Optimal. Leaf size=160 \[ -\frac {279 i \log \left (3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {279 i \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}-\frac {25 i \cosh (c+d x)}{512 d (3+5 i \sinh (c+d x))^2}+\frac {995 i \cosh (c+d x)}{24576 d (3+5 i \sinh (c+d x))} \]
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Rubi [A]
time = 0.08, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2743, 2833, 12,
2739, 630, 31} \begin {gather*} \frac {995 i \cosh (c+d x)}{24576 d (3+5 i \sinh (c+d x))}-\frac {25 i \cosh (c+d x)}{512 d (3+5 i \sinh (c+d x))^2}+\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}-\frac {279 i \log \left (3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {279 i \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 630
Rule 2739
Rule 2743
Rule 2833
Rubi steps
\begin {align*} \int \frac {1}{(3+5 i \sinh (c+d x))^4} \, dx &=\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}+\frac {1}{48} \int \frac {-9+10 i \sinh (c+d x)}{(3+5 i \sinh (c+d x))^3} \, dx\\ &=\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}-\frac {25 i \cosh (c+d x)}{512 d (3+5 i \sinh (c+d x))^2}+\frac {\int \frac {154-75 i \sinh (c+d x)}{(3+5 i \sinh (c+d x))^2} \, dx}{1536}\\ &=\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}-\frac {25 i \cosh (c+d x)}{512 d (3+5 i \sinh (c+d x))^2}+\frac {995 i \cosh (c+d x)}{24576 d (3+5 i \sinh (c+d x))}+\frac {\int -\frac {837}{3+5 i \sinh (c+d x)} \, dx}{24576}\\ &=\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}-\frac {25 i \cosh (c+d x)}{512 d (3+5 i \sinh (c+d x))^2}+\frac {995 i \cosh (c+d x)}{24576 d (3+5 i \sinh (c+d x))}-\frac {279 \int \frac {1}{3+5 i \sinh (c+d x)} \, dx}{8192}\\ &=\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}-\frac {25 i \cosh (c+d x)}{512 d (3+5 i \sinh (c+d x))^2}+\frac {995 i \cosh (c+d x)}{24576 d (3+5 i \sinh (c+d x))}+\frac {(279 i) \text {Subst}\left (\int \frac {1}{3+10 x+3 x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{4096 d}\\ &=\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}-\frac {25 i \cosh (c+d x)}{512 d (3+5 i \sinh (c+d x))^2}+\frac {995 i \cosh (c+d x)}{24576 d (3+5 i \sinh (c+d x))}+\frac {(837 i) \text {Subst}\left (\int \frac {1}{1+3 x} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{32768 d}-\frac {(837 i) \text {Subst}\left (\int \frac {1}{9+3 x} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{32768 d}\\ &=-\frac {279 i \log \left (3+i \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {279 i \log \left (1+3 i \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {5 i \cosh (c+d x)}{48 d (3+5 i \sinh (c+d x))^3}-\frac {25 i \cosh (c+d x)}{512 d (3+5 i \sinh (c+d x))^2}+\frac {995 i \cosh (c+d x)}{24576 d (3+5 i \sinh (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.47, size = 265, normalized size = 1.66 \begin {gather*} \frac {-5022 \text {ArcTan}\left (3 \coth \left (\frac {1}{2} (c+d x)\right )\right )-5022 \text {ArcTan}\left (3 \tanh \left (\frac {1}{2} (c+d x)\right )\right )+2511 i \log (4-5 \cosh (c+d x))-2511 i \log (4+5 \cosh (c+d x))+\frac {4640 i}{\left (3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {1440 i}{\left (\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}+40 \left (\frac {80}{\left (3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {199}{3 \cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )}+\frac {240}{\left (\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {597}{\cosh \left (\frac {1}{2} (c+d x)\right )+3 i \sinh \left (\frac {1}{2} (c+d x)\right )}\right ) \sinh \left (\frac {1}{2} (c+d x)\right )}{589824 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.40, size = 144, normalized size = 0.90
