Optimal. Leaf size=106 \[ \frac {3 a^3 \sin ^5(c+d x)}{35 d}-\frac {2 a^3 \sin ^3(c+d x)}{7 d}+\frac {3 a^3 \sin (c+d x)}{7 d}-\frac {3 i a^3 \cos ^5(c+d x)}{35 d}-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{7 d} \]
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Rubi [A] time = 0.08, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3496, 3486, 2633} \[ \frac {3 a^3 \sin ^5(c+d x)}{35 d}-\frac {2 a^3 \sin ^3(c+d x)}{7 d}+\frac {3 a^3 \sin (c+d x)}{7 d}-\frac {3 i a^3 \cos ^5(c+d x)}{35 d}-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{7 d} \]
Antiderivative was successfully verified.
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Rule 2633
Rule 3486
Rule 3496
Rubi steps
\begin {align*} \int \cos ^7(c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{7 d}+\frac {1}{7} \left (3 a^2\right ) \int \cos ^5(c+d x) (a+i a \tan (c+d x)) \, dx\\ &=-\frac {3 i a^3 \cos ^5(c+d x)}{35 d}-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{7 d}+\frac {1}{7} \left (3 a^3\right ) \int \cos ^5(c+d x) \, dx\\ &=-\frac {3 i a^3 \cos ^5(c+d x)}{35 d}-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{7 d}-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{7 d}\\ &=-\frac {3 i a^3 \cos ^5(c+d x)}{35 d}+\frac {3 a^3 \sin (c+d x)}{7 d}-\frac {2 a^3 \sin ^3(c+d x)}{7 d}+\frac {3 a^3 \sin ^5(c+d x)}{35 d}-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^2}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.73, size = 77, normalized size = 0.73 \[ \frac {a^3 (\sin (3 (c+d x))-i \cos (3 (c+d x))) (-56 i \sin (2 (c+d x))+20 i \sin (4 (c+d x))+84 \cos (2 (c+d x))-15 \cos (4 (c+d x))+35)}{280 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 76, normalized size = 0.72 \[ \frac {{\left (-5 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 28 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 70 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 140 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 35 i \, a^{3}\right )} e^{\left (-i \, d x - i \, c\right )}}{560 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.74, size = 465, normalized size = 4.39 \[ \frac {19635 \, a^{3} e^{\left (5 i \, d x + 3 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 39270 \, a^{3} e^{\left (3 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 19635 \, a^{3} e^{\left (i \, d x - i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 19635 \, a^{3} e^{\left (5 i \, d x + 3 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 39270 \, a^{3} e^{\left (3 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 19635 \, a^{3} e^{\left (i \, d x - i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 19635 \, a^{3} e^{\left (5 i \, d x + 3 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 39270 \, a^{3} e^{\left (3 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 19635 \, a^{3} e^{\left (i \, d x - i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 19635 \, a^{3} e^{\left (5 i \, d x + 3 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 39270 \, a^{3} e^{\left (3 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 19635 \, a^{3} e^{\left (i \, d x - i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 640 i \, a^{3} e^{\left (12 i \, d x + 10 i \, c\right )} - 4864 i \, a^{3} e^{\left (10 i \, d x + 8 i \, c\right )} - 16768 i \, a^{3} e^{\left (8 i \, d x + 6 i \, c\right )} - 39424 i \, a^{3} e^{\left (6 i \, d x + 4 i \, c\right )} - 40320 i \, a^{3} e^{\left (4 i \, d x + 2 i \, c\right )} - 8960 i \, a^{3} e^{\left (2 i \, d x\right )} + 4480 i \, a^{3} e^{\left (-2 i \, c\right )}}{71680 \, {\left (d e^{\left (5 i \, d x + 3 i \, c\right )} + 2 \, d e^{\left (3 i \, d x + i \, c\right )} + d e^{\left (i \, d x - i \, c\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.51, size = 146, normalized size = 1.38 \[ \frac {-i a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )-3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {3 i a^{3} \left (\cos ^{7}\left (d x +c \right )\right )}{7}+\frac {a^{3} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 123, normalized size = 1.16 \[ -\frac {15 i \, a^{3} \cos \left (d x + c\right )^{7} + i \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{3} + {\left (15 \, \sin \left (d x + c\right )^{7} - 42 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3}\right )} a^{3} + {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a^{3}}{35 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.54, size = 134, normalized size = 1.26 \[ -\frac {2\,a^3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {17\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}-\frac {17\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{2}+\frac {31\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{2}-\frac {5\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{2}+\frac {\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,35{}\mathrm {i}}{8}-\frac {\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,35{}\mathrm {i}}{8}+\frac {\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,119{}\mathrm {i}}{8}-\frac {\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,15{}\mathrm {i}}{8}\right )}{35\,d\,\left (\cos \left (3\,c+3\,d\,x\right )-\sin \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 194, normalized size = 1.83 \[ \begin {cases} - \frac {\left (10240 i a^{3} d^{4} e^{8 i c} e^{7 i d x} + 57344 i a^{3} d^{4} e^{6 i c} e^{5 i d x} + 143360 i a^{3} d^{4} e^{4 i c} e^{3 i d x} + 286720 i a^{3} d^{4} e^{2 i c} e^{i d x} - 71680 i a^{3} d^{4} e^{- i d x}\right ) e^{- i c}}{1146880 d^{5}} & \text {for}\: 1146880 d^{5} e^{i c} \neq 0 \\\frac {x \left (a^{3} e^{8 i c} + 4 a^{3} e^{6 i c} + 6 a^{3} e^{4 i c} + 4 a^{3} e^{2 i c} + a^{3}\right ) e^{- i c}}{16} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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