Optimal. Leaf size=80 \[ \frac {b (2 a+3 b) \tanh ^5(c+d x)}{5 d}-\frac {(a+b) (a+3 b) \tanh ^3(c+d x)}{3 d}+\frac {(a+b)^2 \tanh (c+d x)}{d}-\frac {b^2 \tanh ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.08, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {4146, 373} \[ \frac {b (2 a+3 b) \tanh ^5(c+d x)}{5 d}-\frac {(a+b) (a+3 b) \tanh ^3(c+d x)}{3 d}+\frac {(a+b)^2 \tanh (c+d x)}{d}-\frac {b^2 \tanh ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 373
Rule 4146
Rubi steps
\begin {align*} \int \text {sech}^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \left (a+b-b x^2\right )^2 \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left ((a+b)^2+(-a-3 b) (a+b) x^2+b (2 a+3 b) x^4-b^2 x^6\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {(a+b)^2 \tanh (c+d x)}{d}-\frac {(a+b) (a+3 b) \tanh ^3(c+d x)}{3 d}+\frac {b (2 a+3 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 [A] time = 0.02, size = 144, normalized size = 1.80 \[ -\frac {a^2 \tanh ^3(c+d x)}{3 d}+\frac {a^2 \tanh (c+d x)}{d}+\frac {2 a b \tanh ^5(c+d x)}{5 d}-\frac {4 a b \tanh ^3(c+d x)}{3 d}+\frac {2 a b \tanh (c+d x)}{d}-\frac {b^2 \tanh ^7(c+d x)}{7 d}+\frac {3 b^2 \tanh ^5(c+d x)}{5 d}-\frac {b^2 \tanh ^3(c+d x)}{d}+\frac {b^2 \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.39, size = 677, normalized size = 8.46 \[ -\frac {8 \, {\left (2 \, {\left (35 \, a^{2} + 14 \, a b + 6 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 10 \, {\left (35 \, a^{2} + 14 \, a b + 6 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + {\left (35 \, a^{2} - 28 \, a b - 12 \, b^{2}\right )} \sinh \left (d x + c\right )^{5} + 14 \, {\left (25 \, a^{2} + 34 \, a b + 6 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (10 \, {\left (35 \, a^{2} - 28 \, a b - 12 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 105 \, a^{2} + 84 \, a b - 84 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (10 \, {\left (35 \, a^{2} + 14 \, a b + 6 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 21 \, {\left (25 \, a^{2} + 34 \, a b + 6 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 28 \, {\left (25 \, a^{2} + 46 \, a b + 24 \, b^{2}\right )} \cosh \left (d x + c\right ) + {\left (5 \, {\left (35 \, a^{2} - 28 \, a b - 12 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 63 \, {\left (5 \, a^{2} + 4 \, a b - 4 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 70 \, a^{2} + 112 \, a b + 168 \, b^{2}\right )} \sinh \left (d x + c\right )\right )}}{105 \, {\left (d \cosh \left (d x + c\right )^{9} + 9 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{8} + d \sinh \left (d x + c\right )^{9} + 7 \, d \cosh \left (d x + c\right )^{7} + {\left (36 \, d \cosh \left (d x + c\right )^{2} + 7 \, d\right )} \sinh \left (d x + c\right )^{7} + 7 \, {\left (12 \, d \cosh \left (d x + c\right )^{3} + 7 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 22 \, d \cosh \left (d x + c\right )^{5} + {\left (126 \, d \cosh \left (d x + c\right )^{4} + 147 \, d \cosh \left (d x + c\right )^{2} + 20 \, d\right )} \sinh \left (d x + c\right )^{5} + {\left (126 \, d \cosh \left (d x + c\right )^{5} + 245 \, d \cosh \left (d x + c\right )^{3} + 110 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 42 \, d \cosh \left (d x + c\right )^{3} + {\left (84 \, d \cosh \left (d x + c\right )^{6} + 245 \, d \cosh \left (d x + c\right )^{4} + 200 \, d \cosh \left (d x + c\right )^{2} + 28 \, d\right )} \sinh \left (d x + c\right )^{3} + {\left (36 \, d \cosh \left (d x + c\right )^{7} + 147 \, d \cosh \left (d x + c\right )^{5} + 220 \, d \cosh \left (d x + c\right )^{3} + 126 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 56 \, d \cosh \left (d x + c\right ) + {\left (9 \, d \cosh \left (d x + c\right )^{8} + 49 \, d \cosh \left (d x + c\right )^{6} + 100 \, d \cosh \left (d x + c\right )^{4} + 84 \, d \cosh \left (d x + c\right )^{2} + 14 \, d\right )} \sinh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 197, normalized size = 2.46 \[ -\frac {4 \, {\left (105 \, a^{2} e^{\left (10 \, d x + 10 \, c\right )} + 455 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 560 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 770 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} + 1400 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 840 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 630 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 1176 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 504 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 245 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 392 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 168 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 35 \, a^{2} + 56 \, a b + 24 \, b^{2}\right )}}{105 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 102, normalized size = 1.28 \[ \frac {a^{2} \left (\frac {2}{3}+\frac {\mathrm {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )+2 a b \left (\frac {8}{15}+\frac {\mathrm {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \mathrm {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )+b^{2} \left (\frac {16}{35}+\frac {\mathrm {sech}\left (d x +c \right )^{6}}{7}+\frac {6 \mathrm {sech}\left (d x +c \right )^{4}}{35}+\frac {8 \mathrm {sech}\left (d x +c \right )^{2}}{35}\right ) \tanh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 671, normalized size = 8.39 \[ \frac {32}{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 {21 \, 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 {35 \, 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 {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 )} + \frac {32}{15} \, a b {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\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 )}} + \frac {10 \, e^{\left (-4 \, d x - 4 \, c\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 )}} + \frac {1}{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 )} + \frac {4}{3} \, a^{2} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\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 )}} + \frac {1}{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 )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.42, size = 692, normalized size = 8.65 \[ -\frac {\frac {32\,a\,\left (a+2\,b\right )}{105\,d}+\frac {8\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{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\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{7\,d}+\frac {8\,a^2\,{\mathrm {e}}^{10\,c+10\,d\,x}}{7\,d}+\frac {16\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (3\,a^2+8\,a\,b+8\,b^2\right )}{7\,d}+\frac {32\,a\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a+2\,b\right )}{7\,d}+\frac {32\,a\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (a+2\,b\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\,a^2}{21\,d}+\frac {20\,a^2\,{\mathrm {e}}^{8\,c+8\,d\,x}}{21\,d}+\frac {8\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (3\,a^2+8\,a\,b+8\,b^2\right )}{7\,d}+\frac {32\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+2\,b\right )}{21\,d}+\frac {64\,a\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a+2\,b\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 {\frac {32\,a\,\left (a+2\,b\right )}{105\,d}+\frac {16\,a^2\,{\mathrm {e}}^{6\,c+6\,d\,x}}{21\,d}+\frac {16\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,a^2+8\,a\,b+8\,b^2\right )}{35\,d}+\frac {64\,a\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a+2\,b\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 (3\,a^2+8\,a\,b+8\,b^2\right )}{35\,d}+\frac {4\,a^2\,{\mathrm {e}}^{4\,c+4\,d\,x}}{7\,d}+\frac {32\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+2\,b\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}-\frac {4\,a^2}{21\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \operatorname {sech}^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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