Optimal. Leaf size=32 \[ -\frac {i \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d} \]
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Rubi [A] time = 0.04, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {3488} \[ -\frac {i \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 3488
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac {i \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 31, normalized size = 0.97 \[ -\frac {i a^3 (\cos (c+d x)+i \sin (c+d x))^3}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 17, normalized size = 0.53 \[ -\frac {i \, a^{3} e^{\left (3 i \, d x + 3 i \, c\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.99, size = 901, normalized size = 28.16 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.46, size = 76, normalized size = 2.38 \[ \frac {\frac {i a^{3} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}-a^{3} \left (\sin ^{3}\left (d x +c \right )\right )-i a^{3} \left (\cos ^{3}\left (d x +c \right )\right )+\frac {a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 75, normalized size = 2.34 \[ -\frac {3 i \, a^{3} \cos \left (d x + c\right )^{3} + 3 \, a^{3} \sin \left (d x + c\right )^{3} + i \, {\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} a^{3} + {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{3}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.33, size = 66, normalized size = 2.06 \[ -\frac {2\,a^3\,\left (3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}{3\,d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,3{}\mathrm {i}+3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 37, normalized size = 1.16 \[ \begin {cases} - \frac {i a^{3} e^{3 i c} e^{3 i d x}}{3 d} & \text {for}\: 3 d \neq 0 \\a^{3} x e^{3 i c} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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