Optimal. Leaf size=120 \[ \frac{15}{16 a^4 d (1+i \tan (c+d x))}+\frac{7}{16 a^4 d (1+i \tan (c+d x))^2}+\frac{\log (\sin (c+d x))}{a^4 d}-\frac{15 i x}{16 a^4}+\frac{1}{4 a d (a+i a \tan (c+d x))^3}+\frac{1}{8 d (a+i a \tan (c+d x))^4} \]
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Rubi [A] time = 0.320424, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3559, 3596, 3531, 3475} \[ \frac{15}{16 a^4 d (1+i \tan (c+d x))}+\frac{7}{16 a^4 d (1+i \tan (c+d x))^2}+\frac{\log (\sin (c+d x))}{a^4 d}-\frac{15 i x}{16 a^4}+\frac{1}{4 a d (a+i a \tan (c+d x))^3}+\frac{1}{8 d (a+i a \tan (c+d x))^4} \]
Antiderivative was successfully verified.
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Rule 3559
Rule 3596
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \frac{\cot (c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=\frac{1}{8 d (a+i a \tan (c+d x))^4}+\frac{\int \frac{\cot (c+d x) (8 a-4 i a \tan (c+d x))}{(a+i a \tan (c+d x))^3} \, dx}{8 a^2}\\ &=\frac{1}{8 d (a+i a \tan (c+d x))^4}+\frac{1}{4 a d (a+i a \tan (c+d x))^3}+\frac{\int \frac{\cot (c+d x) \left (48 a^2-36 i a^2 \tan (c+d x)\right )}{(a+i a \tan (c+d x))^2} \, dx}{48 a^4}\\ &=\frac{7}{16 a^4 d (1+i \tan (c+d x))^2}+\frac{1}{8 d (a+i a \tan (c+d x))^4}+\frac{1}{4 a d (a+i a \tan (c+d x))^3}+\frac{\int \frac{\cot (c+d x) \left (192 a^3-168 i a^3 \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{192 a^6}\\ &=\frac{7}{16 a^4 d (1+i \tan (c+d x))^2}+\frac{1}{8 d (a+i a \tan (c+d x))^4}+\frac{1}{4 a d (a+i a \tan (c+d x))^3}+\frac{15}{16 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{\int \cot (c+d x) \left (384 a^4-360 i a^4 \tan (c+d x)\right ) \, dx}{384 a^8}\\ &=-\frac{15 i x}{16 a^4}+\frac{7}{16 a^4 d (1+i \tan (c+d x))^2}+\frac{1}{8 d (a+i a \tan (c+d x))^4}+\frac{1}{4 a d (a+i a \tan (c+d x))^3}+\frac{15}{16 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{\int \cot (c+d x) \, dx}{a^4}\\ &=-\frac{15 i x}{16 a^4}+\frac{\log (\sin (c+d x))}{a^4 d}+\frac{7}{16 a^4 d (1+i \tan (c+d x))^2}+\frac{1}{8 d (a+i a \tan (c+d x))^4}+\frac{1}{4 a d (a+i a \tan (c+d x))^3}+\frac{15}{16 d \left (a^4+i a^4 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.397964, size = 123, normalized size = 1.02 \[ \frac{\sec ^4(c+d x) (96 i \sin (2 (c+d x))+120 d x \sin (4 (c+d x))-i \sin (4 (c+d x))+112 \cos (2 (c+d x))+128 i \sin (4 (c+d x)) \log (\sin (c+d x))+\cos (4 (c+d x)) (128 \log (\sin (c+d x))-120 i d x+1)+32)}{128 a^4 d (\tan (c+d x)-i)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.089, size = 130, normalized size = 1.1 \begin{align*}{\frac{{\frac{i}{4}}}{d{a}^{4} \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}-{\frac{{\frac{15\,i}{16}}}{d{a}^{4} \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{1}{8\,d{a}^{4} \left ( \tan \left ( dx+c \right ) -i \right ) ^{4}}}-{\frac{7}{16\,d{a}^{4} \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{31\,\ln \left ( \tan \left ( dx+c \right ) -i \right ) }{32\,d{a}^{4}}}-{\frac{\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{32\,d{a}^{4}}}+{\frac{\ln \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{4}}} \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 = 2.32243, size = 274, normalized size = 2.28 \begin{align*} \frac{{\left (-248 i \, d x e^{\left (8 i \, d x + 8 i \, c\right )} + 128 \, e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) + 104 \, e^{\left (6 i \, d x + 6 i \, c\right )} + 32 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (-8 i \, d x - 8 i \, c\right )}}{128 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.90161, size = 221, normalized size = 1.84 \begin{align*} \begin{cases} \frac{\left (106496 a^{12} d^{3} e^{18 i c} e^{- 2 i d x} + 32768 a^{12} d^{3} e^{16 i c} e^{- 4 i d x} + 8192 a^{12} d^{3} e^{14 i c} e^{- 6 i d x} + 1024 a^{12} d^{3} e^{12 i c} e^{- 8 i d x}\right ) e^{- 20 i c}}{131072 a^{16} d^{4}} & \text{for}\: 131072 a^{16} d^{4} e^{20 i c} \neq 0 \\x \left (- \frac{\left (31 i e^{8 i c} + 26 i e^{6 i c} + 16 i e^{4 i c} + 6 i e^{2 i c} + i\right ) e^{- 8 i c}}{16 a^{4}} + \frac{31 i}{16 a^{4}}\right ) & \text{otherwise} \end{cases} - \frac{31 i x}{16 a^{4}} + \frac{\log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28059, size = 138, normalized size = 1.15 \begin{align*} -\frac{\frac{12 \, \log \left (\tan \left (d x + c\right ) + i\right )}{a^{4}} + \frac{372 \, \log \left (\tan \left (d x + c\right ) - i\right )}{a^{4}} - \frac{384 \, \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}} - \frac{775 \, \tan \left (d x + c\right )^{4} - 3460 i \, \tan \left (d x + c\right )^{3} - 5898 \, \tan \left (d x + c\right )^{2} + 4612 i \, \tan \left (d x + c\right ) + 1447}{a^{4}{\left (\tan \left (d x + c\right ) - i\right )}^{4}}}{384 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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