Optimal. Leaf size=82 \[ -\frac {i (a+i a \tan (c+d x))^8}{2 a^3 d}+\frac {4 i (a+i a \tan (c+d x))^9}{9 a^4 d}-\frac {i (a+i a \tan (c+d x))^{10}}{10 a^5 d} \]
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Rubi [A]
time = 0.05, 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 = {3568, 45}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rubi steps
\begin {align*} \int \sec ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac {i \text {Subst}\left (\int (a-x)^2 (a+x)^7 \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac {i \text {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 = 1.33, size = 154, normalized size = 1.88 \begin {gather*} \frac {a^5 \sec (c) \sec ^{10}(c+d x) (126 i \cos (c)+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)+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))}{360 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 294 vs. \(2 (70 ) = 140\).
time = 0.27, size = 295, normalized size = 3.60
method | result | size |
risch | \(\frac {128 i a^{5} \left (120 \,{\mathrm e}^{14 i \left (d x +c \right )}+210 \,{\mathrm e}^{12 i \left (d x +c \right )}+252 \,{\mathrm e}^{10 i \left (d x +c \right )}+210 \,{\mathrm e}^{8 i \left (d x +c \right )}+120 \,{\mathrm e}^{6 i \left (d x +c \right )}+45 \,{\mathrm e}^{4 i \left (d x +c \right )}+10 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{45 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{10}}\) | \(102\) |
derivativedivides | \(\frac {i a^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin ^{6}\left (d x +c \right )}{20 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{60 \cos \left (d x +c \right )^{6}}\right )+5 a^{5} \left (\frac {\sin ^{5}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \left (\sin ^{5}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{5}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{5}}\right )-10 i a^{5} \left (\frac {\sin ^{4}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}\right )-10 a^{5} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+\frac {5 i a^{5}}{6 \cos \left (d x +c \right )^{6}}-a^{5} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(295\) |
default | \(\frac {i a^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin ^{6}\left (d x +c \right )}{20 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{60 \cos \left (d x +c \right )^{6}}\right )+5 a^{5} \left (\frac {\sin ^{5}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \left (\sin ^{5}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{5}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{5}}\right )-10 i a^{5} \left (\frac {\sin ^{4}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}\right )-10 a^{5} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+\frac {5 i a^{5}}{6 \cos \left (d x +c \right )^{6}}-a^{5} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(295\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 108, normalized size = 1.32 \begin {gather*} -\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} \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. 229 vs. \(2 (64) = 128\).
time = 0.36, size = 229, normalized size = 2.79 \begin {gather*} -\frac {128 \, {\left (-120 i \, a^{5} e^{\left (14 i \, d x + 14 i \, c\right )} - 210 i \, a^{5} e^{\left (12 i \, d x + 12 i \, c\right )} - 252 i \, a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} - 210 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} - 120 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 45 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} - 10 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{5}\right )}}{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 )}} \end {gather*}
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 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.91, size = 108, normalized size = 1.32 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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