Optimal. Leaf size=98 \[ -\frac {279 \text {ArcTan}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{16384 d}+\frac {5 \sinh (c+d x)}{48 d (3+5 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (3+5 \cosh (c+d x))^2}+\frac {995 \sinh (c+d x)}{24576 d (3+5 \cosh (c+d x))} \]
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Rubi [A]
time = 0.07, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2743, 2833, 12,
2738, 212} \begin {gather*} -\frac {279 \text {ArcTan}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{16384 d}+\frac {995 \sinh (c+d x)}{24576 d (5 \cosh (c+d x)+3)}-\frac {25 \sinh (c+d x)}{512 d (5 \cosh (c+d x)+3)^2}+\frac {5 \sinh (c+d x)}{48 d (5 \cosh (c+d x)+3)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 212
Rule 2738
Rule 2743
Rule 2833
Rubi steps
\begin {align*} \int \frac {1}{(3+5 \cosh (c+d x))^4} \, dx &=\frac {5 \sinh (c+d x)}{48 d (3+5 \cosh (c+d x))^3}+\frac {1}{48} \int \frac {-9+10 \cosh (c+d x)}{(3+5 \cosh (c+d x))^3} \, dx\\ &=\frac {5 \sinh (c+d x)}{48 d (3+5 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (3+5 \cosh (c+d x))^2}+\frac {\int \frac {154-75 \cosh (c+d x)}{(3+5 \cosh (c+d x))^2} \, dx}{1536}\\ &=\frac {5 \sinh (c+d x)}{48 d (3+5 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (3+5 \cosh (c+d x))^2}+\frac {995 \sinh (c+d x)}{24576 d (3+5 \cosh (c+d x))}+\frac {\int -\frac {837}{3+5 \cosh (c+d x)} \, dx}{24576}\\ &=\frac {5 \sinh (c+d x)}{48 d (3+5 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (3+5 \cosh (c+d x))^2}+\frac {995 \sinh (c+d x)}{24576 d (3+5 \cosh (c+d x))}-\frac {279 \int \frac {1}{3+5 \cosh (c+d x)} \, dx}{8192}\\ &=\frac {5 \sinh (c+d x)}{48 d (3+5 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (3+5 \cosh (c+d x))^2}+\frac {995 \sinh (c+d x)}{24576 d (3+5 \cosh (c+d x))}+\frac {(279 i) \text {Subst}\left (\int \frac {1}{8-2 x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{4096 d}\\ &=-\frac {279 \tan ^{-1}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{16384 d}+\frac {5 \sinh (c+d x)}{48 d (3+5 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (3+5 \cosh (c+d x))^2}+\frac {995 \sinh (c+d x)}{24576 d (3+5 \cosh (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 65, normalized size = 0.66 \begin {gather*} \frac {-837 \text {ArcTan}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )+\frac {5 (8141+9540 \cosh (c+d x)+4975 \cosh (2 (c+d x))) \sinh (c+d x)}{(3+5 \cosh (c+d x))^3}}{49152 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.86, size = 75, normalized size = 0.77
method | result | size |
derivativedivides | \(\frac {-\frac {-\frac {745 \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {265 \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96}-\frac {295 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}}{8 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+4\right )^{3}}-\frac {279 \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{16384}}{d}\) | \(75\) |
default | \(\frac {-\frac {-\frac {745 \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {265 \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96}-\frac {295 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}}{8 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+4\right )^{3}}-\frac {279 \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{16384}}{d}\) | \(75\) |
risch | \(-\frac {20925 \,{\mathrm e}^{5 d x +5 c}+62775 \,{\mathrm e}^{4 d x +4 c}+111042 \,{\mathrm e}^{3 d x +3 c}+119310 \,{\mathrm e}^{2 d x +2 c}+68625 \,{\mathrm e}^{d x +c}+24875}{12288 d \left (5 \,{\mathrm e}^{2 d x +2 c}+6 \,{\mathrm e}^{d x +c}+5\right )^{3}}+\frac {279 i \ln \left ({\mathrm e}^{d x +c}+\frac {3}{5}-\frac {4 i}{5}\right )}{32768 d}-\frac {279 i \ln \left ({\mathrm e}^{d x +c}+\frac {3}{5}+\frac {4 i}{5}\right )}{32768 d}\) | \(118\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 152, normalized size = 1.55 \begin {gather*} \frac {279 \, \arctan \left (\frac {5}{4} \, e^{\left (-d x - c\right )} + \frac {3}{4}\right )}{16384 \, d} + \frac {68625 \, e^{\left (-d x - c\right )} + 119310 \, e^{\left (-2 \, d x - 2 \, c\right )} + 111042 \, e^{\left (-3 \, d x - 3 \, c\right )} + 62775 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20925 \, e^{\left (-5 \, d x - 5 \, c\right )} + 24875}{12288 \, d {\left (450 \, e^{\left (-d x - c\right )} + 915 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1116 \, e^{\left (-3 \, d x - 3 \, c\right )} + 915 \, e^{\left (-4 \, d x - 4 \, c\right )} + 450 \, e^{\left (-5 \, d x - 5 \, c\right )} + 125 \, e^{\left (-6 \, d x - 6 \, c\right )} + 125\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. 793 vs.
