Optimal. Leaf size=351 \[ \frac {15 a^2 b \text {ArcTan}(\sinh (c+d x))}{2 d}+\frac {1155 b^3 \text {ArcTan}(\sinh (c+d x))}{128 d}-\frac {a^3 \cosh (c+d x)}{d}-\frac {12 a b^2 \cosh (c+d x)}{d}+\frac {a^3 \cosh ^3(c+d x)}{3 d}+\frac {a b^2 \cosh ^3(c+d x)}{d}-\frac {18 a b^2 \text {sech}(c+d x)}{d}+\frac {4 a b^2 \text {sech}^3(c+d x)}{d}-\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {15 a^2 b \sinh (c+d x)}{2 d}-\frac {1155 b^3 \sinh (c+d x)}{128 d}+\frac {5 a^2 b \sinh ^3(c+d x)}{2 d}+\frac {385 b^3 \sinh ^3(c+d x)}{128 d}-\frac {3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}-\frac {231 b^3 \sinh ^3(c+d x) \tanh ^2(c+d x)}{128 d}-\frac {33 b^3 \sinh ^3(c+d x) \tanh ^4(c+d x)}{64 d}-\frac {11 b^3 \sinh ^3(c+d x) \tanh ^6(c+d x)}{48 d}-\frac {b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d} \]
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Rubi [A]
time = 0.25, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3747, 2713,
2672, 294, 308, 209, 2670, 276} \begin {gather*} \frac {a^3 \cosh ^3(c+d x)}{3 d}-\frac {a^3 \cosh (c+d x)}{d}+\frac {15 a^2 b \text {ArcTan}(\sinh (c+d x))}{2 d}+\frac {5 a^2 b \sinh ^3(c+d x)}{2 d}-\frac {15 a^2 b \sinh (c+d x)}{2 d}-\frac {3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}+\frac {a b^2 \cosh ^3(c+d x)}{d}-\frac {12 a b^2 \cosh (c+d x)}{d}-\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {4 a b^2 \text {sech}^3(c+d x)}{d}-\frac {18 a b^2 \text {sech}(c+d x)}{d}+\frac {1155 b^3 \text {ArcTan}(\sinh (c+d x))}{128 d}+\frac {385 b^3 \sinh ^3(c+d x)}{128 d}-\frac {1155 b^3 \sinh (c+d x)}{128 d}-\frac {b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}-\frac {11 b^3 \sinh ^3(c+d x) \tanh ^6(c+d x)}{48 d}-\frac {33 b^3 \sinh ^3(c+d x) \tanh ^4(c+d x)}{64 d}-\frac {231 b^3 \sinh ^3(c+d x) \tanh ^2(c+d x)}{128 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 276
Rule 294
Rule 308
Rule 2670
Rule 2672
Rule 2713
Rule 3747
Rubi steps
\begin {align*} \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx &=i \int \left (-i a^3 \sinh ^3(c+d x)-3 i a^2 b \sinh ^3(c+d x) \tanh ^3(c+d x)-3 i a b^2 \sinh ^3(c+d x) \tanh ^6(c+d x)-i b^3 \sinh ^3(c+d x) \tanh ^9(c+d x)\right ) \, dx\\ &=a^3 \int \sinh ^3(c+d x) \, dx+\left (3 a^2 b\right ) \int \sinh ^3(c+d x) \tanh ^3(c+d x) \, dx+\left (3 a b^2\right ) \int \sinh ^3(c+d x) \tanh ^6(c+d x) \, dx+b^3 \int \sinh ^3(c+d x) \tanh ^9(c+d x) \, dx\\ &=-\frac {a^3 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}+\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^4}{x^6} \, dx,x,\cosh (c+d x)\right )}{d}+\frac {b^3 \text {Subst}\left (\int \frac {x^{12}}{\left (1+x^2\right )^5} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=-\frac {a^3 \cosh (c+d x)}{d}+\frac {a^3 \cosh ^3(c+d x)}{3 d}-\frac {3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}-\frac {b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}+\frac {\left (15 a^2 b\right ) \text {Subst}\left (\int \frac {x^4}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \left (-4+\frac {1}{x^6}-\frac {4}{x^4}+\frac {6}{x^2}+x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}+\frac {\left (11 b^3\right ) \text {Subst}\left (\int \frac {x^{10}}{\left (1+x^2\right )^4} \, dx,x,\sinh (c+d x)\right )}{8 d}\\ &=-\frac {a^3 \cosh (c+d x)}{d}-\frac {12 a b^2 \cosh (c+d x)}{d}+\frac {a^3 \cosh ^3(c+d x)}{3 d}+\frac {a b^2 \cosh ^3(c+d x)}{d}-\frac {18 a b^2 \text {sech}(c+d x)}{d}+\frac {4 a b^2 \text {sech}^3(c+d x)}{d}-\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}-\frac {11 b^3 \sinh ^3(c+d x) \tanh ^6(c+d x)}{48 d}-\frac {b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}+\frac {\left (15 a^2 b\right ) \text {Subst}\left (\int \left (-1+x^2+\frac {1}{1+x^2}\right ) \, dx,x,\sinh (c+d x)\right )}{2 d}+\frac {\left (33 b^3\right ) \text {Subst}\left (\int \frac {x^8}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{16 d}\\ &=-\frac {a^3 \cosh (c+d x)}{d}-\frac {12 a b^2 \cosh (c+d x)}{d}+\frac {a^3 \cosh ^3(c+d x)}{3 d}+\frac {a b^2 \cosh ^3(c+d x)}{d}-\frac {18 a b^2 \text {sech}(c+d x)}{d}+\frac {4 a b^2 \text {sech}^3(c+d x)}{d}-\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {15 a^2 b \sinh (c+d x)}{2 d}+\frac {5 a^2 b \sinh ^3(c+d x)}{2 d}-\frac {3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}-\frac {33 b^3 \sinh ^3(c+d x) \tanh ^4(c+d x)}{64 d}-\frac {11 b^3 \sinh ^3(c+d x) \tanh ^6(c+d x)}{48 d}-\frac {b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}+\frac {\left (15 a^2 b\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 d}+\frac {\left (231 b^3\right ) \text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{64 d}\\ &=\frac {15 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac {a^3 \cosh (c+d x)}{d}-\frac {12 a b^2 \cosh (c+d x)}{d}+\frac {a^3 \cosh ^3(c+d x)}{3 d}+\frac {a b^2 \cosh ^3(c+d x)}{d}-\frac {18 a b^2 \text {sech}(c+d x)}{d}+\frac {4 a b^2 \text {sech}^3(c+d x)}{d}-\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {15 a^2 b \sinh (c+d x)}{2 d}+\frac {5 a^2 b \sinh ^3(c+d x)}{2 d}-\frac {3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}-\frac {231 b^3 \sinh ^3(c+d x) \tanh ^2(c+d x)}{128 d}-\frac {33 b^3 \sinh ^3(c+d x) \tanh ^4(c+d x)}{64 d}-\frac {11 b^3 \sinh ^3(c+d x) \tanh ^6(c+d x)}{48 d}-\frac {b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}+\frac {\left (1155 b^3\right ) \text {Subst}\left (\int \frac {x^4}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{128 d}\\ &=\frac {15 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac {a^3 \cosh (c+d x)}{d}-\frac {12 a b^2 \cosh (c+d x)}{d}+\frac {a^3 \cosh ^3(c+d x)}{3 d}+\frac {a b^2 \cosh ^3(c+d x)}{d}-\frac {18 a b^2 \text {sech}(c+d x)}{d}+\frac {4 a b^2 \text {sech}^3(c+d x)}{d}-\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {15 a^2 b \sinh (c+d x)}{2 d}+\frac {5 a^2 b \sinh ^3(c+d x)}{2 d}-\frac {3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}-\frac {231 b^3 \sinh ^3(c+d x) \tanh ^2(c+d x)}{128 d}-\frac {33 b^3 \sinh ^3(c+d x) \tanh ^4(c+d x)}{64 d}-\frac {11 b^3 \sinh ^3(c+d x) \tanh ^6(c+d x)}{48 d}-\frac {b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}+\frac {\left (1155 b^3\right ) \text {Subst}\left (\int \left (-1+x^2+\frac {1}{1+x^2}\right ) \, dx,x,\sinh (c+d x)\right )}{128 d}\\ &=\frac {15 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}-\frac {a^3 \cosh (c+d x)}{d}-\frac {12 