Optimal. Leaf size=224 \[ \frac {7 x}{64 a^4}+\frac {i a^2}{48 d (a+i a \tan (c+d x))^6}+\frac {3 i a}{80 d (a+i a \tan (c+d x))^5}+\frac {3 i}{64 d (a+i a \tan (c+d x))^4}+\frac {5 i}{96 a d (a+i a \tan (c+d x))^3}-\frac {i}{256 d \left (a^2-i a^2 \tan (c+d x)\right )^2}+\frac {15 i}{256 d \left (a^2+i a^2 \tan (c+d x)\right )^2}-\frac {7 i}{256 d \left (a^4-i a^4 \tan (c+d x)\right )}+\frac {21 i}{256 d \left (a^4+i a^4 \tan (c+d x)\right )} \]
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Rubi [A]
time = 0.11, antiderivative size = 224, 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 = {3568, 46, 212}
\begin {gather*} -\frac {7 i}{256 d \left (a^4-i a^4 \tan (c+d x)\right )}+\frac {21 i}{256 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {7 x}{64 a^4}+\frac {i a^2}{48 d (a+i a \tan (c+d x))^6}-\frac {i}{256 d \left (a^2-i a^2 \tan (c+d x)\right )^2}+\frac {15 i}{256 d \left (a^2+i a^2 \tan (c+d x)\right )^2}+\frac {3 i a}{80 d (a+i a \tan (c+d x))^5}+\frac {3 i}{64 d (a+i a \tan (c+d x))^4}+\frac {5 i}{96 a d (a+i a \tan (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 212
Rule 3568
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=-\frac {\left (i a^5\right ) \text {Subst}\left (\int \frac {1}{(a-x)^3 (a+x)^7} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {\left (i a^5\right ) \text {Subst}\left (\int \left (\frac {1}{128 a^7 (a-x)^3}+\frac {7}{256 a^8 (a-x)^2}+\frac {1}{8 a^3 (a+x)^7}+\frac {3}{16 a^4 (a+x)^6}+\frac {3}{16 a^5 (a+x)^5}+\frac {5}{32 a^6 (a+x)^4}+\frac {15}{128 a^7 (a+x)^3}+\frac {21}{256 a^8 (a+x)^2}+\frac {7}{64 a^8 \left (a^2-x^2\right )}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=\frac {i a^2}{48 d (a+i a \tan (c+d x))^6}+\frac {3 i a}{80 d (a+i a \tan (c+d x))^5}+\frac {3 i}{64 d (a+i a \tan (c+d x))^4}+\frac {5 i}{96 a d (a+i a \tan (c+d x))^3}-\frac {i}{256 d \left (a^2-i a^2 \tan (c+d x)\right )^2}+\frac {15 i}{256 d \left (a^2+i a^2 \tan (c+d x)\right )^2}-\frac {7 i}{256 d \left (a^4-i a^4 \tan (c+d x)\right )}+\frac {21 i}{256 d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac {(7 i) \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,i a \tan (c+d x)\right )}{64 a^3 d}\\ &=\frac {7 x}{64 a^4}+\frac {i a^2}{48 d (a+i a \tan (c+d x))^6}+\frac {3 i a}{80 d (a+i a \tan (c+d x))^5}+\frac {3 i}{64 d (a+i a \tan (c+d x))^4}+\frac {5 i}{96 a d (a+i a \tan (c+d x))^3}-\frac {i}{256 d \left (a^2-i a^2 \tan (c+d x)\right )^2}+\frac {15 i}{256 d \left (a^2+i a^2 \tan (c+d x)\right )^2}-\frac {7 i}{256 d \left (a^4-i a^4 \tan (c+d x)\right )}+\frac {21 i}{256 d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.80, size = 142, normalized size = 0.63 \begin {gather*} \frac {\sec ^4(c+d x) (525 i+1120 i \cos (2 (c+d x))+105 (i+8 d x) \cos (4 (c+d x))-96 i \cos (6 (c+d x))-5 i \cos (8 (c+d x))-560 \sin (2 (c+d x))+105 \sin (4 (c+d x))+840 i d x \sin (4 (c+d x))+144 \sin (6 (c+d x))+10 \sin (8 (c+d x)))}{7680 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 = 143, normalized size = 0.64
method | result | size |
derivativedivides | \(\frac {-\frac {7 i \ln \left (\tan \left (d x +c \right )-i\right )}{128}+\frac {3 i}{64 \left (\tan \left (d x +c \right )-i\right )^{4}}-\frac {i}{48 \left (\tan \left (d x +c \right )-i\right )^{6}}-\frac {15 i}{256 \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {3}{80 \left (\tan \left (d x +c \right )-i\right )^{5}}-\frac {5}{96 \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {21}{256 \left (\tan \left (d x +c \right )-i\right )}+\frac {i}{256 \left (\tan \left (d x +c \right )+i\right )^{2}}+\frac {7 i \ln \left (\tan \left (d x +c \right )+i\right )}{128}+\frac {7}{256 \left (\tan \left (d x +c \right )+i\right )}}{d \,a^{4}}\) | \(143\) |
default | \(\frac {-\frac {7 i \ln \left (\tan \left (d x +c \right )-i\right )}{128}+\frac {3 i}{64 \left (\tan \left (d x +c \right )-i\right )^{4}}-\frac {i}{48 \left (\tan \left (d x +c \right )-i\right )^{6}}-\frac {15 i}{256 \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {3}{80 \left (\tan \left (d x +c \right )-i\right )^{5}}-\frac {5}{96 \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {21}{256 \left (\tan \left (d x +c \right )-i\right )}+\frac {i}{256 \left (\tan \left (d x +c \right )+i\right )^{2}}+\frac {7 i \ln \left (\tan \left (d x +c \right )+i\right )}{128}+\frac {7}{256 \left (\tan \left (d x +c \right )+i\right )}}{d \,a^{4}}\) | \(143\) |
