Optimal. Leaf size=78 \[ \frac {a^4 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}+\frac {2 i \cos (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d} \]
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Rubi [A]
time = 0.06, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3577, 3855}
\begin {gather*} \frac {a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {2 i \cos (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3577
Rule 3855
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx &=-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}-a^2 \int \cos (c+d x) (a+i a \tan (c+d x))^2 \, dx\\ &=-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}+\frac {2 i \cos (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}+a^4 \int \sec (c+d x) \, dx\\ &=\frac {a^4 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {2 i a \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}+\frac {2 i \cos (c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{d}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(246\) vs. \(2(78)=156\).
time = 0.49, size = 246, normalized size = 3.15 \begin {gather*} \frac {a^4 \left (-3 \cos (4 c) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+3 \cos (4 c) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-2 \cos (3 d x) \sin (c)+6 \cos (d x) \sin (3 c)+3 i \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (4 c)-3 i \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (4 c)+\cos (3 c) (6 i \cos (d x)-6 \sin (d x))+6 i \sin (3 c) \sin (d x)-2 i \sin (c) \sin (3 d x)+2 \cos (c) (-i \cos (3 d x)+\sin (3 d x))\right ) (\cos (c+d x)+i \sin (c+d x))^4}{3 d (\cos (d x)+i \sin (d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 113, normalized size = 1.45
method | result | size |
risch | \(-\frac {2 i a^{4} {\mathrm e}^{3 i \left (d x +c \right )}}{3 d}+\frac {2 i a^{4} {\mathrm e}^{i \left (d x +c \right )}}{d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(79\) |
derivativedivides | \(\frac {a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+\frac {4 i a^{4} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}-2 a^{4} \left (\sin ^{3}\left (d x +c \right )\right )-\frac {4 i a^{4} \left (\cos ^{3}\left (d x +c \right )\right )}{3}+\frac {a^{4} \left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(113\) |
default | \(\frac {a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+\frac {4 i a^{4} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}-2 a^{4} \left (\sin ^{3}\left (d x +c \right )\right )-\frac {4 i a^{4} \left (\cos ^{3}\left (d x +c \right )\right )}{3}+\frac {a^{4} \left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(113\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 121, normalized size = 1.55 \begin {gather*} -\frac {8 i \, a^{4} \cos \left (d x + c\right )^{3} + 12 \, a^{4} \sin \left (d x + c\right )^{3} + 8 i \, {\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} a^{4} + {\left (2 \, \sin \left (d x + c\right )^{3} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 6 \, \sin \left (d x + c\right )\right )} a^{4} + 2 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{4}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 68, normalized size = 0.87 \begin {gather*} \frac {-2 i \, a^{4} e^{\left (3 i \, d x + 3 i \, c\right )} + 6 i \, a^{4} e^{\left (i \, d x + i \, c\right )} + 3 \, a^{4} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 3 \, a^{4} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.27, size = 109, normalized size = 1.40 \begin {gather*} \frac {a^{4} \left (- \log {\left (e^{i d x} - i e^{- i c} \right )} + \log {\left (e^{i d x} + i e^{- i c} \right )}\right )}{d} + \begin {cases} \frac {- 2 i a^{4} d e^{3 i c} e^{3 i d x} + 6 i a^{4} d e^{i c} e^{i d x}}{3 d^{2}} & \text {for}\: d^{2} \neq 0 \\x \left (2 a^{4} e^{3 i c} - 2 a^{4} e^{i c}\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. 1299 vs. \(2 (68) = 136\).
time = 1.64, size = 1299, normalized size = 16.65 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.56, size = 88, normalized size = 1.13 \begin {gather*} \frac {2\,a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {\frac {8\,a^4}{3}-a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,8{}\mathrm {i}}{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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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