Optimal. Leaf size=141 \[ -\frac{i}{32 d \left (a^3-i a^3 \tan (c+d x)\right )}+\frac{i}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{5 x}{32 a^3}+\frac{i a}{16 d (a+i a \tan (c+d x))^4}+\frac{i}{12 d (a+i a \tan (c+d x))^3}+\frac{3 i}{32 a d (a+i a \tan (c+d x))^2} \]
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Rubi [A] time = 0.0861982, antiderivative size = 141, 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 = {3487, 44, 206} \[ -\frac{i}{32 d \left (a^3-i a^3 \tan (c+d x)\right )}+\frac{i}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{5 x}{32 a^3}+\frac{i a}{16 d (a+i a \tan (c+d x))^4}+\frac{i}{12 d (a+i a \tan (c+d x))^3}+\frac{3 i}{32 a d (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 44
Rule 206
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=-\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x)^2 (a+x)^5} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{32 a^5 (a-x)^2}+\frac{1}{4 a^2 (a+x)^5}+\frac{1}{4 a^3 (a+x)^4}+\frac{3}{16 a^4 (a+x)^3}+\frac{1}{8 a^5 (a+x)^2}+\frac{5}{32 a^5 \left (a^2-x^2\right )}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=\frac{i a}{16 d (a+i a \tan (c+d x))^4}+\frac{i}{12 d (a+i a \tan (c+d x))^3}+\frac{3 i}{32 a d (a+i a \tan (c+d x))^2}-\frac{i}{32 d \left (a^3-i a^3 \tan (c+d x)\right )}+\frac{i}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{(5 i) \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,i a \tan (c+d x)\right )}{32 a^2 d}\\ &=\frac{5 x}{32 a^3}+\frac{i a}{16 d (a+i a \tan (c+d x))^4}+\frac{i}{12 d (a+i a \tan (c+d x))^3}+\frac{3 i}{32 a d (a+i a \tan (c+d x))^2}-\frac{i}{32 d \left (a^3-i a^3 \tan (c+d x)\right )}+\frac{i}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.223901, size = 115, normalized size = 0.82 \[ \frac{\sec ^3(c+d x) (-60 i \sin (c+d x)-120 d x \sin (3 (c+d x))+20 i \sin (3 (c+d x))+15 i \sin (5 (c+d x))-180 \cos (c+d x)+20 i (6 d x+i) \cos (3 (c+d x))+9 \cos (5 (c+d x)))}{768 a^3 d (\tan (c+d x)-i)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.097, size = 137, normalized size = 1. \begin{align*}{\frac{-{\frac{5\,i}{64}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) }{{a}^{3}d}}+{\frac{{\frac{i}{16}}}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{4}}}-{\frac{{\frac{3\,i}{32}}}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{1}{12\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}+{\frac{1}{8\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{{\frac{5\,i}{64}}\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{{a}^{3}d}}+{\frac{1}{32\,{a}^{3}d \left ( \tan \left ( dx+c \right ) +i \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 = 2.25761, size = 247, normalized size = 1.75 \begin{align*} \frac{{\left (120 \, d x e^{\left (8 i \, d x + 8 i \, c\right )} - 12 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 120 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 60 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 20 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (-8 i \, d x - 8 i \, c\right )}}{768 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.04198, size = 226, normalized size = 1.6 \begin{align*} \begin{cases} \frac{\left (- 100663296 i a^{12} d^{4} e^{22 i c} e^{2 i d x} + 1006632960 i a^{12} d^{4} e^{18 i c} e^{- 2 i d x} + 503316480 i a^{12} d^{4} e^{16 i c} e^{- 4 i d x} + 167772160 i a^{12} d^{4} e^{14 i c} e^{- 6 i d x} + 25165824 i a^{12} d^{4} e^{12 i c} e^{- 8 i d x}\right ) e^{- 20 i c}}{6442450944 a^{15} d^{5}} & \text{for}\: 6442450944 a^{15} d^{5} e^{20 i c} \neq 0 \\x \left (\frac{\left (e^{10 i c} + 5 e^{8 i c} + 10 e^{6 i c} + 10 e^{4 i c} + 5 e^{2 i c} + 1\right ) e^{- 8 i c}}{32 a^{3}} - \frac{5}{32 a^{3}}\right ) & \text{otherwise} \end{cases} + \frac{5 x}{32 a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17716, size = 161, normalized size = 1.14 \begin{align*} -\frac{-\frac{60 i \, \log \left (-i \, \tan \left (d x + c\right ) + 1\right )}{a^{3}} + \frac{60 i \, \log \left (-i \, \tan \left (d x + c\right ) - 1\right )}{a^{3}} - \frac{12 \,{\left (5 \, \tan \left (d x + c\right ) + 7 i\right )}}{a^{3}{\left (i \, \tan \left (d x + c\right ) - 1\right )}} + \frac{-125 i \, \tan \left (d x + c\right )^{4} - 596 \, \tan \left (d x + c\right )^{3} + 1110 i \, \tan \left (d x + c\right )^{2} + 996 \, \tan \left (d x + c\right ) - 405 i}{a^{3}{\left (\tan \left (d x + c\right ) - i\right )}^{4}}}{768 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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