Optimal. Leaf size=82 \[ \frac {i (a-i a \tan (c+d x))^7}{7 a^9 d}-\frac {2 i (a-i a \tan (c+d x))^6}{3 a^8 d}+\frac {4 i (a-i a \tan (c+d x))^5}{5 a^7 d} \]
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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} \[ \frac {i (a-i a \tan (c+d x))^7}{7 a^9 d}-\frac {2 i (a-i a \tan (c+d x))^6}{3 a^8 d}+\frac {4 i (a-i a \tan (c+d x))^5}{5 a^7 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rubi steps
\begin {align*} \int \frac {\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^2} \, dx &=-\frac {i \operatorname {Subst}\left (\int (a-x)^4 (a+x)^2 \, dx,x,i a \tan (c+d x)\right )}{a^9 d}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (4 a^2 (a-x)^4-4 a (a-x)^5+(a-x)^6\right ) \, dx,x,i a \tan (c+d x)\right )}{a^9 d}\\ &=\frac {4 i (a-i a \tan (c+d x))^5}{5 a^7 d}-\frac {2 i (a-i a \tan (c+d x))^6}{3 a^8 d}+\frac {i (a-i a \tan (c+d x))^7}{7 a^9 d}\\ \end {align*}
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Mathematica [A] time = 0.55, size = 90, normalized size = 1.10 \[ \frac {\sec (c) \sec ^7(c+d x) (-35 \sin (2 c+d x)+42 \sin (2 c+3 d x)+14 \sin (4 c+5 d x)+2 \sin (6 c+7 d x)-35 i \cos (2 c+d x)+35 \sin (d x)-35 i \cos (d x))}{210 a^2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 138, normalized size = 1.68 \[ \frac {2688 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 896 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 128 i}{105 \, {\left (a^{2} d e^{\left (14 i \, d x + 14 i \, c\right )} + 7 \, a^{2} d e^{\left (12 i \, d x + 12 i \, c\right )} + 21 \, a^{2} d e^{\left (10 i \, d x + 10 i \, c\right )} + 35 \, a^{2} d e^{\left (8 i \, d x + 8 i \, c\right )} + 35 \, a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 21 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 7 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 77, normalized size = 0.94 \[ -\frac {15 \, \tan \left (d x + c\right )^{7} + 35 i \, \tan \left (d x + c\right )^{6} + 21 \, \tan \left (d x + c\right )^{5} + 105 i \, \tan \left (d x + c\right )^{4} - 35 \, \tan \left (d x + c\right )^{3} + 105 i \, \tan \left (d x + c\right )^{2} - 105 \, \tan \left (d x + c\right )}{105 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 78, normalized size = 0.95 \[ \frac {\tan \left (d x +c \right )-\frac {\left (\tan ^{7}\left (d x +c \right )\right )}{7}-\frac {i \left (\tan ^{6}\left (d x +c \right )\right )}{3}-\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}-i \left (\tan ^{4}\left (d x +c \right )\right )+\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-i \left (\tan ^{2}\left (d x +c \right )\right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 77, normalized size = 0.94 \[ -\frac {15 \, \tan \left (d x + c\right )^{7} + 35 i \, \tan \left (d x + c\right )^{6} + 21 \, \tan \left (d x + c\right )^{5} + 105 i \, \tan \left (d x + c\right )^{4} - 35 \, \tan \left (d x + c\right )^{3} + 105 i \, \tan \left (d x + c\right )^{2} - 105 \, \tan \left (d x + c\right )}{105 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.46, size = 93, normalized size = 1.13 \[ \frac {{\cos \left (c+d\,x\right )}^7\,35{}\mathrm {i}+64\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^6+32\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^4+24\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^2-\cos \left (c+d\,x\right )\,35{}\mathrm {i}-15\,\sin \left (c+d\,x\right )}{105\,a^2\,d\,{\cos \left (c+d\,x\right )}^7} \]
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 \[ - \frac {\int \frac {\sec ^{10}{\left (c + d x \right )}}{\tan ^{2}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} - 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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