Optimal. Leaf size=55 \[ \frac {i (a+i a \tan (c+d x))^6}{6 a^3 d}-\frac {2 i (a+i a \tan (c+d x))^5}{5 a^2 d} \]
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Rubi [A] time = 0.04, antiderivative size = 55, 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} \[ \frac {i (a+i a \tan (c+d x))^6}{6 a^3 d}-\frac {2 i (a+i a \tan (c+d x))^5}{5 a^2 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac {i \operatorname {Subst}\left (\int (a-x) (a+x)^4 \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (2 a (a+x)^4-(a+x)^5\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {2 i (a+i a \tan (c+d x))^5}{5 a^2 d}+\frac {i (a+i a \tan (c+d x))^6}{6 a^3 d}\\ \end {align*}
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Mathematica [A] time = 1.30, size = 97, normalized size = 1.76 \[ \frac {a^3 \sec (c) \sec ^6(c+d x) (15 \sin (c+2 d x)-15 \sin (3 c+2 d x)+12 \sin (3 c+4 d x)+2 \sin (5 c+6 d x)+15 i \cos (c+2 d x)+15 i \cos (3 c+2 d x)-20 \sin (c)+20 i \cos (c))}{60 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 139, normalized size = 2.53 \[ \frac {480 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} + 640 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 480 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 192 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 32 i \, a^{3}}{15 \, {\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.62, size = 82, normalized size = 1.49 \[ -\frac {5 i \, a^{3} \tan \left (d x + c\right )^{6} + 18 \, a^{3} \tan \left (d x + c\right )^{5} - 15 i \, a^{3} \tan \left (d x + c\right )^{4} + 20 \, a^{3} \tan \left (d x + c\right )^{3} - 45 i \, a^{3} \tan \left (d x + c\right )^{2} - 30 \, a^{3} \tan \left (d x + c\right )}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.44, size = 128, normalized size = 2.33 \[ \frac {-i a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{4}}\right )-3 a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+\frac {3 i a^{3}}{4 \cos \left (d x +c \right )^{4}}-a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 82, normalized size = 1.49 \[ \frac {-10 i \, a^{3} \tan \left (d x + c\right )^{6} - 36 \, a^{3} \tan \left (d x + c\right )^{5} + 30 i \, a^{3} \tan \left (d x + c\right )^{4} - 40 \, a^{3} \tan \left (d x + c\right )^{3} + 90 i \, a^{3} \tan \left (d x + c\right )^{2} + 60 \, a^{3} \tan \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.26, size = 114, normalized size = 2.07 \[ -\frac {a^3\,\sin \left (c+d\,x\right )\,\left (-30\,{\cos \left (c+d\,x\right )}^5-{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )\,45{}\mathrm {i}+20\,{\cos \left (c+d\,x\right )}^3\,{\sin \left (c+d\,x\right )}^2-{\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^3\,15{}\mathrm {i}+18\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^4+{\sin \left (c+d\,x\right )}^5\,5{}\mathrm {i}\right )}{30\,d\,{\cos \left (c+d\,x\right )}^6} \]
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 \[ - i a^{3} \left (\int i \sec ^{4}{\left (c + d x \right )}\, dx + \int \left (- 3 \tan {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\right )\, dx + \int \tan ^{3}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\, dx + \int \left (- 3 i \tan ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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