Optimal. Leaf size=27 \[ -\frac {i a^5}{2 d (a-i a \tan (c+d x))^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 32}
\begin {gather*} -\frac {i a^5}{2 d (a-i a \tan (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 3568
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac {\left (i a^5\right ) \text {Subst}\left (\int \frac {1}{(a-x)^3} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {i a^5}{2 d (a-i a \tan (c+d x))^2}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 50, normalized size = 1.85 \begin {gather*} \frac {a^3 (3 \cos (c+d x)-i \sin (c+d x)) (-i \cos (3 (c+d x))+\sin (3 (c+d x)))}{8 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 113 vs. \(2 (23 ) = 46\).
time = 0.20, size = 114, normalized size = 4.22
method | result | size |
risch | \(-\frac {i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}}{8 d}-\frac {i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{4 d}\) | \(38\) |
derivativedivides | \(\frac {-\frac {i a^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4}-3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-\frac {3 i a^{3} \left (\cos ^{4}\left (d x +c \right )\right )}{4}+a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(114\) |
default | \(\frac {-\frac {i a^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4}-3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-\frac {3 i a^{3} \left (\cos ^{4}\left (d x +c \right )\right )}{4}+a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(114\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 57 vs. \(2 (21) = 42\).
time = 0.50, size = 57, normalized size = 2.11 \begin {gather*} -\frac {-i \, a^{3} \tan \left (d x + c\right )^{2} - 2 \, a^{3} \tan \left (d x + c\right ) + i \, a^{3}}{2 \, {\left (\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 34, normalized size = 1.26 \begin {gather*} \frac {-i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 2 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 80 vs. \(2 (22) = 44\).
time = 0.18, size = 80, normalized size = 2.96 \begin {gather*} \begin {cases} \frac {- 4 i a^{3} d e^{4 i c} e^{4 i d x} - 8 i a^{3} d e^{2 i c} e^{2 i d x}}{32 d^{2}} & \text {for}\: d^{2} \neq 0 \\x \left (\frac {a^{3} e^{4 i c}}{2} + \frac {a^{3} e^{2 i c}}{2}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 135 vs. \(2 (21) = 42\).
time = 0.73, size = 135, normalized size = 5.00 \begin {gather*} -\frac {i \, a^{3} e^{\left (12 i \, d x + 8 i \, c\right )} + 6 i \, a^{3} e^{\left (10 i \, d x + 6 i \, c\right )} + 14 i \, a^{3} e^{\left (8 i \, d x + 4 i \, c\right )} + 16 i \, a^{3} e^{\left (6 i \, d x + 2 i \, c\right )} + 2 i \, a^{3} e^{\left (2 i \, d x - 2 i \, c\right )} + 9 i \, a^{3} e^{\left (4 i \, d x\right )}}{8 \, {\left (d e^{\left (8 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 2 i \, c\right )} + 4 \, d e^{\left (2 i \, d x - 2 i \, c\right )} + 6 \, d e^{\left (4 i \, d x\right )} + d e^{\left (-4 i \, c\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.34, size = 36, normalized size = 1.33 \begin {gather*} -\frac {a^3\,\left (\frac {{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}}{4}\right )\,1{}\mathrm {i}}{2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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