Optimal. Leaf size=169 \[ \frac {3 x}{32 a^4}+\frac {i a}{20 d (a+i a \tan (c+d x))^5}+\frac {i}{16 d (a+i a \tan (c+d x))^4}+\frac {i}{16 a d (a+i a \tan (c+d x))^3}+\frac {i}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}-\frac {i}{64 d \left (a^4-i a^4 \tan (c+d x)\right )}+\frac {5 i}{64 d \left (a^4+i a^4 \tan (c+d x)\right )} \]
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Rubi [A]
time = 0.08, antiderivative size = 169, 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 {i}{64 d \left (a^4-i a^4 \tan (c+d x)\right )}+\frac {5 i}{64 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {3 x}{32 a^4}+\frac {i}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}+\frac {i a}{20 d (a+i a \tan (c+d x))^5}+\frac {i}{16 d (a+i a \tan (c+d x))^4}+\frac {i}{16 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 ^2(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=-\frac {\left (i a^3\right ) \text {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)^6} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {\left (i a^3\right ) \text {Subst}\left (\int \left (\frac {1}{64 a^6 (a-x)^2}+\frac {1}{4 a^2 (a+x)^6}+\frac {1}{4 a^3 (a+x)^5}+\frac {3}{16 a^4 (a+x)^4}+\frac {1}{8 a^5 (a+x)^3}+\frac {5}{64 a^6 (a+x)^2}+\frac {3}{32 a^6 \left (a^2-x^2\right )}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=\frac {i a}{20 d (a+i a \tan (c+d x))^5}+\frac {i}{16 d (a+i a \tan (c+d x))^4}+\frac {i}{16 a d (a+i a \tan (c+d x))^3}+\frac {i}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}-\frac {i}{64 d \left (a^4-i a^4 \tan (c+d x)\right )}+\frac {5 i}{64 d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac {(3 i) \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,i a \tan (c+d x)\right )}{32 a^3 d}\\ &=\frac {3 x}{32 a^4}+\frac {i a}{20 d (a+i a \tan (c+d x))^5}+\frac {i}{16 d (a+i a \tan (c+d x))^4}+\frac {i}{16 a d (a+i a \tan (c+d x))^3}+\frac {i}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}-\frac {i}{64 d \left (a^4-i a^4 \tan (c+d x)\right )}+\frac {5 i}{64 d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 120, normalized size = 0.71 \begin {gather*} \frac {\sec ^4(c+d x) (100 i+200 i \cos (2 (c+d x))+15 (i+8 d x) \cos (4 (c+d x))-8 i \cos (6 (c+d x))-100 \sin (2 (c+d x))+15 \sin (4 (c+d x))+120 i d x \sin (4 (c+d x))+12 \sin (6 (c+d x)))}{1280 a^4 d (-i+\tan (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 115, normalized size = 0.68
method | result | size |
derivativedivides | \(\frac {-\frac {3 i \ln \left (\tan \left (d x +c \right )-i\right )}{64}+\frac {i}{16 \left (\tan \left (d x +c \right )-i\right )^{4}}-\frac {i}{16 \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {1}{20 \left (\tan \left (d x +c \right )-i\right )^{5}}-\frac {1}{16 \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {5}{64 \left (\tan \left (d x +c \right )-i\right )}+\frac {3 i \ln \left (\tan \left (d x +c \right )+i\right )}{64}+\frac {1}{64 \tan \left (d x +c \right )+64 i}}{d \,a^{4}}\) | \(115\) |
default | \(\frac {-\frac {3 i \ln \left (\tan \left (d x +c \right )-i\right )}{64}+\frac {i}{16 \left (\tan \left (d x +c \right )-i\right )^{4}}-\frac {i}{16 \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {1}{20 \left (\tan \left (d x +c \right )-i\right )^{5}}-\frac {1}{16 \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {5}{64 \left (\tan \left (d x +c \right )-i\right )}+\frac {3 i \ln \left (\tan \left (d x +c \right )+i\right )}{64}+\frac {1}{64 \tan \left (d x +c \right )+64 i}}{d \,a^{4}}\) | \(115\) |
risch | \(\frac {3 x}{32 a^{4}}+\frac {5 i {\mathrm e}^{-4 i \left (d x +c \right )}}{64 a^{4} d}+\frac {5 i {\mathrm e}^{-6 i \left (d x +c \right )}}{128 a^{4} d}+\frac {3 i {\mathrm e}^{-8 i \left (d x +c \right )}}{256 a^{4} d}+\frac {i {\mathrm e}^{-10 i \left (d x +c \right )}}{640 a^{4} d}+\frac {7 i \cos \left (2 d x +2 c \right )}{64 a^{4} d}+\frac {\sin \left (2 d x +2 c \right )}{8 a^{4} d}\) | \(115\) |
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 = 0.42, size = 87, normalized size = 0.51 \begin {gather*} \frac {{\left (120 \, d x e^{\left (10 i \, d x + 10 i \, c\right )} - 10 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 150 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 100 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 50 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 15 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i\right )} e^{\left (-10 i \, d x - 10 i \, c\right )}}{1280 \, a^{4} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.33, size = 258, normalized size = 1.53 \begin {gather*} \begin {cases} \frac {\left (- 171798691840 i a^{20} d^{5} e^{32 i c} e^{2 i d x} + 2576980377600 i a^{20} d^{5} e^{28 i c} e^{- 2 i d x} + 1717986918400 i a^{20} d^{5} e^{26 i c} e^{- 4 i d x} + 858993459200 i a^{20} d^{5} e^{24 i c} e^{- 6 i d x} + 257698037760 i a^{20} d^{5} e^{22 i c} e^{- 8 i d x} + 34359738368 i a^{20} d^{5} e^{20 i c} e^{- 10 i d x}\right ) e^{- 30 i c}}{21990232555520 a^{24} d^{6}} & \text {for}\: a^{24} d^{6} e^{30 i c} \neq 0 \\x \left (\frac {\left (e^{12 i c} + 6 e^{10 i c} + 15 e^{8 i c} + 20 e^{6 i c} + 15 e^{4 i c} + 6 e^{2 i c} + 1\right ) e^{- 10 i c}}{64 a^{4}} - \frac {3}{32 a^{4}}\right ) & \text {otherwise} \end {cases} + \frac {3 x}{32 a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.85, size = 123, normalized size = 0.73 \begin {gather*} -\frac {-\frac {60 i \, \log \left (\tan \left (d x + c\right ) + i\right )}{a^{4}} + \frac {60 i \, \log \left (\tan \left (d x + c\right ) - i\right )}{a^{4}} + \frac {20 \, {\left (3 i \, \tan \left (d x + c\right ) - 4\right )}}{a^{4} {\left (\tan \left (d x + c\right ) + i\right )}} + \frac {-137 i \, \tan \left (d x + c\right )^{5} - 785 \, \tan \left (d x + c\right )^{4} + 1850 i \, \tan \left (d x + c\right )^{3} + 2290 \, \tan \left (d x + c\right )^{2} - 1565 i \, \tan \left (d x + c\right ) - 541}{a^{4} {\left (\tan \left (d x + c\right ) - i\right )}^{5}}}{1280 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.13, size = 90, normalized size = 0.53 \begin {gather*} \frac {3\,x}{32\,a^4}-\frac {-\frac {3\,{\mathrm {tan}\left (c+d\,x\right )}^5}{32}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,3{}\mathrm {i}}{8}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{2}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}}{8}+\frac {47\,\mathrm {tan}\left (c+d\,x\right )}{160}-\frac {3}{10}{}\mathrm {i}}{a^4\,d\,{\left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}^5\,\left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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