Optimal. Leaf size=83 \[ -\frac {8 i a^6}{d (a-i a \tan (c+d x))}+\frac {i a^5 \tan ^2(c+d x)}{2 d}+\frac {5 a^5 \tan (c+d x)}{d}+\frac {12 i a^5 \log (\cos (c+d x))}{d}-12 a^5 x \]
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Rubi [A] time = 0.06, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3487, 43} \[ \frac {i a^5 \tan ^2(c+d x)}{2 d}-\frac {8 i a^6}{d (a-i a \tan (c+d x))}+\frac {5 a^5 \tan (c+d x)}{d}+\frac {12 i a^5 \log (\cos (c+d x))}{d}-12 a^5 x \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac {\left (i a^3\right ) \operatorname {Subst}\left (\int \frac {(a+x)^3}{(a-x)^2} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {\left (i a^3\right ) \operatorname {Subst}\left (\int \left (5 a+\frac {8 a^3}{(a-x)^2}-\frac {12 a^2}{a-x}+x\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-12 a^5 x+\frac {12 i a^5 \log (\cos (c+d x))}{d}+\frac {5 a^5 \tan (c+d x)}{d}+\frac {i a^5 \tan ^2(c+d x)}{2 d}-\frac {8 i a^6}{d (a-i a \tan (c+d x))}\\ \end {align*}
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Mathematica [B] time = 6.84, size = 649, normalized size = 7.82 \[ \frac {(4 \cos (3 c)-4 i \sin (3 c)) \sin (2 d x) \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{d (\cos (d x)+i \sin (d x))^5}-\frac {12 x \cos (5 c) \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{(\cos (d x)+i \sin (d x))^5}+\frac {(-4 \sin (3 c)-4 i \cos (3 c)) \cos (2 d x) \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{d (\cos (d x)+i \sin (d x))^5}+\frac {12 i x \sin (5 c) \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{(\cos (d x)+i \sin (d x))^5}+\frac {(5 \cos (5 c)-5 i \sin (5 c)) \sin (d x) \cos ^4(c+d x) (a+i a \tan (c+d x))^5}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) (\cos (d x)+i \sin (d x))^5}+\frac {\left (\frac {1}{2} \sin (5 c)+\frac {1}{2} i \cos (5 c)\right ) \cos ^3(c+d x) (a+i a \tan (c+d x))^5}{d (\cos (d x)+i \sin (d x))^5}+\frac {x \cos ^5(c+d x) \left (36 i \sin ^5(c)+24 i \sin ^3(c)-6 \cos ^5(c)+6 \cos ^3(c)+6 \sin ^5(c) \tan (c)+6 \sin ^3(c) \tan (c)+36 i \sin (c) \cos ^4(c)+90 \sin ^2(c) \cos ^3(c)-120 i \sin ^3(c) \cos ^2(c)-24 i \sin (c) \cos ^2(c)-90 \sin ^4(c) \cos (c)-36 \sin ^2(c) \cos (c)-i \tan (c) (12 \cos (5 c)-12 i \sin (5 c))\right ) (a+i a \tan (c+d x))^5}{(\cos (d x)+i \sin (d x))^5}+\frac {6 i \cos (5 c) \cos ^5(c+d x) (a+i a \tan (c+d x))^5 \log \left (\cos ^2(c+d x)\right )}{d (\cos (d x)+i \sin (d x))^5}+\frac {6 \sin (5 c) \cos ^5(c+d x) (a+i a \tan (c+d x))^5 \log \left (\cos ^2(c+d x)\right )}{d (\cos (d x)+i \sin (d x))^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 123, normalized size = 1.48 \[ \frac {-4 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 8 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 8 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + 10 i \, a^{5} + {\left (12 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 24 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + 12 i \, a^{5}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.60, size = 145, normalized size = 1.75 \[ \frac {12 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 24 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 4 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 8 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 8 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + 12 i \, a^{5} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 10 i \, a^{5}}{d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.42, size = 175, normalized size = 2.11 \[ \frac {i a^{5} \left (\sin ^{4}\left (d x +c \right )\right )}{2 d}-\frac {5 i a^{5} \left (\cos ^{2}\left (d x +c \right )\right )}{2 d}+\frac {i a^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {6 i a^{5} \left (\sin ^{2}\left (d x +c \right )\right )}{d}+\frac {5 a^{5} \left (\sin ^{5}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )}+\frac {5 a^{5} \cos \left (d x +c \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{d}+\frac {13 a^{5} \sin \left (d x +c \right ) \cos \left (d x +c \right )}{d}-12 a^{5} x -\frac {12 a^{5} c}{d}+\frac {12 i a^{5} \ln \left (\cos \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 86, normalized size = 1.04 \[ -\frac {-i \, a^{5} \tan \left (d x + c\right )^{2} + 24 \, {\left (d x + c\right )} a^{5} + 12 i \, a^{5} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 10 \, a^{5} \tan \left (d x + c\right ) - \frac {16 \, {\left (a^{5} \tan \left (d x + c\right ) - i \, a^{5}\right )}}{\tan \left (d x + c\right )^{2} + 1}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.30, size = 70, normalized size = 0.84 \[ \frac {8\,a^5}{d\,\left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}-\frac {a^5\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,12{}\mathrm {i}}{d}+\frac {5\,a^5\,\mathrm {tan}\left (c+d\,x\right )}{d}+\frac {a^5\,{\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 134, normalized size = 1.61 \[ \frac {12 i a^{5} \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {- 12 a^{5} e^{2 i c} e^{2 i d x} - 10 a^{5}}{i d e^{4 i c} e^{4 i d x} + 2 i d e^{2 i c} e^{2 i d x} + i d} + \begin {cases} - \frac {4 i a^{5} e^{2 i c} e^{2 i d x}}{d} & \text {for}\: d \neq 0 \\8 a^{5} x e^{2 i c} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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