\(2 (87) = 174\).
time = 0.37, size = 793, normalized size = 8.09 \begin {gather*} -\frac {83700 \, \cosh \left (d x + c\right )^{5} + 83700 \, {\left (5 \, \cosh \left (d x + c\right ) + 3\right )} \sinh \left (d x + c\right )^{4} + 83700 \, \sinh \left (d x + c\right )^{5} + 251100 \, \cosh \left (d x + c\right )^{4} + 2232 \, {\left (375 \, \cosh \left (d x + c\right )^{2} + 450 \, \cosh \left (d x + c\right ) + 199\right )} \sinh \left (d x + c\right )^{3} + 444168 \, \cosh \left (d x + c\right )^{3} + 24 \, {\left (34875 \, \cosh \left (d x + c\right )^{3} + 62775 \, \cosh \left (d x + c\right )^{2} + 55521 \, \cosh \left (d x + c\right ) + 19885\right )} \sinh \left (d x + c\right )^{2} + 837 \, {\left (125 \, \cosh \left (d x + c\right )^{6} + 150 \, {\left (5 \, \cosh \left (d x + c\right ) + 3\right )} \sinh \left (d x + c\right )^{5} + 125 \, \sinh \left (d x + c\right )^{6} + 450 \, \cosh \left (d x + c\right )^{5} + 15 \, {\left (125 \, \cosh \left (d x + c\right )^{2} + 150 \, \cosh \left (d x + c\right ) + 61\right )} \sinh \left (d x + c\right )^{4} + 915 \, \cosh \left (d x + c\right )^{4} + 4 \, {\left (625 \, \cosh \left (d x + c\right )^{3} + 1125 \, \cosh \left (d x + c\right )^{2} + 915 \, \cosh \left (d x + c\right ) + 279\right )} \sinh \left (d x + c\right )^{3} + 1116 \, \cosh \left (d x + c\right )^{3} + 3 \, {\left (625 \, \cosh \left (d x + c\right )^{4} + 1500 \, \cosh \left (d x + c\right )^{3} + 1830 \, \cosh \left (d x + c\right )^{2} + 1116 \, \cosh \left (d x + c\right ) + 305\right )} \sinh \left (d x + c\right )^{2} + 915 \, \cosh \left (d x + c\right )^{2} + 6 \, {\left (125 \, \cosh \left (d x + c\right )^{5} + 375 \, \cosh \left (d x + c\right )^{4} + 610 \, \cosh \left (d x + c\right )^{3} + 558 \, \cosh \left (d x + c\right )^{2} + 305 \, \cosh \left (d x + c\right ) + 75\right )} \sinh \left (d x + c\right ) + 450 \, \cosh \left (d x + c\right ) + 125\right )} \arctan \left (\frac {5}{4} \, \cosh \left (d x + c\right ) + \frac {5}{4} \, \sinh \left (d x + c\right ) + \frac {3}{4}\right ) + 477240 \, \cosh \left (d x + c\right )^{2} + 12 \, {\left (34875 \, \cosh \left (d x + c\right )^{4} + 83700 \, \cosh \left (d x + c\right )^{3} + 111042 \, \cosh \left (d x + c\right )^{2} + 79540 \, \cosh \left (d x + c\right ) + 22875\right )} \sinh \left (d x + c\right ) + 274500 \, \cosh \left (d x + c\right ) + 99500}{49152 \, {\left (125 \, d \cosh \left (d x + c\right )^{6} + 125 \, d \sinh \left (d x + c\right )^{6} + 450 \, d \cosh \left (d x + c\right )^{5} + 150 \, {\left (5 \, d \cosh \left (d x + c\right ) + 3 \, d\right )} \sinh \left (d x + c\right )^{5} + 915 \, d \cosh \left (d x + c\right )^{4} + 15 \, {\left (125 \, d \cosh \left (d x + c\right )^{2} + 150 \, d \cosh \left (d x + c\right ) + 61 \, d\right )} \sinh \left (d x + c\right )^{4} + 1116 \, d \cosh \left (d x + c\right )^{3} + 4 \, {\left (625 \, d \cosh \left (d x + c\right )^{3} + 1125 \, d \cosh \left (d x + c\right )^{2} + 915 \, d \cosh \left (d x + c\right ) + 279 \, d\right )} \sinh \left (d x + c\right )^{3} + 915 \, d \cosh \left (d x + c\right )^{2} + 3 \, {\left (625 \, d \cosh \left (d x + c\right )^{4} + 1500 \, d \cosh \left (d x + c\right )^{3} + 1830 \, d \cosh \left (d x + c\right )^{2} + 1116 \, d \cosh \left (d x + c\right ) + 305 \, d\right )} \sinh \left (d x + c\right )^{2} + 450 \, d \cosh \left (d x + c\right ) + 6 \, {\left (125 \, d \cosh \left (d x + c\right )^{5} + 375 \, d \cosh \left (d x + c\right )^{4} + 610 \, d \cosh \left (d x + c\right )^{3} + 558 \, d \cosh \left (d x + c\right )^{2} + 305 \, d \cosh \left (d x + c\right ) + 75 \, d\right )} \sinh \left (d x + c\right ) + 125 \, d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 4.95, size = 908, normalized size = 9.27 \begin {gather*} \begin {cases} \frac {x}{625 \cosh ^{4}{\left (d x + \log {\left (- 3 e^{- d x} - 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 1500 \cosh ^{3}{\left (d x + \log {\left (- 3 e^{- d x} - 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 1350 \cosh ^{2}{\left (d x + \log {\left (- 3 e^{- d x} - 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 540 \cosh {\left (d x + \log {\left (- 3 e^{- d x} - 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 81} & \text {for}\: c = \log {\left (\left (- \frac {3}{5} - \frac {4 i}{5}\right ) e^{- d