Optimal. Leaf size=75 \[ \frac {a \tan ^7(c+d x)}{7 d}+\frac {3 a \tan ^5(c+d x)}{5 d}+\frac {a \tan ^3(c+d x)}{d}+\frac {a \tan (c+d x)}{d}+\frac {i a \sec ^8(c+d x)}{8 d} \]
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Rubi [A] time = 0.04, 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 = {3486, 3767} \[ \frac {a \tan ^7(c+d x)}{7 d}+\frac {3 a \tan ^5(c+d x)}{5 d}+\frac {a \tan ^3(c+d x)}{d}+\frac {a \tan (c+d x)}{d}+\frac {i a \sec ^8(c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 3767
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 \operatorname {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 = 0.12, size = 63, normalized size = 0.84 \[ \frac {a \left (\frac {1}{7} \tan ^7(c+d x)+\frac {3}{5} \tan ^5(c+d x)+\tan ^3(c+d x)+\tan (c+d x)\right )}{d}+\frac {i a \sec ^8(c+d x)}{8 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 153, normalized size = 2.04 \[ \frac {2240 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} + 1792 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 896 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 256 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 32 i \, a}{35 \, {\left (d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 92, normalized size = 1.23 \[ -\frac {-35 i \, a \tan \left (d x + c\right )^{8} - 40 \, a \tan \left (d x + c\right )^{7} - 140 i \, a \tan \left (d x + c\right )^{6} - 168 \, a \tan \left (d x + c\right )^{5} - 210 i \, a \tan \left (d x + c\right )^{4} - 280 \, a \tan \left (d x + c\right )^{3} - 140 i \, a \tan \left (d x + c\right )^{2} - 280 \, a \tan \left (d x + c\right )}{280 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 59, normalized size = 0.79 \[ \frac {\frac {i a}{8 \cos \left (d x +c \right )^{8}}-a \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 92, normalized size = 1.23 \[ \frac {35 i \, a \tan \left (d x + c\right )^{8} + 40 \, a \tan \left (d x + c\right )^{7} + 140 i \, a \tan \left (d x + c\right )^{6} + 168 \, a \tan \left (d x + c\right )^{5} + 210 i \, a \tan \left (d x + c\right )^{4} + 280 \, a \tan \left (d x + c\right )^{3} + 140 i \, a \tan \left (d x + c\right )^{2} + 280 \, a \tan \left (d x + c\right )}{280 \, d} \]
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 \[ \frac {a\,\sin \left (c+d\,x\right )\,\left (280\,{\cos \left (c+d\,x\right )}^7+{\cos \left (c+d\,x\right )}^6\,\sin \left (c+d\,x\right )\,140{}\mathrm {i}+280\,{\cos \left (c+d\,x\right )}^5\,{\sin \left (c+d\,x\right )}^2+{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^3\,210{}\mathrm {i}+168\,{\cos \left (c+d\,x\right )}^3\,{\sin \left (c+d\,x\right )}^4+{\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^5\,140{}\mathrm {i}+40\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^6+{\sin \left (c+d\,x\right )}^7\,35{}\mathrm {i}\right )}{280\,d\,{\cos \left (c+d\,x\right )}^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.06, size = 68, normalized size = 0.91 \[ \begin {cases} \frac {a \left (\frac {\tan ^{7}{\left (c + d x \right )}}{7} + \frac {3 \tan ^{5}{\left (c + d x \right )}}{5} + \tan ^{3}{\left (c + d x \right )} + \tan {\left (c + d x \right )}\right ) + \frac {i a \sec ^{8}{\left (c + d x \right )}}{8}}{d} & \text {for}\: d \neq 0 \\x \left (i a \tan {\relax (c )} + a\right ) \sec ^{8}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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