + \frac {4}{35} \, b^{2} {\left (\frac {7 \, e^{\left (-2 \, d x - 2 \, c\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 )}} - \frac {14 \, e^{\left (-4 \, d x - 4 \, c\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 )}} + \frac {70 \, e^{\left (-6 \, d x - 6 \, c\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 )}} - \frac {35 \, e^{\left (-8 \, d x - 8 \, c\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 )}} + \frac {35 \, e^{\left (-10 \, d x - 10 \, c\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 )}} + \frac {1}{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 )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 1022, normalized size = 13.27 \[ \frac {\frac {4\,\left (7\,a^2+a\,b+8\,b^2\right )}{105\,d}-\frac {4\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (a\,b-a^2\right )}{7\,d}-\frac {16\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (-2\,a^2+a\,b+3\,b^2\right )}{35\,d}+\frac {16\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (2\,a^2+a\,b-b^2\right )}{21\,d}+\frac {8\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (7\,a^2+a\,b+8\,b^2\right )}{35\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1}-\frac {\frac {4\,\left (-2\,a^2+a\,b+3\,b^2\right )}{35\,d}+\frac {4\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a\,b-a^2\right )}{7\,d}-\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (2\,a^2+a\,b-b^2\right )}{7\,d}-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (7\,a^2+a\,b+8\,b^2\right )}{35\,d}}{4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}+a^2\,x+\frac {\frac {4\,\left (7\,a^2+a\,b+8\,b^2\right )}{105\,d}-\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a\,b-a^2\right )}{7\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a^2+a\,b-b^2\right )}{21\,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 {\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a^2+a\,b-b^2\right )}{7\,d}-\frac {4\,{\mathrm {e}}^{12\,c+12\,d\,x}\,\left (a\,b-a^2\right )}{7\,d}-\frac {4\,\left (a\,b-a^2\right )}{7\,d}-\frac {16\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (-2\,a^2+a\,b+3\,b^2\right )}{7\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (7\,a^2+a\,b+8\,b^2\right )}{7\,d}+\frac {8\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (2\,a^2+a\,b-b^2\right )}{7\,d}+\frac {4\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (7\,a^2+a\,b+8\,b^2\right )}{7\,d}}{7\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}+35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}+7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}+1}+\frac {\frac {4\,\left (2\,a^2+a\,b-b^2\right )}{21\,d}-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a\,b-a^2\right )}{7\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}+\frac {\frac {4\,\left (2\,a^2+a\,b-b^2\right )}{21\,d}-\frac {4\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (a\,b-a^2\right )}{7\,d}-\frac {8\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (-2\,a^2+a\,b+3\,b^2\right )}{7\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (7\,a^2+a\,b+8\,b^2\right )}{21\,d}+\frac {20\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (2\,a^2+a\,b-b^2\right )}{21\,d}+\frac {8\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (7\,a^2+a\,b+8\,b^2\right )}{21\,d}}{6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1}-\frac {4\,\left (a\,b-a^2\right )}{7\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \tanh ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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