Optimal. Leaf size=126 \[ \frac {3}{16 a^4 d (1+i \tan (c+d x))}+\frac {i x}{16 a^4}-\frac {1}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}+\frac {\tan ^4(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {i \tan ^3(c+d x)}{12 a d (a+i a \tan (c+d x))^3} \]
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Rubi [A] time = 0.18, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3547, 3546, 3540, 3526, 8} \[ \frac {3}{16 a^4 d (1+i \tan (c+d x))}-\frac {1}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}+\frac {i x}{16 a^4}+\frac {\tan ^4(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {i \tan ^3(c+d x)}{12 a d (a+i a \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3526
Rule 3540
Rule 3546
Rule 3547
Rubi steps
\begin {align*} \int \frac {\tan ^3(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=\frac {\tan ^4(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {\int \frac {\tan ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx}{2 a}\\ &=\frac {\tan ^4(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {i \tan ^3(c+d x)}{12 a d (a+i a \tan (c+d x))^3}-\frac {i \int \frac {\tan ^2(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{4 a^2}\\ &=\frac {\tan ^4(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {i \tan ^3(c+d x)}{12 a d (a+i a \tan (c+d x))^3}-\frac {1}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}-\frac {i \int \frac {a-2 i a \tan (c+d x)}{a+i a \tan (c+d x)} \, dx}{8 a^4}\\ &=\frac {3}{16 a^4 d (1+i \tan (c+d x))}+\frac {\tan ^4(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {i \tan ^3(c+d x)}{12 a d (a+i a \tan (c+d x))^3}-\frac {1}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}+\frac {i \int 1 \, dx}{16 a^4}\\ &=\frac {i x}{16 a^4}+\frac {3}{16 a^4 d (1+i \tan (c+d x))}+\frac {\tan ^4(c+d x)}{8 d (a+i a \tan (c+d x))^4}+\frac {i \tan ^3(c+d x)}{12 a d (a+i a \tan (c+d x))^3}-\frac {1}{16 d \left (a^2+i a^2 \tan (c+d x)\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 95, normalized size = 0.75 \[ \frac {\sec ^4(c+d x) (32 i \sin (2 (c+d x))-24 d x \sin (4 (c+d x))-3 i \sin (4 (c+d x))+16 \cos (2 (c+d x))+3 (1+8 i d x) \cos (4 (c+d x)))}{384 a^4 d (\tan (c+d x)-i)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 54, normalized size = 0.43 \[ \frac {{\left (24 i \, d x e^{\left (8 i \, d x + 8 i \, c\right )} + 24 \, e^{\left (6 i \, d x + 6 i \, c\right )} - 8 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3\right )} e^{\left (-8 i \, d x - 8 i \, c\right )}}{384 \, a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.49, size = 88, normalized size = 0.70 \[ -\frac {\frac {12 \, \log \left (\tan \left (d x + c\right ) + i\right )}{a^{4}} - \frac {12 \, \log \left (\tan \left (d x + c\right ) - i\right )}{a^{4}} + \frac {25 \, \tan \left (d x + c\right )^{4} - 124 i \, \tan \left (d x + c\right )^{3} - 54 \, \tan \left (d x + c\right )^{2} - 4 i \, \tan \left (d x + c\right ) - 7}{a^{4} {\left (\tan \left (d x + c\right ) - i\right )}^{4}}}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 116, normalized size = 0.92 \[ -\frac {\ln \left (\tan \left (d x +c \right )+i\right )}{32 d \,a^{4}}+\frac {i}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )}-\frac {5 i}{12 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{3}}+\frac {1}{8 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{4}}-\frac {7}{16 d \,a^{4} \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {\ln \left (\tan \left (d x +c \right )-i\right )}{32 d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.90, size = 60, normalized size = 0.48 \[ \frac {x\,1{}\mathrm {i}}{16\,a^4}+\frac {\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,1{}\mathrm {i}}{16}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{4}+\frac {\mathrm {tan}\left (c+d\,x\right )\,13{}\mathrm {i}}{48}+\frac {1}{12}}{a^4\,d\,{\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.45, size = 158, normalized size = 1.25 \[ \begin {cases} \frac {\left (6144 a^{8} d^{2} e^{14 i c} e^{- 2 i d x} - 2048 a^{8} d^{2} e^{10 i c} e^{- 6 i d x} + 768 a^{8} d^{2} e^{8 i c} e^{- 8 i d x}\right ) e^{- 16 i c}}{98304 a^{12} d^{3}} & \text {for}\: 98304 a^{12} d^{3} e^{16 i c} \neq 0 \\x \left (\frac {\left (i e^{8 i c} - 2 i e^{6 i c} + 2 i e^{2 i c} - i\right ) e^{- 8 i c}}{16 a^{4}} - \frac {i}{16 a^{4}}\right ) & \text {otherwise} \end {cases} + \frac {i x}{16 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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