Optimal. Leaf size=27 \[ -\frac {i (a+i a \tan (c+d x))^9}{9 a d} \]
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Rubi [A] time = 0.04, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3487, 32} \[ -\frac {i (a+i a \tan (c+d x))^9}{9 a d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3487
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {i \operatorname {Subst}\left (\int (a+x)^8 \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=-\frac {i (a+i a \tan (c+d x))^9}{9 a d}\\ \end {align*}
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Mathematica [B] time = 3.65, size = 212, normalized size = 7.85 \[ \frac {a^8 \sec (c) \sec ^9(c+d x) (-126 \sin (2 c+d x)+84 \sin (2 c+3 d x)-84 \sin (4 c+3 d x)+36 \sin (4 c+5 d x)-36 \sin (6 c+5 d x)+9 \sin (6 c+7 d x)-9 \sin (8 c+7 d x)+2 \sin (8 c+9 d x)+126 i \cos (2 c+d x)+84 i \cos (2 c+3 d x)+84 i \cos (4 c+3 d x)+36 i \cos (4 c+5 d x)+36 i \cos (6 c+5 d x)+9 i \cos (6 c+7 d x)+9 i \cos (8 c+7 d x)+126 \sin (d x)+126 i \cos (d x))}{18 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 231, normalized size = 8.56 \[ \frac {4608 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} + 18432 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} + 43008 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} + 64512 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} + 64512 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} + 43008 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} + 18432 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} + 4608 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} + 512 i \, a^{8}}{9 \, {\left (d e^{\left (18 i \, d x + 18 i \, c\right )} + 9 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 36 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 84 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 126 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 126 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 84 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 36 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 9 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.21, size = 120, normalized size = 4.44 \[ \frac {a^{8} \tan \left (d x + c\right )^{9} - 9 i \, a^{8} \tan \left (d x + c\right )^{8} - 36 \, a^{8} \tan \left (d x + c\right )^{7} + 84 i \, a^{8} \tan \left (d x + c\right )^{6} + 126 \, a^{8} \tan \left (d x + c\right )^{5} - 126 i \, a^{8} \tan \left (d x + c\right )^{4} - 84 \, a^{8} \tan \left (d x + c\right )^{3} + 36 i \, a^{8} \tan \left (d x + c\right )^{2} + 9 \, a^{8} \tan \left (d x + c\right )}{9 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.48, size = 180, normalized size = 6.67 \[ \frac {\frac {a^{8} \left (\sin ^{9}\left (d x +c \right )\right )}{9 \cos \left (d x +c \right )^{9}}-\frac {14 i a^{8} \left (\sin ^{4}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{4}}-\frac {4 a^{8} \left (\sin ^{7}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{7}}+\frac {28 i a^{8} \left (\sin ^{6}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{6}}+\frac {14 a^{8} \left (\sin ^{5}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{5}}-\frac {i a^{8} \left (\sin ^{8}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{8}}-\frac {28 a^{8} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}+\frac {4 i a^{8}}{\cos \left (d x +c \right )^{2}}+a^{8} \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 21, normalized size = 0.78 \[ -\frac {i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{9}}{9 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.83, size = 83, normalized size = 3.07 \[ \frac {a^8\,\left (\sin \left (9\,c+9\,d\,x\right )+\frac {\cos \left (c+d\,x\right )\,63{}\mathrm {i}}{128}+\frac {\cos \left (3\,c+3\,d\,x\right )\,21{}\mathrm {i}}{64}+\frac {\cos \left (5\,c+5\,d\,x\right )\,9{}\mathrm {i}}{64}+\frac {\cos \left (7\,c+7\,d\,x\right )\,9{}\mathrm {i}}{256}-\frac {\cos \left (9\,c+9\,d\,x\right )\,255{}\mathrm {i}}{256}\right )}{9\,d\,{\cos \left (c+d\,x\right )}^9} \]
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 \[ a^{8} \left (\int \left (- 28 \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\right )\, dx + \int 70 \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \left (- 28 \tan ^{6}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\right )\, dx + \int \tan ^{8}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 8 i \tan {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \left (- 56 i \tan ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\right )\, dx + \int 56 i \tan ^{5}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \left (- 8 i \tan ^{7}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\right )\, dx + \int \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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