x} \right )} \\- \frac {\log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )}}{625 d \cosh ^{4}{\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 1500 d \cosh ^{3}{\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 1350 d \cosh ^{2}{\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 540 d \cosh {\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 81 d} + \frac {\log {\left (5 \right )}}{625 d \cosh ^{4}{\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 1500 d \cosh ^{3}{\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 1350 d \cosh ^{2}{\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 540 d \cosh {\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 81 d} & \text {for}\: c = \log {\left (\left (- \frac {3}{5} + \frac {4 i}{5}\right ) e^{- d x} \right )} \\\frac {x}{\left (5 \cosh {\left (c \right )} + 3\right )^{4}} & \text {for}\: d = 0 \\- \frac {837 \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )}}{49152 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 589824 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2359296 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3145728 d} + \frac {4470 \tanh ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{49152 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 589824 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2359296 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3145728 d} - \frac {10044 \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )}}{49152 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 589824 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2359296 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3145728 d} + \frac {16960 \tanh ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{49152 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 589824 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2359296 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3145728 d} - \frac {40176 \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )}}{49152 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 589824 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2359296 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3145728 d} + \frac {28320 \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{49152 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 589824 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2359296 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3145728 d} - \frac {53568 \operatorname {atan}{\left (\frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )}}{49152 d \tanh ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 589824 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2359296 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3145728 d} & \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 = 98, normalized size = 1.00 \begin {gather*} -\frac {\frac {4 \, {\left (20925 \, e^{\left (5 \, d x + 5 \, c\right )} + 62775 \, e^{\left (4 \, d x + 4 \, c\right )} + 111042 \, e^{\left (3 \, d x + 3 \, c\right )} + 119310 \, e^{\left (2 \, d x + 2 \, c\right )} + 68625 \, e^{\left (d x + c\right )} + 24875\right )}}{{\left (5 \, e^{\left (2 \, d x + 2 \, c\right )} + 6 \, e^{\left (d x + c\right )} + 5\right )}^{3}} + 837 \, \arctan \left (\frac {5}{4} \, e^{\left (d x + c\right )} + \frac {3}{4}\right )}{49152 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.95, size = 223, normalized size = 2.28 \begin {gather*} \frac {\frac {39\,{\mathrm {e}}^{c+d\,x}}{50\,d}+\frac {7}{30\,d}}{450\,{\mathrm {e}}^{c+d\,x}+915\,{\mathrm {e}}^{2\,c+2\,d\,x}+1116\,{\mathrm {e}}^{3\,c+3\,d\,x}+915\,{\mathrm {e}}^{4\,c+4\,d\,x}+450\,{\mathrm {e}}^{5\,c+5\,d\,x}+125\,{\mathrm {e}}^{6\,c+6\,d\,x}+125}-\frac {\frac {93\,{\mathrm {e}}^{c+d\,x}}{640\,d}+\frac {791}{3200\,d}}{60\,{\mathrm {e}}^{c+d\,x}+86\,{\mathrm {e}}^{2\,c+2\,d\,x}+60\,{\mathrm {e}}^{3\,c+3\,d\,x}+25\,{\mathrm {e}}^{4\,c+4\,d\,x}+25}-\frac {279\,\mathrm {atan}\left (\left (\frac {3}{4\,d}+\frac {5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}{4\,d}\right )\,\sqrt {d^2}\right )}{16384\,\sqrt {d^2}}-\frac {\frac {279\,{\mathrm {e}}^{c+d\,x}}{4096\,d}+\frac {837}{20480\,d}}{6\,{\mathrm {e}}^{c+d\,x}+5\,{\mathrm {e}}^{2\,c+2\,d\,x}+5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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