a b^2 \cosh (c+d x)}{d}+\frac {a^3 \cosh ^3(c+d x)}{3 d}+\frac {a b^2 \cosh ^3(c+d x)}{d}-\frac {18 a b^2 \text {sech}(c+d x)}{d}+\frac {4 a b^2 \text {sech}^3(c+d x)}{d}-\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {15 a^2 b \sinh (c+d x)}{2 d}-\frac {1155 b^3 \sinh (c+d x)}{128 d}+\frac {5 a^2 b \sinh ^3(c+d x)}{2 d}+\frac {385 b^3 \sinh ^3(c+d x)}{128 d}-\frac {3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}-\frac {231 b^3 \sinh ^3(c+d x) \tanh ^2(c+d x)}{128 d}-\frac {33 b^3 \sinh ^3(c+d x) \tanh ^4(c+d x)}{64 d}-\frac {11 b^3 \sinh ^3(c+d x) \tanh ^6(c+d x)}{48 d}-\frac {b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}+\frac {\left (1155 b^3\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{128 d}\\ &=\frac {15 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {1155 b^3 \tan ^{-1}(\sinh (c+d x))}{128 d}-\frac {a^3 \cosh (c+d x)}{d}-\frac {12 a b^2 \cosh (c+d x)}{d}+\frac {a^3 \cosh ^3(c+d x)}{3 d}+\frac {a b^2 \cosh ^3(c+d x)}{d}-\frac {18 a b^2 \text {sech}(c+d x)}{d}+\frac {4 a b^2 \text {sech}^3(c+d x)}{d}-\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {15 a^2 b \sinh (c+d x)}{2 d}-\frac {1155 b^3 \sinh (c+d x)}{128 d}+\frac {5 a^2 b \sinh ^3(c+d x)}{2 d}+\frac {385 b^3 \sinh ^3(c+d x)}{128 d}-\frac {3 a^2 b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}-\frac {231 b^3 \sinh ^3(c+d x) \tanh ^2(c+d x)}{128 d}-\frac {33 b^3 \sinh ^3(c+d x) \tanh ^4(c+d x)}{64 d}-\frac {11 b^3 \sinh ^3(c+d x) \tanh ^6(c+d x)}{48 d}-\frac {b^3 \sinh ^3(c+d x) \tanh ^8(c+d x)}{8 d}\\ \end {align*}
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Mathematica [A]
time = 6.45, size = 291, normalized size = 0.83 \begin {gather*} \frac {15 b \left (64 a^2+77 b^2\right ) \text {ArcTan}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}-\frac {3 a \left (a^2+15 b^2\right ) \cosh (c+d x)}{4 d}+\frac {a \left (a^2+3 b^2\right ) \cosh (3 (c+d x))}{12 d}-\frac {18 a b^2 \text {sech}(c+d x)}{d}+\frac {4 a b^2 \text {sech}^3(c+d x)}{d}-\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {3 b \left (9 a^2+7 b^2\right ) \sinh (c+d x)}{4 d}-\frac {3 \text {sech}^2(c+d x) \left (64 a^2 b \sinh (c+d x)+255 b^3 \sinh (c+d x)\right )}{128 d}+\frac {b \left (3 a^2+b^2\right ) \sinh (3 (c+d x))}{12 d}+\frac {515 b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{192 d}-\frac {41 b^3 \text {sech}^5(c+d x) \tanh (c+d x)}{48 d}+\frac {b^3 \text {sech}^7(c+d x) \tanh (c+d x)}{8 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 2.84, size = 675, normalized size = 1.92
method | result | size |
risch | \(\frac {{\mathrm e}^{3 d x +3 c} a^{3}}{24 d}+\frac {{\mathrm e}^{3 d x +3 c} a^{2} b}{8 d}+\frac {a \,b^{2} {\mathrm e}^{3 d x +3 c}}{8 d}+\frac {{\mathrm e}^{3 d x +3 c} b^{3}}{24 d}-\frac {3 \,{\mathrm e}^{d x +c} a^{3}}{8 d}-\frac {27 \,{\mathrm e}^{d x +c} a^{2} b}{8 d}-\frac {45 a \,{\mathrm e}^{d x +c} b^{2}}{8 d}-\frac {21 b^{3} {\mathrm e}^{d x +c}}{8 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{3}}{8 d}+\frac {27 \,{\mathrm e}^{-d x -c} a^{2} b}{8 d}-\frac {45 a \,{\mathrm e}^{-d x -c} b^{2}}{8 d}+\frac {21 \,{\mathrm e}^{-d x -c} b^{3}}{8 d}+\frac {{\mathrm e}^{-3 d x -3 c} a^{3}}{24 d}-\frac {{\mathrm e}^{-3 d x -3 c} a^{2} b}{8 d}+\frac {a \,b^{2} {\mathrm e}^{-3 d x -3 c}}{8 d}-\frac {{\mathrm e}^{-3 d x -3 c} b^{3}}{24 d}-\frac {b \,{\mathrm e}^{d x +c} \left (2880 a^{2} {\mathrm e}^{14 d x +14 c}+34560 a b \,{\mathrm e}^{14 d x +14 c}+11475 b^{2} {\mathrm e}^{14 d x +14 c}+14400 a^{2} {\mathrm e}^{12 d x +12 c}+211200 a b \,{\mathrm e}^{12 d x +12 c}+36775 b^{2} {\mathrm e}^{12 d x +12 c}+25920 a^{2} {\mathrm e}^{10 d x +10 c}+590592 a b \,{\mathrm e}^{10 d x +10 c}+67715 b^{2} {\mathrm e}^{10 d x +10 c}+14400 a^{2} {\mathrm e}^{8 d x +8 c}+957696 a b \,{\mathrm e}^{8 d x +8 c}+27055 b^{2} {\mathrm e}^{8 d x +8 c}-14400 a^{2} {\mathrm e}^{6 d x +6 c}+957696 a b \,{\mathrm e}^{6 d x +6 c}-27055 b^{2} {\mathrm e}^{6 d x +6 c}-25920 a^{2} {\mathrm e}^{4 d x +4 c}+590592 a b \,{\mathrm e}^{4 d x +4 c}-67715 b^{2} {\mathrm e}^{4 d x +4 c}-14400 a^{2} {\mathrm e}^{2 d x +2 c}+211200 a b \,{\mathrm e}^{2 d x +2 c}-36775 b^{2} {\mathrm e}^{2 d x +2 c}-2880 a^{2}+34560 a b -11475 b^{2}\right )}{960 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{8}}-\frac {1155 i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{128 d}+\frac {1155 i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{128 d}-\frac {15 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{2 d}+\frac {15 i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{2 d}\) | \(675\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 604, normalized size = 1.72 \begin {gather*} \frac {1}{192} \, b^{3} {\left (\frac {8 \, {\left (63 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d} - \frac {3465 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {440 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6103 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21019 \, e^{\left (-6 \, d x - 6 \, c\right )} + 41207 \, e^{\left (-8 \, d x - 8 \, c\right )} + 40243 \, e^{\left (-10 \, d x - 10 \, c\right )} + 22589 \, e^{\left (-12 \, d x - 12 \, c\right )} + 505 \, e^{\left (-14 \, d x - 14 \, c\right )} - 3331 \, e^{\left (-16 \, d x - 16 \, c\right )} - 1791 \, e^{\left (-18 \, d x - 18 \, c\right )} - 8}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + 8 \, e^{\left (-5 \, d x - 5 \, c\right )} + 28 \, e^{\left (-7 \, d x - 7 \, c\right )} + 56 \, e^{\left (-9 \, d x - 9 \, c\right )} + 70 \, e^{\left (-11 \, d x - 11 \, c\right )} + 56 \, e^{\left (-13 \, d x - 13 \, c\right )} + 28 \, e^{\left (-15 \, d x - 15 \, c\right )} + 8 \, e^{\left (-17 \, d x - 17 \, c\right )} + e^{\left (-19 \, d x - 19 \, c\right )}\right )}}\right )} - \frac {1}{40} \, a b^{2} {\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^{2} b {\left (\frac {27 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} - \frac {120 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {25 \, e^{\left (-2 \, d x - 2 \, c\right )} + 77 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, e^{\left (-6 \, d x - 6 \, c\right )} - 1}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + 2 \, e^{\left (-5 \, d x - 5 \, c\right )} + e^{\left (-7 \, d x - 7 \, 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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 8462 vs.
