Optimal. Leaf size=76 \[ -\frac {a \sin ^7(c+d x)}{7 d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^3(c+d x)}{d}+\frac {a \sin (c+d x)}{d}-\frac {i a \cos ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3486, 2633} \[ -\frac {a \sin ^7(c+d x)}{7 d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^3(c+d x)}{d}+\frac {a \sin (c+d x)}{d}-\frac {i a \cos ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 2633
Rule 3486
Rubi steps
\begin {align*} \int \cos ^7(c+d x) (a+i a \tan (c+d x)) \, dx &=-\frac {i a \cos ^7(c+d x)}{7 d}+a \int \cos ^7(c+d x) \, dx\\ &=-\frac {i a \cos ^7(c+d x)}{7 d}-\frac {a \operatorname {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=-\frac {i a \cos ^7(c+d x)}{7 d}+\frac {a \sin (c+d x)}{d}-\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 76, normalized size = 1.00 \[ -\frac {a \sin ^7(c+d x)}{7 d}+\frac {3 a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^3(c+d x)}{d}+\frac {a \sin (c+d x)}{d}-\frac {i a \cos ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 90, normalized size = 1.18 \[ \frac {{\left (-5 i \, a e^{\left (12 i \, d x + 12 i \, c\right )} - 42 i \, a e^{\left (10 i \, d x + 10 i \, c\right )} - 175 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} - 700 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 525 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 70 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i \, a\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{2240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.25, size = 244, normalized size = 3.21 \[ -\frac {{\left (1015 \, a e^{\left (5 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 700 \, a e^{\left (5 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 1015 \, a e^{\left (5 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 700 \, a e^{\left (5 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 315 \, a e^{\left (5 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 315 \, a e^{\left (5 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 20 i \, a e^{\left (12 i \, d x + 8 i \, c\right )} + 168 i \, a e^{\left (10 i \, d x + 6 i \, c\right )} + 700 i \, a e^{\left (8 i \, d x + 4 i \, c\right )} + 2800 i \, a e^{\left (6 i \, d x + 2 i \, c\right )} - 280 i \, a e^{\left (2 i \, d x - 2 i \, c\right )} - 2100 i \, a e^{\left (4 i \, d x\right )} - 28 i \, a e^{\left (-4 i \, c\right )}\right )} e^{\left (-5 i \, d x - i \, c\right )}}{8960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 57, normalized size = 0.75 \[ \frac {-\frac {i a \left (\cos ^{7}\left (d x +c \right )\right )}{7}+\frac {a \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 58, normalized size = 0.76 \[ -\frac {5 i \, a \cos \left (d x + c\right )^{7} + {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a}{35 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.07, size = 93, normalized size = 1.22 \[ -\frac {2\,a\,\left (-\frac {1225\,\sin \left (c+d\,x\right )}{128}-\frac {245\,\sin \left (3\,c+3\,d\,x\right )}{128}-\frac {49\,\sin \left (5\,c+5\,d\,x\right )}{128}-\frac {5\,\sin \left (7\,c+7\,d\,x\right )}{128}+\frac {\cos \left (c+d\,x\right )\,175{}\mathrm {i}}{128}+\frac {\cos \left (3\,c+3\,d\,x\right )\,105{}\mathrm {i}}{128}+\frac {\cos \left (5\,c+5\,d\,x\right )\,35{}\mathrm {i}}{128}+\frac {\cos \left (7\,c+7\,d\,x\right )\,5{}\mathrm {i}}{128}\right )}{35\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.61, size = 257, normalized size = 3.38 \[ \begin {cases} - \frac {\left (107374182400 i a d^{6} e^{16 i c} e^{7 i d x} + 901943132160 i a d^{6} e^{14 i c} e^{5 i d x} + 3758096384000 i a d^{6} e^{12 i c} e^{3 i d x} + 15032385536000 i a d^{6} e^{10 i c} e^{i d x} - 11274289152000 i a d^{6} e^{8 i c} e^{- i d x} - 1503238553600 i a d^{6} e^{6 i c} e^{- 3 i d x} - 150323855360 i a d^{6} e^{4 i c} e^{- 5 i d x}\right ) e^{- 9 i c}}{48103633715200 d^{7}} & \text {for}\: 48103633715200 d^{7} e^{9 i c} \neq 0 \\\frac {x \left (a e^{12 i c} + 6 a e^{10 i c} + 15 a e^{8 i c} + 20 a e^{6 i c} + 15 a e^{4 i c} + 6 a e^{2 i c} + a\right ) e^{- 5 i c}}{64} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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