Optimal. Leaf size=156 \[ \frac{2 \sin ^5(c+d x)}{33 a^4 d}-\frac{20 \sin ^3(c+d x)}{99 a^4 d}+\frac{10 \sin (c+d x)}{33 a^4 d}+\frac{4 i \cos ^5(c+d x)}{33 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{7 i \cos ^3(c+d x)}{99 a d (a+i a \tan (c+d x))^3}+\frac{i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4} \]
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Rubi [A] time = 0.147537, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3502, 3500, 2633} \[ \frac{2 \sin ^5(c+d x)}{33 a^4 d}-\frac{20 \sin ^3(c+d x)}{99 a^4 d}+\frac{10 \sin (c+d x)}{33 a^4 d}+\frac{4 i \cos ^5(c+d x)}{33 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{7 i \cos ^3(c+d x)}{99 a d (a+i a \tan (c+d x))^3}+\frac{i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 3500
Rule 2633
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=\frac{i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}+\frac{7 \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx}{11 a}\\ &=\frac{i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}+\frac{7 i \cos ^3(c+d x)}{99 a d (a+i a \tan (c+d x))^3}+\frac{14 \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{33 a^2}\\ &=\frac{i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}+\frac{7 i \cos ^3(c+d x)}{99 a d (a+i a \tan (c+d x))^3}+\frac{4 i \cos ^5(c+d x)}{33 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{10 \int \cos ^5(c+d x) \, dx}{33 a^4}\\ &=\frac{i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}+\frac{7 i \cos ^3(c+d x)}{99 a d (a+i a \tan (c+d x))^3}+\frac{4 i \cos ^5(c+d x)}{33 d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac{10 \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{33 a^4 d}\\ &=\frac{10 \sin (c+d x)}{33 a^4 d}-\frac{20 \sin ^3(c+d x)}{99 a^4 d}+\frac{2 \sin ^5(c+d x)}{33 a^4 d}+\frac{i \cos ^3(c+d x)}{11 d (a+i a \tan (c+d x))^4}+\frac{7 i \cos ^3(c+d x)}{99 a d (a+i a \tan (c+d x))^3}+\frac{4 i \cos ^5(c+d x)}{33 d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.353103, size = 117, normalized size = 0.75 \[ -\frac{i \sec ^4(c+d x) (-231 i \sin (c+d x)-891 i \sin (3 (c+d x))+385 i \sin (5 (c+d x))+21 i \sin (7 (c+d x))-924 \cos (c+d x)-1188 \cos (3 (c+d x))+308 \cos (5 (c+d x))+12 \cos (7 (c+d x)))}{6336 a^4 d (\tan (c+d x)-i)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.107, size = 240, normalized size = 1.5 \begin{align*} 2\,{\frac{1}{{a}^{4}d} \left ({\frac{4\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{10}}}-{\frac{{\frac{67\,i}{4}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{4}}}-{\frac{22\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{8}}}+{\frac{{\frac{385\,i}{12}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{6}}}+{\frac{{\frac{201\,i}{64}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}-{\frac{8}{11\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{11}}}+{\frac{104}{9\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{9}}}-{\frac{61}{2\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{7}}}+{\frac{105}{4\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{5}}}-{\frac{267}{32\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{3}}}+{\frac{15}{16\,\tan \left ( 1/2\,dx+c/2 \right ) -16\,i}}-{\frac{{\frac{i}{64}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{2}}}-{\frac{1}{96\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{3}}}+1/16\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{-1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.32921, size = 336, normalized size = 2.15 \begin{align*} \frac{{\left (-33 i \, e^{\left (14 i \, d x + 14 i \, c\right )} - 693 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 2079 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 1155 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 693 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 297 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 77 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 9 i\right )} e^{\left (-11 i \, d x - 11 i \, c\right )}}{12672 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.02176, size = 301, normalized size = 1.93 \begin{align*} \begin{cases} \frac{\left (- 167196136166129664 i a^{28} d^{7} e^{39 i c} e^{3 i d x} - 3511118859488722944 i a^{28} d^{7} e^{37 i c} e^{i d x} + 10533356578466168832 i a^{28} d^{7} e^{35 i c} e^{- i d x} + 5851864765814538240 i a^{28} d^{7} e^{33 i c} e^{- 3 i d x} + 3511118859488722944 i a^{28} d^{7} e^{31 i c} e^{- 5 i d x} + 1504765225495166976 i a^{28} d^{7} e^{29 i c} e^{- 7 i d x} + 390124317720969216 i a^{28} d^{7} e^{27 i c} e^{- 9 i d x} + 45598946227126272 i a^{28} d^{7} e^{25 i c} e^{- 11 i d x}\right ) e^{- 36 i c}}{64203316287793790976 a^{32} d^{8}} & \text{for}\: 64203316287793790976 a^{32} d^{8} e^{36 i c} \neq 0 \\\frac{x \left (e^{14 i c} + 7 e^{12 i c} + 21 e^{10 i c} + 35 e^{8 i c} + 35 e^{6 i c} + 21 e^{4 i c} + 7 e^{2 i c} + 1\right ) e^{- 11 i c}}{128 a^{4}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15726, size = 266, normalized size = 1.71 \begin{align*} \frac{\frac{33 \,{\left (12 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 21 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 11\right )}}{a^{4}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right )}^{3}} + \frac{5940 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 39501 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 141075 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 313236 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 479556 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 516054 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 397914 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 214500 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 79024 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 17765 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2155}{a^{4}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{11}}}{3168 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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