\(2 (325) = 650\).
time = 0.44, size = 8462, normalized size = 24.11 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.82, size = 580, normalized size = 1.65 \begin {gather*} \frac {40 \, a^{3} e^{\left (3 \, d x + 3 \, c\right )} + 120 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 120 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 40 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 360 \, a^{3} e^{\left (d x + c\right )} - 3240 \, a^{2} b e^{\left (d x + c\right )} - 5400 \, a b^{2} e^{\left (d x + c\right )} - 2520 \, b^{3} e^{\left (d x + c\right )} + 225 \, {\left (64 \, a^{2} b + 77 \, b^{3}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) - 40 \, {\left (9 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 81 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 135 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 63 \, 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 )} - \frac {2880 \, a^{2} b e^{\left (15 \, d x + 15 \, c\right )} + 34560 \, a b^{2} e^{\left (15 \, d x + 15 \, c\right )} + 11475 \, b^{3} e^{\left (15 \, d x + 15 \, c\right )} + 14400 \, a^{2} b e^{\left (13 \, d x + 13 \, c\right )} + 211200 \, a b^{2} e^{\left (13 \, d x + 13 \, c\right )} + 36775 \, b^{3} e^{\left (13 \, d x + 13 \, c\right )} + 25920 \, a^{2} b e^{\left (11 \, d x + 11 \, c\right )} + 590592 \, a b^{2} e^{\left (11 \, d x + 11 \, c\right )} + 67715 \, b^{3} e^{\left (11 \, d x + 11 \, c\right )} + 14400 \, a^{2} b e^{\left (9 \, d x + 9 \, c\right )} + 957696 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} + 27055 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} - 14400 \, a^{2} b e^{\left (7 \, d x + 7 \, c\right )} + 957696 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 27055 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} - 25920 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} + 590592 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 67715 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} - 14400 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 211200 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 36775 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 2880 \, a^{2} b e^{\left (d x + c\right )} + 34560 \, a b^{2} e^{\left (d x + c\right )} - 11475 \, b^{3} e^{\left (d x + c\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{960 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.60, size = 757, normalized size = 2.16 \begin {gather*} \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 {15\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (77\,b^3\,\sqrt {d^2}+64\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {4096\,a^4\,b^2+9856\,a^2\,b^4+5929\,b^6}}\right )\,\sqrt {4096\,a^4\,b^2+9856\,a^2\,b^4+5929\,b^6}}{64\,\sqrt {d^2}}-\frac {3\,{\mathrm {e}}^{-c-d\,x}\,{\left (a-b\right )}^2\,\left (a-7\,b\right )}{8\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (11005\,b^3+6144\,a\,b^2\right )}{120\,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 {{\mathrm {e}}^{c+d\,x}\,\left (3365\,b^3+768\,a\,b^2\right )}{20\,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 {596\,b^3\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (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\right )}-\frac {3\,{\mathrm {e}}^{c+d\,x}\,{\left (a+b\right )}^2\,\left (a+7\,b\right )}{8\,d}-\frac {3\,{\mathrm {e}}^{c+d\,x}\,\left (64\,a^2\,b+768\,a\,b^2+255\,b^3\right )}{64\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (1625\,b^3+144\,a\,b^2\right )}{15\,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 {112\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (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\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (576\,a^2\,b+3072\,a\,b^2+4355\,b^3\right )}{96\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}+\frac {32\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (8\,{\mathrm {e}}^{2\,c+2\,d\,x}+28\,{\mathrm {e}}^{4\,c+4\,d\,x}+56\,{\mathrm {e}}^{6\,c+6\,d\,x}+70\,{\mathrm {e}}^{8\,c+8\,d\,x}+56\,{\mathrm {e}}^{10\,c+10\,d\,x}+28\,{\mathrm {e}}^{12\,c+12\,d\,x}+8\,{\mathrm {e}}^{14\,c+14\,d\,x}+{\mathrm {e}}^{16\,c+16\,d\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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