Optimal. Leaf size=82 \[ -\frac {i (a+i a \tan (c+d x))^{10}}{10 a^5 d}+\frac {4 i (a+i a \tan (c+d x))^9}{9 a^4 d}-\frac {i (a+i a \tan (c+d x))^8}{2 a^3 d} \]
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Rubi [A] time = 0.06, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3487, 43} \[ -\frac {i (a+i a \tan (c+d x))^{10}}{10 a^5 d}+\frac {4 i (a+i a \tan (c+d x))^9}{9 a^4 d}-\frac {i (a+i a \tan (c+d x))^8}{2 a^3 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rubi steps
\begin {align*} \int \sec ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac {i \operatorname {Subst}\left (\int (a-x)^2 (a+x)^7 \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (4 a^2 (a+x)^7-4 a (a+x)^8+(a+x)^9\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac {i (a+i a \tan (c+d x))^8}{2 a^3 d}+\frac {4 i (a+i a \tan (c+d x))^9}{9 a^4 d}-\frac {i (a+i a \tan (c+d x))^{10}}{10 a^5 d}\\ \end {align*}
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Mathematica [A] time = 2.84, size = 154, normalized size = 1.88 \[ \frac {a^5 \sec (c) \sec ^{10}(c+d x) (105 \sin (c+2 d x)-105 \sin (3 c+2 d x)+60 \sin (3 c+4 d x)-60 \sin (5 c+4 d x)+45 \sin (5 c+6 d x)+10 \sin (7 c+8 d x)+\sin (9 c+10 d x)+105 i \cos (c+2 d x)+105 i \cos (3 c+2 d x)+60 i \cos (3 c+4 d x)+60 i \cos (5 c+4 d x)-126 \sin (c)+126 i \cos (c))}{360 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 229, normalized size = 2.79 \[ \frac {15360 i \, a^{5} e^{\left (14 i \, d x + 14 i \, c\right )} + 26880 i \, a^{5} e^{\left (12 i \, d x + 12 i \, c\right )} + 32256 i \, a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} + 26880 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} + 15360 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} + 5760 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 1280 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + 128 i \, a^{5}}{45 \, {\left (d e^{\left (20 i \, d x + 20 i \, c\right )} + 10 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 45 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 120 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 210 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 252 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 210 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 120 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 45 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 10 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.23, size = 108, normalized size = 1.32 \[ -\frac {-9 i \, a^{5} \tan \left (d x + c\right )^{10} - 50 \, a^{5} \tan \left (d x + c\right )^{9} + 90 i \, a^{5} \tan \left (d x + c\right )^{8} + 210 i \, a^{5} \tan \left (d x + c\right )^{6} + 252 \, a^{5} \tan \left (d x + c\right )^{5} + 240 \, a^{5} \tan \left (d x + c\right )^{3} - 225 i \, a^{5} \tan \left (d x + c\right )^{2} - 90 \, a^{5} \tan \left (d x + c\right )}{90 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.52, size = 295, normalized size = 3.60 \[ \frac {i a^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin ^{6}\left (d x +c \right )}{20 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{60 \cos \left (d x +c \right )^{6}}\right )+5 a^{5} \left (\frac {\sin ^{5}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \left (\sin ^{5}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{5}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{5}}\right )-10 i a^{5} \left (\frac {\sin ^{4}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}\right )-10 a^{5} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+\frac {5 i a^{5}}{6 \cos \left (d x +c \right )^{6}}-a^{5} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 108, normalized size = 1.32 \[ \frac {126 i \, a^{5} \tan \left (d x + c\right )^{10} + 700 \, a^{5} \tan \left (d x + c\right )^{9} - 1260 i \, a^{5} \tan \left (d x + c\right )^{8} - 2940 i \, a^{5} \tan \left (d x + c\right )^{6} - 3528 \, a^{5} \tan \left (d x + c\right )^{5} - 3360 \, a^{5} \tan \left (d x + c\right )^{3} + 3150 i \, a^{5} \tan \left (d x + c\right )^{2} + 1260 \, a^{5} \tan \left (d x + c\right )}{1260 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.33, size = 151, normalized size = 1.84 \[ \frac {a^5\,\sin \left (c+d\,x\right )\,\left (90\,{\cos \left (c+d\,x\right )}^9+{\cos \left (c+d\,x\right )}^8\,\sin \left (c+d\,x\right )\,225{}\mathrm {i}-240\,{\cos \left (c+d\,x\right )}^7\,{\sin \left (c+d\,x\right )}^2-252\,{\cos \left (c+d\,x\right )}^5\,{\sin \left (c+d\,x\right )}^4-{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^5\,210{}\mathrm {i}-{\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^7\,90{}\mathrm {i}+50\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^8+{\sin \left (c+d\,x\right )}^9\,9{}\mathrm {i}\right )}{90\,d\,{\cos \left (c+d\,x\right )}^{10}} \]
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 \[ i a^{5} \left (\int \left (- i \sec ^{6}{\left (c + d x \right )}\right )\, dx + \int 5 \tan {\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int \left (- 10 \tan ^{3}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\right )\, dx + \int \tan ^{5}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int 10 i \tan ^{2}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int \left (- 5 i \tan ^{4}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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