Optimal. Leaf size=99 \[ \frac {i \cos ^6(c+d x)}{6 a d}+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 a d}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{24 a d}+\frac {5 \sin (c+d x) \cos (c+d x)}{16 a d}+\frac {5 x}{16 a} \]
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Rubi [A] time = 0.15, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3092, 3090, 2635, 8, 2565, 30} \[ \frac {i \cos ^6(c+d x)}{6 a d}+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 a d}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{24 a d}+\frac {5 \sin (c+d x) \cos (c+d x)}{16 a d}+\frac {5 x}{16 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2565
Rule 2635
Rule 3090
Rule 3092
Rubi steps
\begin {align*} \int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx &=-\frac {i \int \cos ^5(c+d x) (i a \cos (c+d x)+a \sin (c+d x)) \, dx}{a^2}\\ &=-\frac {i \int \left (i a \cos ^6(c+d x)+a \cos ^5(c+d x) \sin (c+d x)\right ) \, dx}{a^2}\\ &=-\frac {i \int \cos ^5(c+d x) \sin (c+d x) \, dx}{a}+\frac {\int \cos ^6(c+d x) \, dx}{a}\\ &=\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d}+\frac {5 \int \cos ^4(c+d x) \, dx}{6 a}+\frac {i \operatorname {Subst}\left (\int x^5 \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac {i \cos ^6(c+d x)}{6 a d}+\frac {5 \cos ^3(c+d x) \sin (c+d x)}{24 a d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d}+\frac {5 \int \cos ^2(c+d x) \, dx}{8 a}\\ &=\frac {i \cos ^6(c+d x)}{6 a d}+\frac {5 \cos (c+d x) \sin (c+d x)}{16 a d}+\frac {5 \cos ^3(c+d x) \sin (c+d x)}{24 a d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d}+\frac {5 \int 1 \, dx}{16 a}\\ &=\frac {5 x}{16 a}+\frac {i \cos ^6(c+d x)}{6 a d}+\frac {5 \cos (c+d x) \sin (c+d x)}{16 a d}+\frac {5 \cos ^3(c+d x) \sin (c+d x)}{24 a d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 82, normalized size = 0.83 \[ \frac {45 \sin (2 (c+d x))+9 \sin (4 (c+d x))+\sin (6 (c+d x))+15 i \cos (2 (c+d x))+6 i \cos (4 (c+d x))+i \cos (6 (c+d x))+60 c+60 d x}{192 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 76, normalized size = 0.77 \[ \frac {{\left (120 \, d x e^{\left (6 i \, d x + 6 i \, c\right )} - 3 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 30 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 60 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 15 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{384 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 116, normalized size = 1.17 \[ -\frac {-\frac {30 i \, \log \left (\tan \left (d x + c\right ) + i\right )}{a} + \frac {30 i \, \log \left (\tan \left (d x + c\right ) - i\right )}{a} + \frac {3 \, {\left (-15 i \, \tan \left (d x + c\right )^{2} + 38 \, \tan \left (d x + c\right ) + 25 i\right )}}{a {\left (-i \, \tan \left (d x + c\right ) + 1\right )}^{2}} - \frac {55 i \, \tan \left (d x + c\right )^{3} + 201 \, \tan \left (d x + c\right )^{2} - 255 i \, \tan \left (d x + c\right ) - 117}{a {\left (\tan \left (d x + c\right ) - i\right )}^{3}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 137, normalized size = 1.38 \[ \frac {i}{32 a d \left (\tan \left (d x +c \right )+i\right )^{2}}+\frac {5 i \ln \left (\tan \left (d x +c \right )+i\right )}{32 a d}+\frac {1}{8 a d \left (\tan \left (d x +c \right )+i\right )}-\frac {5 i \ln \left (\tan \left (d x +c \right )-i\right )}{32 a d}-\frac {3 i}{32 a d \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {1}{24 a d \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {3}{16 a d \left (\tan \left (d x +c \right )-i\right )} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.14, size = 164, normalized size = 1.66 \[ \frac {5\,x}{16\,a}+\frac {\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{8}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,3{}\mathrm {i}}{4}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,1{}\mathrm {i}}{12}+\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,1{}\mathrm {i}}{12}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,3{}\mathrm {i}}{4}+\frac {11\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{a\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )}^4\,{\left (1+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.41, size = 223, normalized size = 2.25 \[ \begin {cases} - \frac {\left (50331648 i a^{4} d^{4} e^{16 i c} e^{4 i d x} + 503316480 i a^{4} d^{4} e^{14 i c} e^{2 i d x} - 1006632960 i a^{4} d^{4} e^{10 i c} e^{- 2 i d x} - 251658240 i a^{4} d^{4} e^{8 i c} e^{- 4 i d x} - 33554432 i a^{4} d^{4} e^{6 i c} e^{- 6 i d x}\right ) e^{- 12 i c}}{6442450944 a^{5} d^{5}} & \text {for}\: 6442450944 a^{5} d^{5} e^{12 i c} \neq 0 \\x \left (\frac {\left (e^{10 i c} + 5 e^{8 i c} + 10 e^{6 i c} + 10 e^{4 i c} + 5 e^{2 i c} + 1\right ) e^{- 6 i c}}{32 a} - \frac {5}{16 a}\right ) & \text {otherwise} \end {cases} + \frac {5 x}{16 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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