Optimal. Leaf size=27 \[ -\frac {i (a+i a \tan (c+d x))^6}{6 a d} \]
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Rubi [A] time = 0.04, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3487, 32} \[ -\frac {i (a+i a \tan (c+d x))^6}{6 a d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3487
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac {i \operatorname {Subst}\left (\int (a+x)^5 \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=-\frac {i (a+i a \tan (c+d x))^6}{6 a d}\\ \end {align*}
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Mathematica [B] time = 1.91, size = 134, normalized size = 4.96 \[ \frac {a^5 \sec (c) \sec ^6(c+d x) (15 \sin (c+2 d x)-15 \sin (3 c+2 d x)+6 \sin (3 c+4 d x)-6 \sin (5 c+4 d x)+2 \sin (5 c+6 d x)+15 i \cos (c+2 d x)+15 i \cos (3 c+2 d x)+6 i \cos (3 c+4 d x)+6 i \cos (5 c+4 d x)-20 \sin (c)+20 i \cos (c))}{12 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 153, normalized size = 5.67 \[ \frac {192 i \, a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} + 480 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} + 640 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} + 480 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} + 192 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} + 32 i \, a^{5}}{3 \, {\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.92, size = 82, normalized size = 3.04 \[ -\frac {-i \, a^{5} \tan \left (d x + c\right )^{6} - 6 \, a^{5} \tan \left (d x + c\right )^{5} + 15 i \, a^{5} \tan \left (d x + c\right )^{4} + 20 \, a^{5} \tan \left (d x + c\right )^{3} - 15 i \, a^{5} \tan \left (d x + c\right )^{2} - 6 \, a^{5} \tan \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.44, size = 115, normalized size = 4.26 \[ \frac {\frac {i a^{5} \left (\sin ^{6}\left (d x +c \right )\right )}{6 \cos \left (d x +c \right )^{6}}+\frac {a^{5} \left (\sin ^{5}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{5}}-\frac {5 i a^{5} \left (\sin ^{4}\left (d x +c \right )\right )}{2 \cos \left (d x +c \right )^{4}}-\frac {10 a^{5} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}+\frac {5 i a^{5}}{2 \cos \left (d x +c \right )^{2}}+a^{5} \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 21, normalized size = 0.78 \[ -\frac {i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{6}}{6 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.26, size = 114, normalized size = 4.22 \[ \frac {a^5\,\sin \left (c+d\,x\right )\,\left (6\,{\cos \left (c+d\,x\right )}^5+{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )\,15{}\mathrm {i}-20\,{\cos \left (c+d\,x\right )}^3\,{\sin \left (c+d\,x\right )}^2-{\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^3\,15{}\mathrm {i}+6\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^4+{\sin \left (c+d\,x\right )}^5\,1{}\mathrm {i}\right )}{6\,d\,{\cos \left (c+d\,x\right )}^6} \]
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 \[ i a^{5} \left (\int \left (- i \sec ^{2}{\left (c + d x \right )}\right )\, dx + \int 5 \tan {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \left (- 10 \tan ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\right )\, dx + \int \tan ^{5}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int 10 i \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \left (- 5 i \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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