Optimal. Leaf size=76 \[ \frac {3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {i a \sec ^5(c+d x)}{5 d}+\frac {3 a \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3567, 3853,
3855} \begin {gather*} \frac {i a \sec ^5(c+d x)}{5 d}+\frac {3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac {3 a \tan (c+d x) \sec (c+d x)}{8 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3567
Rule 3853
Rule 3855
Rubi steps
\begin {align*} \int \sec ^5(c+d x) (a+i a \tan (c+d x)) \, dx &=\frac {i a \sec ^5(c+d x)}{5 d}+a \int \sec ^5(c+d x) \, dx\\ &=\frac {i a \sec ^5(c+d x)}{5 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} (3 a) \int \sec ^3(c+d x) \, dx\\ &=\frac {i a \sec ^5(c+d x)}{5 d}+\frac {3 a \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{8} (3 a) \int \sec (c+d x) \, dx\\ &=\frac {3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {i a \sec ^5(c+d x)}{5 d}+\frac {3 a \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 70, normalized size = 0.92 \begin {gather*} \frac {i a \sec ^5(c+d x)}{5 d}+\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {3 a \left (\tanh ^{-1}(\sin (c+d x))+\sec (c+d x) \tan (c+d x)\right )}{8 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 64, normalized size = 0.84
method | result | size |
derivativedivides | \(\frac {\frac {i a}{5 \cos \left (d x +c \right )^{5}}+a \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(64\) |
default | \(\frac {\frac {i a}{5 \cos \left (d x +c \right )^{5}}+a \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(64\) |
risch | \(-\frac {i a \left (15 \,{\mathrm e}^{9 i \left (d x +c \right )}+70 \,{\mathrm e}^{7 i \left (d x +c \right )}-128 \,{\mathrm e}^{5 i \left (d x +c \right )}-70 \,{\mathrm e}^{3 i \left (d x +c \right )}-15 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{20 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}\) | \(116\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 86, normalized size = 1.13 \begin {gather*} -\frac {5 \, a {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - \frac {16 i \, a}{\cos \left (d x + c\right )^{5}}}{80 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 276 vs. \(2 (66) = 132\).
time = 0.37, size = 276, normalized size = 3.63 \begin {gather*} \frac {-30 i \, a e^{\left (9 i \, d x + 9 i \, c\right )} - 140 i \, a e^{\left (7 i \, d x + 7 i \, c\right )} + 256 i \, a e^{\left (5 i \, d x + 5 i \, c\right )} + 140 i \, a e^{\left (3 i \, d x + 3 i \, c\right )} + 30 i \, a e^{\left (i \, d x + i \, c\right )} + 15 \, {\left (a e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 15 \, {\left (a e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, a e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{40 \, {\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} i a \left (\int \left (- i \sec ^{5}{\left (c + d x \right )}\right )\, dx + \int \tan {\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}\, dx\right ) \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. 139 vs. \(2 (66) = 132\).
time = 0.55, size = 139, normalized size = 1.83 \begin {gather*} \frac {15 \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) - 15 \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right ) + \frac {2 \, {\left (25 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 40 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 10 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 80 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 10 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 25 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 i \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{40 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.90, size = 178, normalized size = 2.34 \begin {gather*} \frac {3\,a\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{4\,d}-\frac {-\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}+2{}\mathrm {i}\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}+4{}\mathrm {i}\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {a\,2{}\mathrm {i}}{5}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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