Optimal. Leaf size=139 \[ \frac {56 \sin (c+d x)}{99 a^3 d}-\frac {56 \sin ^3(c+d x)}{99 a^3 d}+\frac {56 \sin ^5(c+d x)}{165 a^3 d}-\frac {8 \sin ^7(c+d x)}{99 a^3 d}+\frac {i \cos ^5(c+d x)}{11 d (a+i a \tan (c+d x))^3}+\frac {16 i \cos ^7(c+d x)}{99 d \left (a^3+i a^3 \tan (c+d x)\right )} \]
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Rubi [A]
time = 0.09, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {8 \sin ^7(c+d x)}{99 a^3 d}+\frac {56 \sin ^5(c+d x)}{165 a^3 d}-\frac {56 \sin ^3(c+d x)}{99 a^3 d}+\frac {56 \sin (c+d x)}{99 a^3 d}+\frac {16 i \cos ^7(c+d x)}{99 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {i \cos ^5(c+d x)}{11 d (a+i a \tan (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rule 3581
Rule 3583
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=\frac {i \cos ^5(c+d x)}{11 d (a+i a \tan (c+d x))^3}+\frac {8 \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{11 a}\\ &=\frac {i \cos ^5(c+d x)}{11 d (a+i a \tan (c+d x))^3}+\frac {16 i \cos ^7(c+d x)}{99 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {56 \int \cos ^7(c+d x) \, dx}{99 a^3}\\ &=\frac {i \cos ^5(c+d x)}{11 d (a+i a \tan (c+d x))^3}+\frac {16 i \cos ^7(c+d x)}{99 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac {56 \text {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{99 a^3 d}\\ &=\frac {56 \sin (c+d x)}{99 a^3 d}-\frac {56 \sin ^3(c+d x)}{99 a^3 d}+\frac {56 \sin ^5(c+d x)}{165 a^3 d}-\frac {8 \sin ^7(c+d x)}{99 a^3 d}+\frac {i \cos ^5(c+d x)}{11 d (a+i a \tan (c+d x))^3}+\frac {16 i \cos ^7(c+d x)}{99 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.76, size = 120, normalized size = 0.86 \begin {gather*} \frac {\sec ^3(c+d x) (-5775-16632 \cos (2 (c+d x))+5940 \cos (4 (c+d x))+440 \cos (6 (c+d x))+27 \cos (8 (c+d x))-11088 i \sin (2 (c+d x))+7920 i \sin (4 (c+d x))+880 i \sin (6 (c+d x))+72 i \sin (8 (c+d x)))}{63360 a^3 d (-i+\tan (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 272 vs. \(2 (123 ) = 246\).
time = 0.30, size = 273, normalized size = 1.96
method | result | size |
risch | \(\frac {i {\mathrm e}^{-7 i \left (d x +c \right )}}{64 a^{3} d}+\frac {i {\mathrm e}^{-9 i \left (d x +c \right )}}{288 a^{3} d}+\frac {i {\mathrm e}^{-11 i \left (d x +c \right )}}{2816 a^{3} d}+\frac {7 i \cos \left (d x +c \right )}{64 a^{3} d}+\frac {21 \sin \left (d x +c \right )}{64 a^{3} d}+\frac {11 i \cos \left (5 d x +5 c \right )}{256 a^{3} d}+\frac {57 \sin \left (5 d x +5 c \right )}{1280 a^{3} d}+\frac {31 i \cos \left (3 d x +3 c \right )}{384 a^{3} d}+\frac {13 \sin \left (3 d x +3 c \right )}{128 a^{3} d}\) | \(155\) |
derivativedivides | \(\frac {\frac {217 i}{6 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {i}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{4}}+\frac {1}{40 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{5}}-\frac {7}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{3}}+\frac {37}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )}-\frac {23 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {4 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {303 i}{64 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {169 i}{8 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {5 i}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{2}}-\frac {8}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {106}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {33}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {623}{20 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {365}{32 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {219}{128 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{a^{3} d}\) | \(273\) |
default | \(\frac {\frac {217 i}{6 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {i}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{4}}+\frac {1}{40 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{5}}-\frac {7}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{3}}+\frac {37}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )}-\frac {23 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {4 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {303 i}{64 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {169 i}{8 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {5 i}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{2}}-\frac {8}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {106}{9 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {33}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {623}{20 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {365}{32 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {219}{128 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{a^{3} d}\) | \(273\) |
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 = 107, normalized size = 0.77 \begin {gather*} \frac {{\left (-99 i \, e^{\left (16 i \, d x + 16 i \, c\right )} - 1320 i \, e^{\left (14 i \, d x + 14 i \, c\right )} - 13860 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 27720 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 11550 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 5544 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 1980 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 440 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 45 i\right )} e^{\left (-11 i \, d x - 11 i \, c\right )}}{126720 \, a^{3} 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. 333 vs. \(2 (122) = 244\).
