Optimal. Leaf size=105 \[ \frac {b^2 (3 a+4 b) \text {sech}^3(c+d x)}{3 d}+\frac {(a+b)^3 \cosh ^3(c+d x)}{3 d}-\frac {(a+b)^2 (a+4 b) \cosh (c+d x)}{d}-\frac {3 b (a+b) (a+2 b) \text {sech}(c+d x)}{d}-\frac {b^3 \text {sech}^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.12, antiderivative size = 105, 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 = {3664, 448} \[ \frac {b^2 (3 a+4 b) \text {sech}^3(c+d x)}{3 d}+\frac {(a+b)^3 \cosh ^3(c+d x)}{3 d}-\frac {(a+b)^2 (a+4 b) \cosh (c+d x)}{d}-\frac {3 b (a+b) (a+2 b) \text {sech}(c+d x)}{d}-\frac {b^3 \text {sech}^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 448
Rule 3664
Rubi steps
\begin {align*} \int \sinh ^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (-1+x^2\right ) \left (a+b-b x^2\right )^3}{x^4} \, dx,x,\text {sech}(c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (3 (-a-2 b) b (a+b)-\frac {(a+b)^3}{x^4}+\frac {(a+b)^2 (a+4 b)}{x^2}+b^2 (3 a+4 b) x^2-b^3 x^4\right ) \, dx,x,\text {sech}(c+d x)\right )}{d}\\ &=-\frac {(a+b)^2 (a+4 b) \cosh (c+d x)}{d}+\frac {(a+b)^3 \cosh ^3(c+d x)}{3 d}-\frac {3 b (a+b) (a+2 b) \text {sech}(c+d x)}{d}+\frac {b^2 (3 a+4 b) \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 = 0.30, size = 91, normalized size = 0.87 \[ \frac {20 b^2 (3 a+4 b) \text {sech}^3(c+d x)-45 (a+b)^2 (a+5 b) \cosh (c+d x)+5 (a+b)^3 \cosh (3 (c+d x))-180 b (a+b) (a+2 b) \text {sech}(c+d x)-12 b^3 \text {sech}^5(c+d x)}{60 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 540, normalized size = 5.14 \[ \frac {5 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{8} + 5 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sinh \left (d x + c\right )^{8} - 20 \, {\left (a^{3} + 12 \, a^{2} b + 21 \, a b^{2} + 10 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} - 20 \, {\left (a^{3} + 12 \, a^{2} b + 21 \, a b^{2} + 10 \, b^{3} - 7 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{6} - 20 \, {\left (11 \, a^{3} + 123 \, a^{2} b + 249 \, a b^{2} + 137 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 10 \, {\left (35 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{4} - 22 \, a^{3} - 246 \, a^{2} b - 498 \, a b^{2} - 274 \, b^{3} - 30 \, {\left (a^{3} + 12 \, a^{2} b + 21 \, a b^{2} + 10 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{4} - 425 \, a^{3} - 5235 \, a^{2} b - 10395 \, a b^{2} - 5649 \, b^{3} - 20 \, {\left (31 \, a^{3} + 372 \, a^{2} b + 747 \, a b^{2} + 390 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 20 \, {\left (7 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{6} - 15 \, {\left (a^{3} + 12 \, a^{2} b + 21 \, a b^{2} + 10 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} - 31 \, a^{3} - 372 \, a^{2} b - 747 \, a b^{2} - 390 \, b^{3} - 6 \, {\left (11 \, a^{3} + 123 \, a^{2} b + 249 \, a b^{2} + 137 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2}}{120 \, {\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 5 \, d \cosh \left (d x + c\right )^{3} + 5 \, {\left (2 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, d \cosh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.66, size = 442, normalized size = 4.21 \[ -\frac {5 \, {\left (9 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 63 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 99 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 45 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )} - 5 \, {\left (a^{3} e^{\left (3 \, d x + 48 \, c\right )} + 3 \, a^{2} b e^{\left (3 \, d x + 48 \, c\right )} + 3 \, a b^{2} e^{\left (3 \, d x + 48 \, c\right )} + b^{3} e^{\left (3 \, d x + 48 \, c\right )} - 9 \, a^{3} e^{\left (d x + 46 \, c\right )} - 63 \, a^{2} b e^{\left (d x + 46 \, c\right )} - 99 \, a b^{2} e^{\left (d x + 46 \, c\right )} - 45 \, b^{3} e^{\left (d x + 46 \, c\right )}\right )} e^{\left (-45 \, c\right )} + \frac {16 \, {\left (45 \, a^{2} b e^{\left (9 \, d x + 9 \, c\right )} + 135 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} + 90 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} + 180 \, a^{2} b e^{\left (7 \, d x + 7 \, c\right )} + 480 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 280 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} + 270 