Optimal. Leaf size=109 \[ -\frac {4 i (a+i a \tan (c+d x))^6}{3 a^4 d}+\frac {12 i (a+i a \tan (c+d x))^7}{7 a^5 d}-\frac {3 i (a+i a \tan (c+d x))^8}{4 a^6 d}+\frac {i (a+i a \tan (c+d x))^9}{9 a^7 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 109, 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))^9}{9 a^7 d}-\frac {3 i (a+i a \tan (c+d x))^8}{4 a^6 d}+\frac {12 i (a+i a \tan (c+d x))^7}{7 a^5 d}-\frac {4 i (a+i a \tan (c+d x))^6}{3 a^4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rubi steps
\begin {align*} \int \sec ^8(c+d x) (a+i a \tan (c+d x))^2 \, dx &=-\frac {i \text {Subst}\left (\int (a-x)^3 (a+x)^5 \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac {i \text {Subst}\left (\int \left (8 a^3 (a+x)^5-12 a^2 (a+x)^6+6 a (a+x)^7-(a+x)^8\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac {4 i (a+i a \tan (c+d x))^6}{3 a^4 d}+\frac {12 i (a+i a \tan (c+d x))^7}{7 a^5 d}-\frac {3 i (a+i a \tan (c+d x))^8}{4 a^6 d}+\frac {i (a+i a \tan (c+d x))^9}{9 a^7 d}\\ \end {align*}
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Mathematica [A]
time = 0.90, size = 99, normalized size = 0.91 \begin {gather*} \frac {a^2 \sec (c) \sec ^9(c+d x) (63 i \cos (d x)+63 i \cos (2 c+d x)+63 \sin (d x)-63 \sin (2 c+d x)+84 \sin (2 c+3 d x)+36 \sin (4 c+5 d x)+9 \sin (6 c+7 d x)+\sin (8 c+9 d x))}{504 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 141, normalized size = 1.29
method | result | size |
risch | \(\frac {64 i a^{2} \left (126 \,{\mathrm e}^{10 i \left (d x +c \right )}+126 \,{\mathrm e}^{8 i \left (d x +c \right )}+84 \,{\mathrm e}^{6 i \left (d x +c \right )}+36 \,{\mathrm e}^{4 i \left (d x +c \right )}+9 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{63 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{9}}\) | \(80\) |
derivativedivides | \(\frac {-a^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )+\frac {i a^{2}}{4 \cos \left (d x +c \right )^{8}}-a^{2} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )}{d}\) | \(141\) |
default | \(\frac {-a^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )+\frac {i a^{2}}{4 \cos \left (d x +c \right )^{8}}-a^{2} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )}{d}\) | \(141\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 108, normalized size = 0.99 \begin {gather*} -\frac {28 \, a^{2} \tan \left (d x + c\right )^{9} - 63 i \, a^{2} \tan \left (d x + c\right )^{8} + 72 \, a^{2} \tan \left (d x + c\right )^{7} - 252 i \, a^{2} \tan \left (d x + c\right )^{6} - 378 i \, a^{2} \tan \left (d x + c\right )^{4} - 168 \, a^{2} \tan \left (d x + c\right )^{3} - 252 i \, a^{2} \tan \left (d x + c\right )^{2} - 252 \, a^{2} \tan \left (d x + c\right )}{252 \, 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. 189 vs. \(2 (85) = 170\).
time = 0.40, size = 189, normalized size = 1.73 \begin {gather*} -\frac {64 \, {\left (-126 i \, a^{2} e^{\left (10 i \, d x + 10 i \, c\right )} - 126 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 84 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 36 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 9 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{2}\right )}}{63 \, {\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 )}} \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*} - a^{2} \left (\int \tan ^{2}{\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\, dx + \int \left (- 2 i \tan {\left (c + d x \right )} \sec ^{8}{\left (c + d x \right )}\right )\, dx + \int \left (- \sec ^{8}{\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 = 0.61, size = 108, normalized size = 0.99 \begin {gather*} -\frac {28 \, a^{2} \tan \left (d x + c\right )^{9} - 63 i \, a^{2} \tan \left (d x + c\right )^{8} + 72 \, a^{2} \tan \left (d x + c\right )^{7} - 252 i \, a^{2} \tan \left (d x + c\right )^{6} - 378 i \, a^{2} \tan \left (d x + c\right )^{4} - 168 \, a^{2} \tan \left (d x + c\right )^{3} - 252 i \, a^{2} \tan \left (d x + c\right )^{2} - 252 \, a^{2} \tan \left (d x + c\right )}{252 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.26, size = 151, normalized size = 1.39 \begin {gather*} \frac {a^2\,\sin \left (c+d\,x\right )\,\left (252\,{\cos \left (c+d\,x\right )}^8+{\cos \left (c+d\,x\right )}^7\,\sin \left (c+d\,x\right )\,252{}\mathrm {i}+168\,{\cos \left (c+d\,x\right )}^6\,{\sin \left (c+d\,x\right )}^2+{\cos \left (c+d\,x\right )}^5\,{\sin \left (c+d\,x\right )}^3\,378{}\mathrm {i}+{\cos \left (c+d\,x\right )}^3\,{\sin \left (c+d\,x\right )}^5\,252{}\mathrm {i}-72\,{\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^6+\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^7\,63{}\mathrm {i}-28\,{\sin \left (c+d\,x\right )}^8\right )}{252\,d\,{\cos \left (c+d\,x\right )}^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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