Optimal. Leaf size=31 \[ -\frac {i (a \cos (c+d x)+i a \sin (c+d x))^4}{4 d} \]
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Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {3071} \[ -\frac {i (a \cos (c+d x)+i a \sin (c+d x))^4}{4 d} \]
Antiderivative was successfully verified.
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Rule 3071
Rubi steps
\begin {align*} \int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx &=-\frac {i (a \cos (c+d x)+i a \sin (c+d x))^4}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 31, normalized size = 1.00 \[ -\frac {i (a \cos (c+d x)+i a \sin (c+d x))^4}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.83, size = 17, normalized size = 0.55 \[ -\frac {i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 52, normalized size = 1.68 \[ -\frac {i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )}}{8 \, d} - \frac {i \, a^{4} e^{\left (-4 i \, d x - 4 i \, c\right )}}{8 \, d} + \frac {a^{4} \sin \left (4 \, d x + 4 \, c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.29, size = 151, normalized size = 4.87 \[ \frac {a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-i a^{4} \left (\sin ^{4}\left (d x +c \right )\right )-6 a^{4} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-i a^{4} \left (\cos ^{4}\left (d x +c \right )\right )+a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 132, normalized size = 4.26 \[ -\frac {i \, a^{4} \cos \left (d x + c\right )^{4}}{d} - \frac {i \, a^{4} \sin \left (d x + c\right )^{4}}{d} + \frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{32 \, d} + \frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) - 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{32 \, d} - \frac {3 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.55, size = 84, normalized size = 2.71 \[ -\frac {2\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,4{}\mathrm {i}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,4{}\mathrm {i}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 37, normalized size = 1.19 \[ \begin {cases} - \frac {i a^{4} e^{4 i c} e^{4 i d x}}{4 d} & \text {for}\: 4 d \neq 0 \\a^{4} x e^{4 i c} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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