Optimal. Leaf size=62 \[ \frac {i a \sec ^6(c+d x)}{6 d}+\frac {a \tan (c+d x)}{d}+\frac {2 a \tan ^3(c+d x)}{3 d}+\frac {a \tan ^5(c+d x)}{5 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3567, 3852}
\begin {gather*} \frac {a \tan ^5(c+d x)}{5 d}+\frac {2 a \tan ^3(c+d x)}{3 d}+\frac {a \tan (c+d x)}{d}+\frac {i a \sec ^6(c+d x)}{6 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3567
Rule 3852
Rubi steps
\begin {align*} \int \sec ^6(c+d x) (a+i a \tan (c+d x)) \, dx &=\frac {i a \sec ^6(c+d x)}{6 d}+a \int \sec ^6(c+d x) \, dx\\ &=\frac {i a \sec ^6(c+d x)}{6 d}-\frac {a \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac {i a \sec ^6(c+d x)}{6 d}+\frac {a \tan (c+d x)}{d}+\frac {2 a \tan ^3(c+d x)}{3 d}+\frac {a \tan ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 55, normalized size = 0.89 \begin {gather*} \frac {i a \sec ^6(c+d x)}{6 d}+\frac {a \left (\tan (c+d x)+\frac {2}{3} \tan ^3(c+d x)+\frac {1}{5} \tan ^5(c+d x)\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 49, normalized size = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {i a}{6 \cos \left (d x +c \right )^{6}}-a \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(49\) |
default | \(\frac {\frac {i a}{6 \cos \left (d x +c \right )^{6}}-a \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(49\) |
risch | \(\frac {16 i a \left (20 \,{\mathrm e}^{6 i \left (d x +c \right )}+15 \,{\mathrm e}^{4 i \left (d x +c \right )}+6 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 70, normalized size = 1.13 \begin {gather*} \frac {5 i \, a \tan \left (d x + c\right )^{6} + 6 \, a \tan \left (d x + c\right )^{5} + 15 i \, a \tan \left (d x + c\right )^{4} + 20 \, a \tan \left (d x + c\right )^{3} + 15 i \, a \tan \left (d x + c\right )^{2} + 30 \, a \tan \left (d x + c\right )}{30 \, 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. 117 vs. \(2 (54) = 108\).
time = 0.34, size = 117, normalized size = 1.89 \begin {gather*} -\frac {16 \, {\left (-20 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 15 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 6 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a\right )}}{15 \, {\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, 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 [A]
time = 2.46, size = 60, normalized size = 0.97 \begin {gather*} \begin {cases} \frac {a \left (\frac {\tan ^{5}{\left (c + d x \right )}}{5} + \frac {2 \tan ^{3}{\left (c + d x \right )}}{3} + \tan {\left (c + d x \right )}\right ) + \frac {i a \sec ^{6}{\left (c + d x \right )}}{6}}{d} & \text {for}\: d \neq 0 \\x \left (i a \tan {\left (c \right )} + a\right ) \sec ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.53, size = 70, normalized size = 1.13 \begin {gather*} -\frac {-5 i \, a \tan \left (d x + c\right )^{6} - 6 \, a \tan \left (d x + c\right )^{5} - 15 i \, a \tan \left (d x + c\right )^{4} - 20 \, a \tan \left (d x + c\right )^{3} - 15 i \, a \tan \left (d x + c\right )^{2} - 30 \, a \tan \left (d x + c\right )}{30 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.24, size = 112, normalized size = 1.81 \begin {gather*} \frac {a\,\sin \left (c+d\,x\right )\,\left (30\,{\cos \left (c+d\,x\right )}^5+{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )\,15{}\mathrm {i}+20\,{\cos \left (c+d\,x\right )}^3\,{\sin \left (c+d\,x\right )}^2+{\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^3\,15{}\mathrm {i}+6\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^4+{\sin \left (c+d\,x\right )}^5\,5{}\mathrm {i}\right )}{30\,d\,{\cos \left (c+d\,x\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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