Optimal. Leaf size=106 \[ \frac {385 x}{32768}-\frac {385 \tanh ^{-1}\left (\frac {\sinh (c+d x)}{3+\cosh (c+d x)}\right )}{16384 d}-\frac {\sinh (c+d x)}{16 d (5+3 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))^2}-\frac {311 \sinh (c+d x)}{8192 d (5+3 \cosh (c+d x))} \]
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Rubi [A]
time = 0.07, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2743, 2833, 12,
2736} \begin {gather*} -\frac {311 \sinh (c+d x)}{8192 d (3 \cosh (c+d x)+5)}-\frac {25 \sinh (c+d x)}{512 d (3 \cosh (c+d x)+5)^2}-\frac {\sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)^3}-\frac {385 \tanh ^{-1}\left (\frac {\sinh (c+d x)}{\cosh (c+d x)+3}\right )}{16384 d}+\frac {385 x}{32768} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2736
Rule 2743
Rule 2833
Rubi steps
\begin {align*} \int \frac {1}{(5+3 \cosh (c+d x))^4} \, dx &=-\frac {\sinh (c+d x)}{16 d (5+3 \cosh (c+d x))^3}-\frac {1}{48} \int \frac {-15+6 \cosh (c+d x)}{(5+3 \cosh (c+d x))^3} \, dx\\ &=-\frac {\sinh (c+d x)}{16 d (5+3 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))^2}+\frac {\int \frac {186-75 \cosh (c+d x)}{(5+3 \cosh (c+d x))^2} \, dx}{1536}\\ &=-\frac {\sinh (c+d x)}{16 d (5+3 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))^2}-\frac {311 \sinh (c+d x)}{8192 d (5+3 \cosh (c+d x))}-\frac {\int -\frac {1155}{5+3 \cosh (c+d x)} \, dx}{24576}\\ &=-\frac {\sinh (c+d x)}{16 d (5+3 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))^2}-\frac {311 \sinh (c+d x)}{8192 d (5+3 \cosh (c+d x))}+\frac {385 \int \frac {1}{5+3 \cosh (c+d x)} \, dx}{8192}\\ &=\frac {385 x}{32768}-\frac {385 \tanh ^{-1}\left (\frac {\sinh (c+d x)}{3+\cosh (c+d x)}\right )}{16384 d}-\frac {\sinh (c+d x)}{16 d (5+3 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))^2}-\frac {311 \sinh (c+d x)}{8192 d (5+3 \cosh (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 68, normalized size = 0.64 \begin {gather*} \frac {770 \tanh ^{-1}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )-\frac {9 (4883 \sinh (c+d x)+2340 \sinh (2 (c+d x))+311 \sinh (3 (c+d x)))}{(5+3 \cosh (c+d x))^3}}{32768 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.86, size = 124, normalized size = 1.17
method | result | size |
risch | \(\frac {10395 \,{\mathrm e}^{5 d x +5 c}+86625 \,{\mathrm e}^{4 d x +4 c}+239470 \,{\mathrm e}^{3 d x +3 c}+218466 \,{\mathrm e}^{2 d x +2 c}+73575 \,{\mathrm e}^{d x +c}+8397}{12288 d \left (3 \,{\mathrm e}^{2 d x +2 c}+10 \,{\mathrm e}^{d x +c}+3\right )^{3}}+\frac {385 \ln \left ({\mathrm e}^{d x +c}+\frac {1}{3}\right )}{32768 d}-\frac {385 \ln \left (3+{\mathrm e}^{d x +c}\right )}{32768 d}\) | \(112\) |
derivativedivides | \(\frac {\frac {9}{2048 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{3}}+\frac {81}{4096 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}+\frac {639}{16384 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}-\frac {385 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768}+\frac {9}{2048 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{3}}-\frac {81}{4096 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}+\frac {639}{16384 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}+\frac {385 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768}}{d}\) | \(124\) |