time = 0.53, size = 333, normalized size = 2.40 \begin {gather*} \begin {cases} \frac {\left (- 626985510622986240 i a^{24} d^{8} e^{41 i c} e^{5 i d x} - 8359806808306483200 i a^{24} d^{8} e^{39 i c} e^{3 i d x} - 87777971487218073600 i a^{24} d^{8} e^{37 i c} e^{i d x} + 175555942974436147200 i a^{24} d^{8} e^{35 i c} e^{- i d x} + 73148309572681728000 i a^{24} d^{8} e^{33 i c} e^{- 3 i d x} + 35111188594887229440 i a^{24} d^{8} e^{31 i c} e^{- 5 i d x} + 12539710212459724800 i a^{24} d^{8} e^{29 i c} e^{- 7 i d x} + 2786602269435494400 i a^{24} d^{8} e^{27 i c} e^{- 9 i d x} + 284993413919539200 i a^{24} d^{8} e^{25 i c} e^{- 11 i d x}\right ) e^{- 36 i c}}{802541453597422387200 a^{27} d^{9}} & \text {for}\: a^{27} d^{9} e^{36 i c} \neq 0 \\\frac {x \left (e^{16 i c} + 8 e^{14 i c} + 28 e^{12 i c} + 56 e^{10 i c} + 70 e^{8 i c} + 56 e^{6 i c} + 28 e^{4 i c} + 8 e^{2 i c} + 1\right ) e^{- 11 i c}}{256 a^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.81, size = 223, normalized size = 1.60 \begin {gather*} \frac {\frac {33 \, {\left (555 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1920 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2710 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1760 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 463\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right )}^{5}} + \frac {108405 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 784080 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 2901195 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 6652800 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 10407474 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 11435424 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 8949270 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 4899840 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1816265 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 411664 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 47279}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{11}}}{63360 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.52, size = 136, normalized size = 0.98 \begin {gather*} \frac {\left (\frac {\cos \left (7\,c+7\,d\,x\right )}{64}+\frac {\cos \left (9\,c+9\,d\,x\right )}{288}+\frac {\cos \left (11\,c+11\,d\,x\right )}{2816}-\frac {\sin \left (7\,c+7\,d\,x\right )\,1{}\mathrm {i}}{64}-\frac {\sin \left (9\,c+9\,d\,x\right )\,1{}\mathrm {i}}{288}-\frac {\sin \left (11\,c+11\,d\,x\right )\,1{}\mathrm {i}}{2816}+\frac {\sqrt {224}\,\cos \left (5\,c+5\,d\,x+\mathrm {atanh}\left (\frac {57}{55}\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{1280}+\frac {\sqrt {560}\,\cos \left (3\,c+3\,d\,x+\mathrm {atanh}\left (\frac {39}{31}\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{384}+\frac {\sqrt {2}\,\cos \left (c+d\,x+\mathrm {atanh}\left (3\right )\,1{}\mathrm {i}\right )\,7{}\mathrm {i}}{32}\right )\,1{}\mathrm {i}}{a^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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