method | result | size |
risch | \(\frac {i \left (-62775 i {\mathrm e}^{4 d x +4 c}+20925 \,{\mathrm e}^{5 d x +5 c}+119310 i {\mathrm e}^{2 d x +2 c}-111042 \,{\mathrm e}^{3 d x +3 c}-24875 i+68625 \,{\mathrm e}^{d x +c}\right )}{12288 d \left (5 \,{\mathrm e}^{2 d x +2 c}-5-6 i {\mathrm e}^{d x +c}\right )^{3}}+\frac {279 i \ln \left ({\mathrm e}^{d x +c}-\frac {4}{5}-\frac {3 i}{5}\right )}{32768 d}-\frac {279 i \ln \left ({\mathrm e}^{d x +c}+\frac {4}{5}-\frac {3 i}{5}\right )}{32768 d}\) | \(123\) |
derivativedivides | \(\frac {\frac {275 i}{27648 \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}+\frac {279 i \ln \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}{32768}-\frac {125}{20736 \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{3}}+\frac {3505}{221184 \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}-\frac {279 i \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )}{32768}+\frac {75 i}{1024 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )^{2}}-\frac {125}{768 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )^{3}}+\frac {345}{8192 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )}}{d}\) | \(144\) |
default | \(\frac {\frac {275 i}{27648 \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}+\frac {279 i \ln \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}{32768}-\frac {125}{20736 \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{3}}+\frac {3505}{221184 \left (3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )}-\frac {279 i \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )}{32768}+\frac {75 i}{1024 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )^{2}}-\frac {125}{768 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )^{3}}+\frac {345}{8192 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-3 i\right )}}{d}\) | \(144\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 167, normalized size = 1.04 \begin {gather*} \frac {279 i \, \log \left (\frac {5 \, e^{\left (-d x - c\right )} + 3 i - 4}{5 \, e^{\left (-d x - c\right )} + 3 i + 4}\right )}{32768 \, d} + \frac {68625 i \, e^{\left (-d x - c\right )} + 119310 \, e^{\left (-2 \, d x - 2 \, c\right )} - 111042 i \, e^{\left (-3 \, d x - 3 \, c\right )} - 62775 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20925 i \, e^{\left (-5 \, d x - 5 \, c\right )} - 24875}{-12288 \, d {\left (-450 i \, e^{\left (-d x - c\right )} - 915 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1116 i \, e^{\left (-3 \, d x - 3 \, c\right )} + 915 \, e^{\left (-4 \, d x - 4 \, c\right )} - 450 i \, 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] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 283 vs. \(2 (126) = 252\).
time = 0.49, size = 283, normalized size = 1.77 \begin {gather*} -\frac {837 \, {\left (125 i \, e^{\left (6 \, d x + 6 \, c\right )} + 450 \, e^{\left (5 \, d x + 5 \, c\right )} - 915 i \, e^{\left (4 \, d x + 4 \, c\right )} - 1116 \, e^{\left (3 \, d x + 3 \, c\right )} + 915 i \, e^{\left (2 \, d x + 2 \, c\right )} + 450 \, e^{\left (d x + c\right )} - 125 i\right )} \log \left (e^{\left (d x + c\right )} - \frac {3}{5} i + \frac {4}{5}\right ) + 837 \, {\left (-125 i \, e^{\left (6 \, d x + 6 \, c\right )} - 450 \, e^{\left (5 \, d x + 5 \, c\right )} + 915 i \, e^{\left (4 \, d x + 4 \, c\right )} + 1116 \, e^{\left (3 \, d x + 3 \, c\right )} - 915 i \, e^{\left (2 \, d x + 2 \, c\right )} - 450 \, e^{\left (d x + c\right )} + 125 i\right )} \log \left (e^{\left (d x + c\right )} - \frac {3}{5} i - \frac {4}{5}\right ) - 167400 i \, e^{\left (5 \, d x + 5 \, c\right )} - 502200 \, e^{\left (4 \, d x + 4 \, c\right )} + 888336 i \, e^{\left (3 \, d x + 3 \, c\right )} + 954480 \, e^{\left (2 \, d x + 2 \, c\right )} - 549000 i \, e^{\left (d x + c\right )} - 199000}{98304 \, {\left (125 \, d e^{\left (6 \, d x + 6 \, c\right )} - 450 i \, d e^{\left (5 \, d x + 5 \, c\right )} - 915 \, d e^{\left (4 \, d x + 4 \, c\right )} + 1116 i \, d e^{\left (3 \, d x + 3 \, c\right )} + 915 \, d e^{\left (2 \, d x + 2 \, c\right )} - 450 i \, d e^{\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 [A]
time = 0.37, size = 197, normalized size = 1.23 \begin {gather*} \frac {20925 i e^{5 c} e^{5 d x} + 62775 e^{4 c} e^{4 d x} - 111042 i e^{3 c} e^{3 d x} - 119310 e^{2 c} e^{2 d x} + 68625 i e^{c} e^{d x} + 24875}{1536000 d e^{6 c} e^{6 d x} - 5529600 i d e^{5 c} e^{5 d x} - 11243520 d e^{4 c} e^{4 d x} + 13713408 i d e^{3 c} e^{3 d x} + 11243520 d e^{2 c} e^{2 d x} - 5529600 i d e^{c} e^{d x} - 1536000 d} + \frac {\operatorname {RootSum} {\left (1073741824 z^{2} + 77841, \left ( i \mapsto i \log {\left (\frac {\left (131072 i i - 837 i\right ) e^{- c}}{1395} + e^{d x} \right )} \right )\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 111, normalized size = 0.69 \begin {gather*} \frac {\frac {8 \, {\left (20925 i \, e^{\left (5 \, d x + 5 \, c\right )} + 62775 \, e^{\left (4 \, d x + 4 \, c\right )} - 111042 i \, e^{\left (3 \, d x + 3 \, c\right )} - 119310 \, e^{\left (2 \, d x + 2 \, c\right )} + 68625 i \, e^{\left (d x + c\right )} + 24875\right )}}{{\left (5 \, e^{\left (2 \, d x + 2 \, c\right )} - 6 i \, e^{\left (d x + c\right )} - 5\right )}^{3}} - 837 i \, \log \left (-\left (i - 2\right ) \, e^{\left (d x + c\right )} - 2 i + 1\right ) + 837 i \, \log \left (-\left (2 i - 1\right ) \, e^{\left (d x + c\right )} + i - 2\right )}{98304 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.26, size = 237, normalized size = 1.48 \begin {gather*} -\frac {\frac {837}{102400\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,279{}\mathrm {i}}{20480\,d}}{1-{\mathrm {e}}^{2\,c+2\,d\,x}+\frac {{\mathrm {e}}^{c+d\,x}\,6{}\mathrm {i}}{5}}+\frac {\frac {7}{3750\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,39{}\mathrm {i}}{6250\,d}}{\frac {183\,{\mathrm {e}}^{4\,c+4\,d\,x}}{25}-\frac {183\,{\mathrm {e}}^{2\,c+2\,d\,x}}{25}-{\mathrm {e}}^{6\,c+6\,d\,x}+1+\frac {{\mathrm {e}}^{c+d\,x}\,18{}\mathrm {i}}{5}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,1116{}\mathrm {i}}{125}+\frac {{\mathrm {e}}^{5\,c+5\,d\,x}\,18{}\mathrm {i}}{5}}-\frac {\ln \left (-\frac {1395}{4}+{\mathrm {e}}^{c+d\,x}\,\left (-279-\frac {837}{4}{}\mathrm {i}\right )\right )\,279{}\mathrm {i}}{32768\,d}+\frac {\ln \left (\frac {1395}{4}+{\mathrm {e}}^{c+d\,x}\,\left (-279+\frac {837}{4}{}\mathrm {i}\right )\right )\,279{}\mathrm {i}}{32768\,d}-\frac {\frac {791}{80000\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,93{}\mathrm {i}}{16000\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-\frac {86\,{\mathrm {e}}^{2\,c+2\,d\,x}}{25}+1+\frac {{\mathrm {e}}^{c+d\,x}\,12{}\mathrm {i}}{5}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,12{}\mathrm {i}}{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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