default | \(\frac {\frac {9}{2048 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{3}}+\frac {81}{4096 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}+\frac {639}{16384 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}-\frac {385 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768}+\frac {9}{2048 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{3}}-\frac {81}{4096 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}+\frac {639}{16384 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}+\frac {385 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768}}{d}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 169, normalized size = 1.59 \begin {gather*} -\frac {385 \, \log \left (3 \, e^{\left (-d x - c\right )} + 1\right )}{32768 \, d} + \frac {385 \, \log \left (e^{\left (-d x - c\right )} + 3\right )}{32768 \, d} - \frac {73575 \, e^{\left (-d x - c\right )} + 218466 \, e^{\left (-2 \, d x - 2 \, c\right )} + 239470 \, e^{\left (-3 \, d x - 3 \, c\right )} + 86625 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10395 \, e^{\left (-5 \, d x - 5 \, c\right )} + 8397}{12288 \, d {\left (270 \, e^{\left (-d x - c\right )} + 981 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1540 \, e^{\left (-3 \, d x - 3 \, c\right )} + 981 \, e^{\left (-4 \, d x - 4 \, c\right )} + 270 \, e^{\left (-5 \, d x - 5 \, c\right )} + 27 \, e^{\left (-6 \, d x - 6 \, c\right )} + 27\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. 1078 vs.
\(2 (96) = 192\).
time = 0.44, size = 1078, normalized size = 10.17 \begin {gather*} \frac {83160 \, \cosh \left (d x + c\right )^{5} + 138600 \, {\left (3 \, \cosh \left (d x + c\right ) + 5\right )} \sinh \left (d x + c\right )^{4} + 83160 \, \sinh \left (d x + c\right )^{5} + 693000 \, \cosh \left (d x + c\right )^{4} + 6160 \, {\left (135 \, \cosh \left (d x + c\right )^{2} + 450 \, \cosh \left (d x + c\right ) + 311\right )} \sinh \left (d x + c\right )^{3} + 1915760 \, \cosh \left (d x + c\right )^{3} + 48 \, {\left (17325 \, \cosh \left (d x + c\right )^{3} + 86625 \, \cosh \left (d x + c\right )^{2} + 119735 \, \cosh \left (d x + c\right ) + 36411\right )} \sinh \left (d x + c\right )^{2} + 1747728 \, \cosh \left (d x + c\right )^{2} + 1155 \, {\left (27 \, \cosh \left (d x + c\right )^{6} + 54 \, {\left (3 \, \cosh \left (d x + c\right ) + 5\right )} \sinh \left (d x + c\right )^{5} + 27 \, \sinh \left (d x + c\right )^{6} + 270 \, \cosh \left (d x + c\right )^{5} + 9 \, {\left (45 \, \cosh \left (d x + c\right )^{2} + 150 \, \cosh \left (d x + c\right ) + 109\right )} \sinh \left (d x + c\right )^{4} + 981 \, \cosh \left (d x + c\right )^{4} + 4 \, {\left (135 \, \cosh \left (d x + c\right )^{3} + 675 \, \cosh \left (d x + c\right )^{2} + 981 \, \cosh \left (d x + c\right ) + 385\right )} \sinh \left (d x + c\right )^{3} + 1540 \, \cosh \left (d x + c\right )^{3} + 3 \, {\left (135 \, \cosh \left (d x + c\right )^{4} + 900 \, \cosh \left (d x + c\right )^{3} + 1962 \, \cosh \left (d x + c\right )^{2} + 1540 \, \cosh \left (d x + c\right ) + 327\right )} \sinh \left (d x + c\right )^{2} + 981 \, \cosh \left (d x + c\right )^{2} + 6 \, {\left (27 \, \cosh \left (d x + c\right )^{5} + 225 \, \cosh \left (d x + c\right )^{4} + 654 \, \cosh \left (d x + c\right )^{3} + 770 \, \cosh \left (d x + c\right )^{2} + 327 \, \cosh \left (d x + c\right ) + 45\right )} \sinh \left (d x + c\right ) + 270 \, \cosh \left (d x + c\right ) + 27\right )} \log \left (3 \, \cosh \left (d x + c\right ) + 3 \, \sinh \left (d x + c\right ) + 1\right ) - 1155 \, {\left (27 \, \cosh \left (d x + c\right )^{6} + 54 \, {\left (3 \, \cosh \left (d x + c\right ) + 5\right )} \sinh \left (d x + c\right )^{5} + 27 \, \sinh \left (d x + c\right )^{6} + 270 \, \cosh \left (d x + c\right )^{5} + 9 \, {\left (45 \, \cosh \left (d x + c\right )^{2} + 150 \, \cosh \left (d x + c\right ) + 109\right )} \sinh \left (d x + c\right )^{4} + 981 \, \cosh \left (d x + c\right )^{4} + 4 \, {\left (135 \, \cosh \left (d x + c\right )^{3} + 675 \, \cosh \left (d x + c\right )^{2} + 981 \, \cosh \left (d x + c\right ) + 385\right )} \sinh \left (d x + c\right )^{3} + 1540 \, \cosh \left (d x + c\right )^{3} + 3 \, {\left (135 \, \cosh \left (d x + c\right )^{4} + 900 \, \cosh \left (d x + c\right )^{3} + 1962 \, \cosh \left (d x + c\right )^{2} + 1540 \, \cosh \left (d x + c\right ) + 327\right )} \sinh \left (d x + c\right )^{2} + 981 \, \cosh \left (d x + c\right )^{2} + 6 \, {\left (27 \, \cosh \left (d x + c\right )^{5} + 225 \, \cosh \left (d x + c\right )^{4} + 654 \, \cosh \left (d x + c\right )^{3} + 770 \, \cosh \left (d x + c\right )^{2} + 327 \, \cosh \left (d x + c\right ) + 45\right )} \sinh \left (d x + c\right ) + 270 \, \cosh \left (d x + c\right ) + 27\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 3\right ) + 24 \, {\left (17325 \, \cosh \left (d x + c\right )^{4} + 115500 \, \cosh \left (d x + c\right )^{3} + 239470 \, \cosh \left (d x + c\right )^{2} + 145644 \, \cosh \left (d x + c\right ) + 24525\right )} \sinh \left (d x + c\right ) + 588600 \, \cosh \left (d x + c\right ) + 67176}{98304 \, {\left (27 \, d \cosh \left (d x + c\right )^{6} + 27 \, d \sinh \left (d x + c\right )^{6} + 270 \, d \cosh \left (d x + c\right )^{5} + 54 \, {\left (3 \, d \cosh \left (d x + c\right ) + 5 \, d\right )} \sinh \left (d x + c\right )^{5} + 981 \, d \cosh \left (d x + c\right )^{4} + 9 \, {\left (45 \, d \cosh \left (d x + c\right )^{2} + 150 \, d \cosh \left (d x + c\right ) + 109 \, d\right )} \sinh \left (d x + c\right )^{4} + 1540 \, d \cosh \left (d x + c\right )^{3} + 4 \, {\left (135 \, d \cosh \left (d x + c\right )^{3} + 675 \, d \cosh \left (d x + c\right )^{2} + 981 \, d \cosh \left (d x + c\right ) + 385 \, d\right )} \sinh \left (d x + c\right )^{3} + 981 \, d \cosh \left (d x + c\right )^{2} + 3 \, {\left (135 \, d \cosh \left (d x + c\right )^{4} + 900 \, d \cosh \left (d x + c\right )^{3} + 1962 \, d \cosh \left (d x + c\right )^{2} + 1540 \, d \cosh \left (d x + c\right ) + 327 \, d\right )} \sinh \left (d x + c\right )^{2} + 270 \, d \cosh \left (d x + c\right ) + 6 \, {\left (27 \, d \cosh \left (d x + c\right )^{5} + 225 \, d \cosh \left (d x + c\right )^{4} + 654 \, d \cosh \left (d x + c\right )^{3} + 770 \, d \cosh \left (d x + c\right )^{2} + 327 \, d \cosh \left (d x + c\right ) + 45 \, d\right )} \sinh \left (d x + c\right ) + 27 \, d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 784 vs.
