Optimal. Leaf size=156 \[ \frac {10 \sin (c+d x)}{33 a^4 d}-\frac {20 \sin ^3(c+d x)}{99 a^4 d}+\frac {2 \sin ^5(c+d x)}{33 a^4 d}+\frac {i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}+\frac {7 i \cos ^3(c+d x)}{99 a d (a+i a \tan (c+d x))^3}+\frac {4 i \cos ^5(c+d x)}{33 d \left (a^4+i a^4 \tan (c+d x)\right )} \]
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Rubi [A]
time = 0.12, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3583, 3581,
2713} \begin {gather*} \frac {2 \sin ^5(c+d x)}{33 a^4 d}-\frac {20 \sin ^3(c+d x)}{99 a^4 d}+\frac {10 \sin (c+d x)}{33 a^4 d}+\frac {4 i \cos ^5(c+d x)}{33 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {7 i \cos ^3(c+d x)}{99 a d (a+i a \tan (c+d x))^3}+\frac {i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rule 3581
Rule 3583
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=\frac {i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}+\frac {7 \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx}{11 a}\\ &=\frac {i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}+\frac {7 i \cos ^3(c+d x)}{99 a d (a+i a \tan (c+d x))^3}+\frac {14 \int \frac {\cos ^3(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{33 a^2}\\ &=\frac {i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}+\frac {7 i \cos ^3(c+d x)}{99 a d (a+i a \tan (c+d x))^3}+\frac {4 i \cos ^5(c+d x)}{33 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {10 \int \cos ^5(c+d x) \, dx}{33 a^4}\\ &=\frac {i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}+\frac {7 i \cos ^3(c+d x)}{99 a d (a+i a \tan (c+d x))^3}+\frac {4 i \cos ^5(c+d x)}{33 d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac {10 \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{33 a^4 d}\\ &=\frac {10 \sin (c+d x)}{33 a^4 d}-\frac {20 \sin ^3(c+d x)}{99 a^4 d}+\frac {2 \sin ^5(c+d x)}{33 a^4 d}+\frac {i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}+\frac {7 i \cos ^3(c+d x)}{99 a d (a+i a \tan (c+d x))^3}+\frac {4 i \cos ^5(c+d x)}{33 d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.55, size = 117, normalized size = 0.75 \begin {gather*} -\frac {i \sec ^4(c+d x) (-924 \cos (c+d x)-1188 \cos (3 (c+d x))+308 \cos (5 (c+d x))+12 \cos (7 (c+d x))-231 i \sin (c+d x)-891 i \sin (3 (c+d x))+385 i \sin (5 (c+d x))+21 i \sin (7 (c+d x)))}{6336 a^4 d (-i+\tan (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 240, normalized size = 1.54
method | result | size |
risch | \(\frac {7 i {\mathrm e}^{-5 i \left (d x +c \right )}}{128 a^{4} d}+\frac {3 i {\mathrm e}^{-7 i \left (d x +c \right )}}{128 a^{4} d}+\frac {7 i {\mathrm e}^{-9 i \left (d x +c \right )}}{1152 a^{4} d}+\frac {i {\mathrm e}^{-11 i \left (d x +c \right )}}{1408 a^{4} d}+\frac {7 i \cos \left (d x +c \right )}{64 a^{4} d}+\frac {7 \sin \left (d x +c \right )}{32 a^{4} d}+\frac {17 i \cos \left (3 d x +3 c \right )}{192 a^{4} d}+\frac {3 \sin \left (3 d x +3 c \right )}{32 a^{4} d}\) | \(138\) |
derivativedivides | \(\frac {-\frac {i}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{2}}-\frac {1}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{3}}+\frac {2}{16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+16 i}+\frac {8 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {67 i}{2 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {44 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {385 i}{6 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {201 i}{32 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {16}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {208}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {61}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {105}{2 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {267}{16 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {15}{8 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{a^{4} d}\) | \(240\) |
default | \(\frac {-\frac {i}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{2}}-\frac {1}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{3}}+\frac {2}{16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+16 i}+\frac {8 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {67 i}{2 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {44 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {385 i}{6 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {201 i}{32 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {16}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {208}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {61}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {105}{2 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {267}{16 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {15}{8 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{a^{4} d}\) | \(240\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 96, normalized size = 0.62 \begin {gather*} \frac {{\left (-33 i \, e^{\left (14 i \, d x + 14 i \, c\right )} - 693 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 2079 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 1155 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 693 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 297 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 77 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 9 i\right )} e^{\left (-11 i \, d x - 11 i \, c\right )}}{12672 \, a^{4} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 299 vs. \(2 (136) = 272\).
time = 0.46, size = 299, normalized size = 1.92 \begin {gather*} \begin {cases} \frac {\left (- 167196136166129664 i a^{28} d^{7} e^{39 i c} e^{3 i d x} - 3511118859488722944 i a^{28} d^{7} e^{37 i c} e^{i d x} + 10533356578466168832 i a^{28} d^{7} e^{35 i c} e^{- i d x} + 5851864765814538240 i a^{28} d^{7} e^{33 i c} e^{- 3 i d x} + 3511118859488722944 i a^{28} d^{7} e^{31 i c} e^{- 5 i d x} + 1504765225495166976 i a^{28} d^{7} e^{29 i c} e^{- 7 i d x} + 390124317720969216 i a^{28} d^{7} e^{27 i c} e^{- 9 i d x} + 45598946227126272 i a^{28} d^{7} e^{25 i c} e^{- 11 i d x}\right ) e^{- 36 i c}}{64203316287793790976 a^{32} d^{8}} & \text {for}\: a^{32} d^{8} e^{36 i c} \neq 0 \\\frac {x \left (e^{14 i c} + 7 e^{12 i c} + 21 e^{10 i c} + 35 e^{8 i c} + 35 e^{6 i c} + 21 e^{4 i c} + 7 e^{2 i c} + 1\right ) e^{- 11 i c}}{128 a^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.86, size = 197, normalized size = 1.26 \begin {gather*} \frac {\frac {33 \, {\left (12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 21 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 11\right )}}{a^{4} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right )}^{3}} + \frac {5940 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 39501 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 141075 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 313236 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 479556 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 516054 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 397914 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 214500 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 79024 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 17765 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2155}{a^{4} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{11}}}{3168 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.61, size = 216, normalized size = 1.38 \begin {gather*} -\frac {\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {269\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}-\frac {1307\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{64}+\frac {1307\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{64}-\frac {1099\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{32}+\frac {203\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{32}-\frac {21\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{64}+\frac {21\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{64}+\frac {\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,231{}\mathrm {i}}{16}-\frac {\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,231{}\mathrm {i}}{16}+\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,33{}\mathrm {i}-\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )\,5{}\mathrm {i}+\frac {\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,3{}\mathrm {i}}{16}-\frac {\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,3{}\mathrm {i}}{16}\right )\,2{}\mathrm {i}}{99\,a^4\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^{11}\,{\left (\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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