Optimal. Leaf size=74 \[ \frac {3 b^2 (a+b) \tanh ^5(c+d x)}{5 d}-\frac {b (a+b)^2 \tanh ^3(c+d x)}{d}+\frac {(a+b)^3 \tanh (c+d x)}{d}-\frac {b^3 \tanh ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.08, antiderivative size = 74, 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, 194} \[ \frac {3 b^2 (a+b) \tanh ^5(c+d x)}{5 d}-\frac {b (a+b)^2 \tanh ^3(c+d x)}{d}+\frac {(a+b)^3 \tanh (c+d x)}{d}-\frac {b^3 \tanh ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 194
Rule 4146
Rubi steps
\begin {align*} \int \text {sech}^2(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b-b x^2\right )^3 \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^3 \left (1+\frac {b \left (3 a^2+3 a b+b^2\right )}{a^3}\right )-3 b (a+b)^2 x^2+3 b^2 (a+b) x^4-b^3 x^6\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {(a+b)^3 \tanh (c+d x)}{d}-\frac {b (a+b)^2 \tanh ^3(c+d x)}{d}+\frac {3 b^2 (a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^3 \tanh ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [B] time = 1.49, size = 319, normalized size = 4.31 \[ \frac {\text {sech}(c) \text {sech}(c+d x) \left (525 a^3 \sinh (2 c+3 d x)-210 a^3 \sinh (4 c+3 d x)+210 a^3 \sinh (4 c+5 d x)-35 a^3 \sinh (6 c+5 d x)+35 a^3 \sinh (6 c+7 d x)-35 a \left (15 a^2+26 a b+16 b^2\right ) \sinh (2 c+d x)+1260 a^2 b \sinh (2 c+3 d x)-210 a^2 b \sinh (4 c+3 d x)+490 a^2 b \sinh (4 c+5 d x)+70 a^2 b \sinh (6 c+7 d x)+140 \left (5 a^3+11 a^2 b+10 a b^2+4 b^3\right ) \sinh (d x)+1176 a b^2 \sinh (2 c+3 d x)+392 a b^2 \sinh (4 c+5 d x)+56 a b^2 \sinh (6 c+7 d x)+336 b^3 \sinh (2 c+3 d x)+112 b^3 \sinh (4 c+5 d x)+16 b^3 \sinh (6 c+7 d x)\right ) \left (a+b \text {sech}^2(c+d x)\right )^3}{280 d (a \cosh (2 (c+d x))+a+2 b)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 816, normalized size = 11.03 \[ -\frac {4 \, {\left ({\left (35 \, a^{3} + 35 \, a^{2} b + 28 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} - 6 \, {\left (35 \, a^{2} b + 28 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + {\left (35 \, a^{3} + 35 \, a^{2} b + 28 \, a b^{2} + 8 \, b^{3}\right )} \sinh \left (d x + c\right )^{6} + 14 \, {\left (15 \, a^{3} + 25 \, a^{2} b + 14 \, a b^{2} + 4 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + {\left (210 \, a^{3} + 350 \, a^{2} b + 196 \, a b^{2} + 56 \, b^{3} + 15 \, {\left (35 \, a^{3} + 35 \, a^{2} b + 28 \, a b^{2} + 8 \, 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 + 28 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 28 \, {\left (5 \, a^{2} b + 7 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 350 \, a^{3} + 770 \, a^{2} b + 700 \, a b^{2} + 280 \, b^{3} + 7 \, {\left (75 \, a^{3} + 155 \, a^{2} b + 124 \, a b^{2} + 24 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + {\left (15 \, {\left (35 \, a^{3} + 35 \, a^{2} b + 28 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 525 \, a^{3} + 1085 \, a^{2} b + 868 \, a b^{2} + 168 \, b^{3} + 84 \, {\left (15 \, a^{3} + 25 \, a^{2} b + 14 \, a b^{2} + 4 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} - 2 \, {\left (3 \, {\left (35 \, a^{2} b + 28 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 56 \, {\left (5 \, a^{2} b + 7 \, a b^{2} + 2 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 7 \, {\left (25 \, a^{2} b + 44 \, a b^{2} + 24 \, 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 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 302, normalized size = 4.08 \[ -\frac {2 \, {\left (35 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 210 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 210 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 525 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 910 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 560 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 700 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 1540 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 1400 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 560 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 525 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 1260 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 1176 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 336 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 210 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 490 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 392 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 112 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 35 \, a^{3} + 70 \, a^{2} b + 56 \, a b^{2} + 16 \, b^{3}\right )}}{35 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.57, size = 116, normalized size = 1.57 \[ \frac {a^{3} \tanh \left (d x +c \right )+3 a^{2} b \left (\frac {2}{3}+\frac {\mathrm {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )+3 a \,b^{2} \left (\frac {8}{15}+\frac {\mathrm {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \mathrm {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )+b^{3} \left (\frac {16}{35}+\frac {\mathrm {sech}\left (d x +c \right )^{6}}{7}+\frac {6 \mathrm {sech}\left (d x +c \right )^{4}}{35}+\frac {8 \mathrm {sech}\left (d x +c \right )^{2}}{35}\right ) \tanh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 695, normalized size = 9.39 \[ \frac {32}{35} \, b^{3} {\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 {16}{5} \, a b^{2} {\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 )} + 4 \, a^{2} b {\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 )} + \frac {2 \, a^{3}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.50, size = 978, normalized size = 13.22 \[ -\frac {\frac {2\,\left (5\,a^3+18\,a^2\,b+24\,a\,b^2+16\,b^3\right )}{35\,d}+\frac {2\,a^3\,{\mathrm {e}}^{6\,c+6\,d\,x}}{7\,d}+\frac {6\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{35\,d}+\frac {6\,a^2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a+2\,b\right )}{7\,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 {\frac {2\,a^2\,\left (a+2\,b\right )}{7\,d}+\frac {2\,a^3\,{\mathrm {e}}^{2\,c+2\,d\,x}}{7\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {\frac {2\,a\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{35\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (5\,a^3+18\,a^2\,b+24\,a\,b^2+16\,b^3\right )}{35\,d}+\frac {2\,a^3\,{\mathrm {e}}^{8\,c+8\,d\,x}}{7\,d}+\frac {12\,a\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{35\,d}+\frac {8\,a^2\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a+2\,b\right )}{7\,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 {2\,a^3}{7\,d}+\frac {8\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (5\,a^3+18\,a^2\,b+24\,a\,b^2+16\,b^3\right )}{7\,d}+\frac {2\,a^3\,{\mathrm {e}}^{12\,c+12\,d\,x}}{7\,d}+\frac {6\,a\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{7\,d}+\frac {6\,a\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{7\,d}+\frac {12\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+2\,b\right )}{7\,d}+\frac {12\,a^2\,{\mathrm {e}}^{10\,c+10\,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 {2\,a\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{35\,d}+\frac {2\,a^3\,{\mathrm {e}}^{4\,c+4\,d\,x}}{7\,d}+\frac {4\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+2\,b\right )}{7\,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 {2\,a^2\,\left (a+2\,b\right )}{7\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (5\,a^3+18\,a^2\,b+24\,a\,b^2+16\,b^3\right )}{7\,d}+\frac {2\,a^3\,{\mathrm {e}}^{10\,c+10\,d\,x}}{7\,d}+\frac {2\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{7\,d}+\frac {4\,a\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{7\,d}+\frac {10\,a^2\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (a+2\,b\right )}{7\,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 {2\,a^3}{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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3} \operatorname {sech}^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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