risch | \(\frac {7 x}{64 a^{4}}+\frac {7 i {\mathrm e}^{-6 i \left (d x +c \right )}}{192 a^{4} d}+\frac {7 i {\mathrm e}^{-8 i \left (d x +c \right )}}{512 a^{4} d}+\frac {i {\mathrm e}^{-10 i \left (d x +c \right )}}{320 a^{4} d}+\frac {i {\mathrm e}^{-12 i \left (d x +c \right )}}{3072 a^{4} d}+\frac {69 i \cos \left (4 d x +4 c \right )}{1024 a^{4} d}+\frac {71 \sin \left (4 d x +4 c \right )}{1024 a^{4} d}+\frac {3 i \cos \left (2 d x +2 c \right )}{32 a^{4} d}+\frac {\sin \left (2 d x +2 c \right )}{8 a^{4} d}\) | \(150\) |
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 = 0.39, size = 109, normalized size = 0.49 \begin {gather*} \frac {{\left (1680 \, d x e^{\left (12 i \, d x + 12 i \, c\right )} - 15 i \, e^{\left (16 i \, d x + 16 i \, c\right )} - 240 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 1680 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 1050 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 560 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 210 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 48 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i\right )} e^{\left (-12 i \, d x - 12 i \, c\right )}}{15360 \, a^{4} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.42, size = 326, normalized size = 1.46 \begin {gather*} \begin {cases} \frac {\left (- 202661983231672320 i a^{28} d^{7} e^{46 i c} e^{4 i d x} - 3242591731706757120 i a^{28} d^{7} e^{44 i c} e^{2 i d x} + 22698142121947299840 i a^{28} d^{7} e^{40 i c} e^{- 2 i d x} + 14186338826217062400 i a^{28} d^{7} e^{38 i c} e^{- 4 i d x} + 7566047373982433280 i a^{28} d^{7} e^{36 i c} e^{- 6 i d x} + 2837267765243412480 i a^{28} d^{7} e^{34 i c} e^{- 8 i d x} + 648518346341351424 i a^{28} d^{7} e^{32 i c} e^{- 10 i d x} + 67553994410557440 i a^{28} d^{7} e^{30 i c} e^{- 12 i d x}\right ) e^{- 42 i c}}{207525870829232455680 a^{32} d^{8}} & \text {for}\: a^{32} d^{8} e^{42 i c} \neq 0 \\x \left (\frac {\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^{- 12 i c}}{256 a^{4}} - \frac {7}{64 a^{4}}\right ) & \text {otherwise} \end {cases} + \frac {7 x}{64 a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.85, size = 147, normalized size = 0.66 \begin {gather*} -\frac {-\frac {420 i \, \log \left (-i \, \tan \left (d x + c\right ) + 1\right )}{a^{4}} + \frac {420 i \, \log \left (-i \, \tan \left (d x + c\right ) - 1\right )}{a^{4}} + \frac {30 \, {\left (21 i \, \tan \left (d x + c\right )^{2} - 49 \, \tan \left (d x + c\right ) - 29 i\right )}}{a^{4} {\left (\tan \left (d x + c\right ) + i\right )}^{2}} + \frac {-1029 i \, \tan \left (d x + c\right )^{6} - 6804 \, \tan \left (d x + c\right )^{5} + 19035 i \, \tan \left (d x + c\right )^{4} + 29080 \, \tan \left (d x + c\right )^{3} - 25995 i \, \tan \left (d x + c\right )^{2} - 13332 \, \tan \left (d x + c\right ) + 3317 i}{a^{4} {\left (\tan \left (d x + c\right ) - i\right )}^{6}}}{7680 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.94, size = 197, normalized size = 0.88 \begin {gather*} \frac {7\,x}{64\,a^4}+\frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,169{}\mathrm {i}}{960\,a^4}+\frac {4}{15\,a^4}+\frac {119\,{\mathrm {tan}\left (c+d\,x\right )}^2}{240\,a^4}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,889{}\mathrm {i}}{960\,a^4}-\frac {7\,{\mathrm {tan}\left (c+d\,x\right )}^4}{24\,a^4}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^5\,91{}\mathrm {i}}{192\,a^4}-\frac {7\,{\mathrm {tan}\left (c+d\,x\right )}^6}{16\,a^4}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^7\,7{}\mathrm {i}}{64\,a^4}}{d\,\left (-{\mathrm {tan}\left (c+d\,x\right )}^8\,1{}\mathrm {i}-4\,{\mathrm {tan}\left (c+d\,x\right )}^7+{\mathrm {tan}\left (c+d\,x\right )}^6\,4{}\mathrm {i}-4\,{\mathrm {tan}\left (c+d\,x\right )}^5+{\mathrm {tan}\left (c+d\,x\right )}^4\,10{}\mathrm {i}+4\,{\mathrm {tan}\left (c+d\,x\right )}^3+{\mathrm {tan}\left (c+d\,x\right )}^2\,4{}\mathrm {i}+4\,\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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