Optimal. Leaf size=75 \[ \frac {i \cos ^4(c+d x)}{4 a d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac {3 \sin (c+d x) \cos (c+d x)}{8 a d}+\frac {3 x}{8 a} \]
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Rubi [A] time = 0.13, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {i \cos ^4(c+d x)}{4 a d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac {3 \sin (c+d x) \cos (c+d x)}{8 a d}+\frac {3 x}{8 a} \]
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 ^3(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx &=-\frac {i \int \cos ^3(c+d x) (i a \cos (c+d x)+a \sin (c+d x)) \, dx}{a^2}\\ &=-\frac {i \int \left (i a \cos ^4(c+d x)+a \cos ^3(c+d x) \sin (c+d x)\right ) \, dx}{a^2}\\ &=-\frac {i \int \cos ^3(c+d x) \sin (c+d x) \, dx}{a}+\frac {\int \cos ^4(c+d x) \, dx}{a}\\ &=\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac {3 \int \cos ^2(c+d x) \, dx}{4 a}+\frac {i \operatorname {Subst}\left (\int x^3 \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac {i \cos ^4(c+d x)}{4 a d}+\frac {3 \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac {3 \int 1 \, dx}{8 a}\\ &=\frac {3 x}{8 a}+\frac {i \cos ^4(c+d x)}{4 a d}+\frac {3 \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{4 a d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 60, normalized size = 0.80 \[ \frac {8 \sin (2 (c+d x))+\sin (4 (c+d x))+4 i \cos (2 (c+d x))+i \cos (4 (c+d x))+12 c+12 d x}{32 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 54, normalized size = 0.72 \[ \frac {{\left (12 \, d x e^{\left (4 i \, d x + 4 i \, c\right )} - 2 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 6 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{32 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 99, normalized size = 1.32 \[ -\frac {\frac {6 i \, \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a} - \frac {6 i \, \log \left (i \, \tan \left (d x + c\right ) - 1\right )}{a} + \frac {2 \, {\left (3 \, \tan \left (d x + c\right ) + 5 i\right )}}{a {\left (-i \, \tan \left (d x + c\right ) + 1\right )}} + \frac {-9 i \, \tan \left (d x + c\right )^{2} - 26 \, \tan \left (d x + c\right ) + 21 i}{a {\left (\tan \left (d x + c\right ) - i\right )}^{2}}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 98, normalized size = 1.31 \[ \frac {3 i \ln \left (\tan \left (d x +c \right )+i\right )}{16 a d}+\frac {1}{8 a d \left (\tan \left (d x +c \right )+i\right )}-\frac {3 i \ln \left (\tan \left (d x +c \right )-i\right )}{16 a d}-\frac {i}{8 a d \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {1}{4 a d \left (\tan \left (d x +c \right )-i\right )} \]
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 = 3.43, size = 111, normalized size = 1.48 \[ \frac {3\,x}{8\,a}-\frac {\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,1{}\mathrm {i}}{2}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,1{}\mathrm {i}}{2}+\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{a\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )}^2\,{\left (1+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 155, normalized size = 2.07 \[ \begin {cases} - \frac {\left (512 i a^{2} d^{2} e^{8 i c} e^{2 i d x} - 1536 i a^{2} d^{2} e^{4 i c} e^{- 2 i d x} - 256 i a^{2} d^{2} e^{2 i c} e^{- 4 i d x}\right ) e^{- 6 i c}}{8192 a^{3} d^{3}} & \text {for}\: 8192 a^{3} d^{3} e^{6 i c} \neq 0 \\x \left (\frac {\left (e^{6 i c} + 3 e^{4 i c} + 3 e^{2 i c} + 1\right ) e^{- 4 i c}}{8 a} - \frac {3}{8 a}\right ) & \text {otherwise} \end {cases} + \frac {3 x}{8 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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