\(2 (94) = 188\).
time = 3.75, size = 784, normalized size = 7.40 \begin {gather*} \begin {cases} - \frac {385 \log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )} \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 393216 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1572864 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2097152 d} + \frac {4620 \log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )} \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 393216 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1572864 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2097152 d} - \frac {18480 \log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )} \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 393216 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1572864 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2097152 d} + \frac {24640 \log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )}}{32768 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 393216 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1572864 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2097152 d} + \frac {385 \log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )} \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 393216 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1572864 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2097152 d} - \frac {4620 \log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )} \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 393216 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1572864 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2097152 d} + \frac {18480 \log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )} \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 393216 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1572864 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2097152 d} - \frac {24640 \log {\left (\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )}}{32768 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 393216 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1572864 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2097152 d} + \frac {2556 \tanh ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 393216 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1572864 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2097152 d} - \frac {14976 \tanh ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 393216 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1572864 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2097152 d} + \frac {23616 \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 393216 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1572864 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2097152 d} & \text {for}\: d \neq 0 \\\frac {x}{\left (3 \cosh {\left (c \right )} + 5\right )^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 109, normalized size = 1.03 \begin {gather*} \frac {\frac {8 \, {\left (10395 \, e^{\left (5 \, d x + 5 \, c\right )} + 86625 \, e^{\left (4 \, d x + 4 \, c\right )} + 239470 \, e^{\left (3 \, d x + 3 \, c\right )} + 218466 \, e^{\left (2 \, d x + 2 \, c\right )} + 73575 \, e^{\left (d x + c\right )} + 8397\right )}}{{\left (3 \, e^{\left (2 \, d x + 2 \, c\right )} + 10 \, e^{\left (d x + c\right )} + 3\right )}^{3}} + 1155 \, \log \left (3 \, e^{\left (d x + c\right )} + 1\right ) - 1155 \, \log \left (e^{\left (d x + c\right )} + 3\right )}{98304 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 226, normalized size = 2.13 \begin {gather*} \frac {\frac {385\,{\mathrm {e}}^{c+d\,x}}{4096\,d}+\frac {1925}{12288\,d}}{10\,{\mathrm {e}}^{c+d\,x}+3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3}-\frac {385\,\mathrm {atan}\left (\left (\frac {5}{4\,d}+\frac {3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}{4\,d}\right )\,\sqrt {-d^2}\right )}{16384\,\sqrt {-d^2}}-\frac {\frac {385\,{\mathrm {e}}^{c+d\,x}}{1152\,d}+\frac {3461}{3456\,d}}{60\,{\mathrm {e}}^{c+d\,x}+118\,{\mathrm {e}}^{2\,c+2\,d\,x}+60\,{\mathrm {e}}^{3\,c+3\,d\,x}+9\,{\mathrm {e}}^{4\,c+4\,d\,x}+9}+\frac {\frac {365\,{\mathrm {e}}^{c+d\,x}}{54\,d}+\frac {41}{18\,d}}{270\,{\mathrm {e}}^{c+d\,x}+981\,{\mathrm {e}}^{2\,c+2\,d\,x}+1540\,{\mathrm {e}}^{3\,c+3\,d\,x}+981\,{\mathrm {e}}^{4\,c+4\,d\,x}+270\,{\mathrm {e}}^{5\,c+5\,d\,x}+27\,{\mathrm {e}}^{6\,c+6\,d\,x}+27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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