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 = 0.0757118, antiderivative size = 106, normalized size of antiderivative = 1., 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 3496
Rule 3486
Rule 2633
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*}
Mathematica [A] time = 0.573259, 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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Maple [A] time = 0.062, size = 146, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ( -i{a}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{7}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35}} \right ) -3\,{a}^{3} \left ( -1/7\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}\sin \left ( dx+c \right ) +1/35\, \left ( 8/3+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+4/3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) -{\frac{3\,i}{7}}{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}+{\frac{{a}^{3}\sin \left ( dx+c \right ) }{7} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15661, size = 166, normalized size = 1.57 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.07833, size = 219, normalized size = 2.07 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.02439, size = 192, normalized size = 1.81 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.51986, size = 628, normalized size = 5.92 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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