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} + 690 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 428 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 180 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 480 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 280 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 45 \, a^{2} b e^{\left (d x + c\right )} + 135 \, a b^{2} e^{\left (d x + c\right )} + 90 \, b^{3} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.32, size = 239, normalized size = 2.28 \[ \frac {a^{3} \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (d x +c \right )\right )}{3}\right ) \cosh \left (d x +c \right )+3 a^{2} b \left (\frac {\sinh ^{4}\left (d x +c \right )}{3 \cosh \left (d x +c \right )}-\frac {4 \left (\sinh ^{2}\left (d x +c \right )\right )}{3 \cosh \left (d x +c \right )}-\frac {8}{3 \cosh \left (d x +c \right )}\right )+3 a \,b^{2} \left (\frac {\sinh ^{6}\left (d x +c \right )}{3 \cosh \left (d x +c \right )^{3}}-\frac {2 \left (\sinh ^{4}\left (d x +c \right )\right )}{\cosh \left (d x +c \right )^{3}}-\frac {8 \left (\sinh ^{2}\left (d x +c \right )\right )}{\cosh \left (d x +c \right )^{3}}-\frac {16}{3 \cosh \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sinh ^{8}\left (d x +c \right )}{3 \cosh \left (d x +c \right )^{5}}-\frac {8 \left (\sinh ^{6}\left (d x +c \right )\right )}{3 \cosh \left (d x +c \right )^{5}}-\frac {16 \left (\sinh ^{4}\left (d x +c \right )\right )}{\cosh \left (d x +c \right )^{5}}-\frac {64 \left (\sinh ^{2}\left (d x +c \right )\right )}{3 \cosh \left (d x +c \right )^{5}}-\frac {128}{15 \cosh \left (d x +c \right )^{5}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 439, normalized size = 4.18 \[ -\frac {1}{120} \, b^{3} {\left (\frac {5 \, {\left (45 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d} + \frac {200 \, e^{\left (-2 \, d x - 2 \, c\right )} + 2515 \, e^{\left (-4 \, d x - 4 \, c\right )} + 6680 \, e^{\left (-6 \, d x - 6 \, c\right )} + 9073 \, e^{\left (-8 \, d x - 8 \, c\right )} + 5600 \, e^{\left (-10 \, d x - 10 \, c\right )} + 1665 \, e^{\left (-12 \, d x - 12 \, c\right )} - 5}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + 5 \, e^{\left (-5 \, d x - 5 \, c\right )} + 10 \, e^{\left (-7 \, d x - 7 \, c\right )} + 10 \, e^{\left (-9 \, d x - 9 \, c\right )} + 5 \, e^{\left (-11 \, d x - 11 \, c\right )} + e^{\left (-13 \, d x - 13 \, c\right )}\right )}}\right )} - \frac {1}{8} \, a b^{2} {\left (\frac {33 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {30 \, e^{\left (-2 \, d x - 2 \, c\right )} + 240 \, e^{\left (-4 \, d x - 4 \, c\right )} + 322 \, e^{\left (-6 \, d x - 6 \, c\right )} + 177 \, e^{\left (-8 \, d x - 8 \, c\right )} - 1}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )} + 3 \, e^{\left (-7 \, d x - 7 \, c\right )} + e^{\left (-9 \, d x - 9 \, c\right )}\right )}}\right )} - \frac {1}{8} \, a^{2} b {\left (\frac {21 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {20 \, e^{\left (-2 \, d x - 2 \, c\right )} + 69 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} + \frac {1}{24} \, a^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 361, normalized size = 3.44 \[ \frac {{\mathrm {e}}^{-3\,c-3\,d\,x}\,{\left (a+b\right )}^3}{24\,d}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,{\left (a+b\right )}^3}{24\,d}+\frac {8\,{\mathrm {e}}^{c+d\,x}\,\left (4\,b^3+3\,a\,b^2\right )}{3\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}+\frac {64\,b^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {8\,{\mathrm {e}}^{c+d\,x}\,\left (32\,b^3+15\,a\,b^2\right )}{15\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {32\,b^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}-\frac {3\,{\mathrm {e}}^{c+d\,x}\,{\left (a+b\right )}^2\,\left (a+5\,b\right )}{8\,d}-\frac {6\,{\mathrm {e}}^{c+d\,x}\,\left (a^2\,b+3\,a\,b^2+2\,b^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {3\,{\mathrm {e}}^{-c-d\,x}\,{\left (a+b\right )}^2\,\left (a+5\,b\right )}{8\,d} \]
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 \[ \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \sinh ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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