Optimal. Leaf size=99 \[ \frac {a^3 \cosh ^3(c+d x)}{3 d}-\frac {a^2 (a-3 b) \cosh (c+d x)}{d}+\frac {b^2 (3 a-b) \text {sech}^3(c+d x)}{3 d}+\frac {3 a b (a-b) \text {sech}(c+d x)}{d}+\frac {b^3 \text {sech}^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.11, antiderivative size = 99, 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 = {4133, 448} \[ -\frac {a^2 (a-3 b) \cosh (c+d x)}{d}+\frac {a^3 \cosh ^3(c+d x)}{3 d}+\frac {b^2 (3 a-b) \text {sech}^3(c+d x)}{3 d}+\frac {3 a b (a-b) \text {sech}(c+d x)}{d}+\frac {b^3 \text {sech}^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 448
Rule 4133
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right )^3 \sinh ^3(c+d x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right ) \left (b+a x^2\right )^3}{x^6} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (a^2 (a-3 b)+\frac {b^3}{x^6}+\frac {(3 a-b) b^2}{x^4}+\frac {3 a (a-b) b}{x^2}-a^3 x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^2 (a-3 b) \cosh (c+d x)}{d}+\frac {a^3 \cosh ^3(c+d x)}{3 d}+\frac {3 a (a-b) b \text {sech}(c+d x)}{d}+\frac {(3 a-b) b^2 \text {sech}^3(c+d x)}{3 d}+\frac {b^3 \text {sech}^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 1.25, size = 119, normalized size = 1.20 \[ \frac {4 \text {sech}^5(c+d x) \left (a \cosh ^2(c+d x)+b\right )^3 \left (5 a^2 \cosh ^6(c+d x) (a \cosh (2 (c+d x))-5 a+18 b)+10 b^2 (3 a-b) \cosh ^2(c+d x)+90 a b (a-b) \cosh ^4(c+d x)+6 b^3\right )}{15 d (a \cosh (2 (c+d x))+a+2 b)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 403, normalized size = 4.07 \[ \frac {5 \, a^{3} \cosh \left (d x + c\right )^{8} + 5 \, a^{3} \sinh \left (d x + c\right )^{8} - 20 \, {\left (a^{3} - 9 \, a^{2} b\right )} \cosh \left (d x + c\right )^{6} + 20 \, {\left (7 \, a^{3} \cosh \left (d x + c\right )^{2} - a^{3} + 9 \, a^{2} b\right )} \sinh \left (d x + c\right )^{6} - 20 \, {\left (11 \, a^{3} - 90 \, a^{2} b + 36 \, a b^{2}\right )} \cosh \left (d x + c\right )^{4} + 10 \, {\left (35 \, a^{3} \cosh \left (d x + c\right )^{4} - 22 \, a^{3} + 180 \, a^{2} b - 72 \, a b^{2} - 30 \, {\left (a^{3} - 9 \, a^{2} b\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{4} - 425 \, a^{3} + 3960 \, a^{2} b - 1200 \, a b^{2} + 64 \, b^{3} - 20 \, {\left (31 \, a^{3} - 279 \, a^{2} b + 96 \, a b^{2} + 16 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 20 \, {\left (7 \, a^{3} \cosh \left (d x + c\right )^{6} - 15 \, {\left (a^{3} - 9 \, a^{2} b\right )} \cosh \left (d x + c\right )^{4} - 31 \, a^{3} + 279 \, a^{2} b - 96 \, a b^{2} - 16 \, b^{3} - 6 \, {\left (11 \, a^{3} - 90 \, a^{2} b + 36 \, a b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2}}{120 \, {\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 5 \, d \cosh \left (d x + c\right )^{3} + 5 \, {\left (2 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, d \cosh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 193, normalized size = 1.95 \[ \frac {5 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 60 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 180 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + \frac {16 \, {\left (45 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} - 45 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} + 60 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 20 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 48 \, b^{3}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 130, normalized size = 1.31 \[ \frac {a^{3} \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (d x +c \right )\right )}{3}\right ) \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {\sinh ^{2}\left (d x +c \right )}{\cosh \left (d x +c \right )}+\frac {2}{\cosh \left (d x +c \right )}\right )+3 a \,b^{2} \left (-\frac {\sinh ^{2}\left (d x +c \right )}{\cosh \left (d x +c \right )^{3}}-\frac {2}{3 \cosh \left (d x +c \right )^{3}}\right )+b^{3} \left (-\frac {\sinh ^{2}\left (d x +c \right )}{3 \cosh \left (d x +c \right )^{5}}-\frac {2}{15 \cosh \left (d x +c \right )^{5}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 489, normalized size = 4.94 \[ \frac {1}{24} \, a^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {e^{\left (-d x - c\right )}}{d} + \frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}\right )} - 2 \, a b^{2} {\left (\frac {3 \, e^{\left (-d x - 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 {2 \, e^{\left (-3 \, d x - 3 \, 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 {3 \, e^{\left (-5 \, d x - 5 \, 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 )}}\right )} - \frac {8}{15} \, b^{3} {\left (\frac {5 \, e^{\left (-3 \, d x - 3 \, 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 {2 \, e^{\left (-5 \, d x - 5 \, 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 {5 \, e^{\left (-7 \, d x - 7 \, 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 )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 348, normalized size = 3.52 \[ \frac {a^3\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}+\frac {a^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {3\,a^2\,{\mathrm {e}}^{-c-d\,x}\,\left (a-4\,b\right )}{8\,d}+\frac {8\,{\mathrm {e}}^{c+d\,x}\,\left (3\,a\,b^2-b^3\right )}{3\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {64\,b^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {8\,{\mathrm {e}}^{c+d\,x}\,\left (15\,a\,b^2-17\,b^3\right )}{15\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {32\,b^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}-\frac {6\,{\mathrm {e}}^{c+d\,x}\,\left (a\,b^2-a^2\,b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {3\,a^2\,{\mathrm {e}}^{c+d\,x}\,\left (a-4\,b\right )}{8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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