Optimal. Leaf size=75 \[ \frac {i a \sec ^8(c+d x)}{8 d}+\frac {a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{d}+\frac {3 a \tan ^5(c+d x)}{5 d}+\frac {a \tan ^7(c+d x)}{7 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 75, 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 ^7(c+d x)}{7 d}+\frac {3 a \tan ^5(c+d x)}{5 d}+\frac {a \tan ^3(c+d x)}{d}+\frac {a \tan (c+d x)}{d}+\frac {i a \sec ^8(c+d x)}{8 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3567
Rule 3852
Rubi steps
\begin {align*} \int \sec ^8(c+d x) (a+i a \tan (c+d x)) \, dx &=\frac {i a \sec ^8(c+d x)}{8 d}+a \int \sec ^8(c+d x) \, dx\\ &=\frac {i a \sec ^8(c+d x)}{8 d}-\frac {a \text {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac {i a \sec ^8(c+d x)}{8 d}+\frac {a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{d}+\frac {3 a \tan ^5(c+d x)}{5 d}+\frac {a \tan ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 63, normalized size = 0.84 \begin {gather*} \frac {i a \sec ^8(c+d x)}{8 d}+\frac {a \left (\tan (c+d x)+\tan ^3(c+d x)+\frac {3}{5} \tan ^5(c+d x)+\frac {1}{7} \tan ^7(c+d x)\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 59, normalized size = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {i a}{8 \cos \left (d x +c \right )^{8}}-a \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )}{d}\) | \(59\) |
default | \(\frac {\frac {i a}{8 \cos \left (d x +c \right )^{8}}-a \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )}{d}\) | \(59\) |
risch | \(\frac {32 i a \left (70 \,{\mathrm e}^{8 i \left (d x +c \right )}+56 \,{\mathrm e}^{6 i \left (d x +c \right )}+28 \,{\mathrm e}^{4 i \left (d x +c \right )}+8 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{35 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{8}}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 92, normalized size = 1.23 \begin {gather*} \frac {35 i \, a \tan \left (d x + c\right )^{8} + 40 \, a \tan \left (d x + c\right )^{7} + 140 i \, a \tan \left (d x + c\right )^{6} + 168 \, a \tan \left (d x + c\right )^{5} + 210 i \, a \tan \left (d x + c\right )^{4} + 280 \, a \tan \left (d x + c\right )^{3} + 140 i \, a \tan \left (d x + c\right )^{2} + 280 \, a \tan \left (d x + c\right )}{280 \, 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. 153 vs. \(2 (67) = 134\).
time = 0.35, size = 153, normalized size = 2.04 \begin {gather*} -\frac {32 \, {\left (-70 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} - 56 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 28 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 8 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a\right )}}{35 \, {\left (d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, 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 = 3.66, size = 68, normalized size = 0.91 \begin {gather*} \begin {cases} \frac {a \left (\frac {\tan ^{7}{\left (c + d x \right )}}{7} + \frac {3 \tan ^{5}{\left (c + d x \right )}}{5} + \tan ^{3}{\left (c + d x \right )} + \tan {\left (c + d x \right )}\right ) + \frac {i a \sec ^{8}{\left (c + d x \right )}}{8}}{d} & \text {for}\: d \neq 0 \\x \left (i a \tan {\left (c \right )} + a\right ) \sec ^{8}{\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 = 0.59, size = 92, normalized size = 1.23 \begin {gather*} -\frac {-35 i \, a \tan \left (d x + c\right )^{8} - 40 \, a \tan \left (d x + c\right )^{7} - 140 i \, a \tan \left (d x + c\right )^{6} - 168 \, a \tan \left (d x + c\right )^{5} - 210 i \, a \tan \left (d x + c\right )^{4} - 280 \, a \tan \left (d x + c\right )^{3} - 140 i \, a \tan \left (d x + c\right )^{2} - 280 \, a \tan \left (d x + c\right )}{280 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.27, size = 149, normalized size = 1.99 \begin {gather*} \frac {a\,\sin \left (c+d\,x\right )\,\left (280\,{\cos \left (c+d\,x\right )}^7+{\cos \left (c+d\,x\right )}^6\,\sin \left (c+d\,x\right )\,140{}\mathrm {i}+280\,{\cos \left (c+d\,x\right )}^5\,{\sin \left (c+d\,x\right )}^2+{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^3\,210{}\mathrm {i}+168\,{\cos \left (c+d\,x\right )}^3\,{\sin \left (c+d\,x\right )}^4+{\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^5\,140{}\mathrm {i}+40\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^6+{\sin \left (c+d\,x\right )}^7\,35{}\mathrm {i}\right )}{280\,d\,{\cos \left (c+d\,x\right )}^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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