Optimal. Leaf size=52 \[ -\frac {2 \sin ^3(c+d x)}{3 a^2 d}+\frac {\sin (c+d x)}{a^2 d}+\frac {2 i \cos ^3(c+d x)}{3 a^2 d} \]
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Rubi [A] time = 0.11, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {3092, 3090, 2633, 2565, 30, 2564} \[ -\frac {2 \sin ^3(c+d x)}{3 a^2 d}+\frac {\sin (c+d x)}{a^2 d}+\frac {2 i \cos ^3(c+d x)}{3 a^2 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2564
Rule 2565
Rule 2633
Rule 3090
Rule 3092
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx &=-\frac {\int \cos (c+d x) (i a \cos (c+d x)+a \sin (c+d x))^2 \, dx}{a^4}\\ &=-\frac {\int \left (-a^2 \cos ^3(c+d x)+2 i a^2 \cos ^2(c+d x) \sin (c+d x)+a^2 \cos (c+d x) \sin ^2(c+d x)\right ) \, dx}{a^4}\\ &=-\frac {(2 i) \int \cos ^2(c+d x) \sin (c+d x) \, dx}{a^2}+\frac {\int \cos ^3(c+d x) \, dx}{a^2}-\frac {\int \cos (c+d x) \sin ^2(c+d x) \, dx}{a^2}\\ &=\frac {(2 i) \operatorname {Subst}\left (\int x^2 \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac {\operatorname {Subst}\left (\int x^2 \, dx,x,\sin (c+d x)\right )}{a^2 d}-\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a^2 d}\\ &=\frac {2 i \cos ^3(c+d x)}{3 a^2 d}+\frac {\sin (c+d x)}{a^2 d}-\frac {2 \sin ^3(c+d x)}{3 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 73, normalized size = 1.40 \[ \frac {\sin (c+d x)}{2 a^2 d}+\frac {\sin (3 (c+d x))}{6 a^2 d}+\frac {i \cos (c+d x)}{2 a^2 d}+\frac {i \cos (3 (c+d x))}{6 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 30, normalized size = 0.58 \[ \frac {{\left (3 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{6 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 47, normalized size = 0.90 \[ \frac {2 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2\right )}}{3 \, a^{2} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 57, normalized size = 1.10 \[ \frac {\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i}+\frac {2 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}-\frac {4}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{3}}}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 45, normalized size = 0.87 \[ \frac {i \, \cos \left (3 \, d x + 3 \, c\right ) + 3 i \, \cos \left (d x + c\right ) + \sin \left (3 \, d x + 3 \, c\right ) + 3 \, \sin \left (d x + c\right )}{6 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 79, normalized size = 1.52 \[ -\frac {2\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,3{}\mathrm {i}+3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-2{}\mathrm {i}\right )}{3\,a^2\,d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,3{}\mathrm {i}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 94, normalized size = 1.81 \[ \begin {cases} \frac {\left (6 i a^{2} d e^{3 i c} e^{- i d x} + 2 i a^{2} d e^{i c} e^{- 3 i d x}\right ) e^{- 4 i c}}{12 a^{4} d^{2}} & \text {for}\: 12 a^{4} d^{2} e^{4 i c} \neq 0 \\\frac {x \left (e^{2 i c} + 1\right ) e^{- 3 i c}}{2 a^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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