Optimal. Leaf size=112 \[ \frac{B+i A}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{x (A-i B)}{8 a^3}+\frac{-B+i A}{6 d (a+i a \tan (c+d x))^3}+\frac{B+i A}{8 a d (a+i a \tan (c+d x))^2} \]
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Rubi [A] time = 0.0836763, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {3526, 3479, 8} \[ \frac{B+i A}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{x (A-i B)}{8 a^3}+\frac{-B+i A}{6 d (a+i a \tan (c+d x))^3}+\frac{B+i A}{8 a d (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 3526
Rule 3479
Rule 8
Rubi steps
\begin{align*} \int \frac{A+B \tan (c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=\frac{i A-B}{6 d (a+i a \tan (c+d x))^3}+\frac{(A-i B) \int \frac{1}{(a+i a \tan (c+d x))^2} \, dx}{2 a}\\ &=\frac{i A-B}{6 d (a+i a \tan (c+d x))^3}+\frac{i A+B}{8 a d (a+i a \tan (c+d x))^2}+\frac{(A-i B) \int \frac{1}{a+i a \tan (c+d x)} \, dx}{4 a^2}\\ &=\frac{i A-B}{6 d (a+i a \tan (c+d x))^3}+\frac{i A+B}{8 a d (a+i a \tan (c+d x))^2}+\frac{i A+B}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{(A-i B) \int 1 \, dx}{8 a^3}\\ &=\frac{(A-i B) x}{8 a^3}+\frac{i A-B}{6 d (a+i a \tan (c+d x))^3}+\frac{i A+B}{8 a d (a+i a \tan (c+d x))^2}+\frac{i A+B}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.77543, size = 150, normalized size = 1.34 \[ \frac{\sec ^3(c+d x) ((-27 A+3 i B) \cos (c+d x)+2 (6 i A d x-A+6 B d x-i B) \cos (3 (c+d x))-9 i A \sin (c+d x)+2 i A \sin (3 (c+d x))-12 A d x \sin (3 (c+d x))-9 B \sin (c+d x)-2 B \sin (3 (c+d x))+12 i B d x \sin (3 (c+d x)))}{96 a^3 d (\tan (c+d x)-i)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.03, size = 203, normalized size = 1.8 \begin{align*} -{\frac{A}{6\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}-{\frac{{\frac{i}{6}}B}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{3}}}-{\frac{{\frac{i}{16}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) A}{{a}^{3}d}}-{\frac{\ln \left ( \tan \left ( dx+c \right ) -i \right ) B}{16\,{a}^{3}d}}-{\frac{{\frac{i}{8}}A}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}-{\frac{B}{8\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) ^{2}}}+{\frac{A}{8\,{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) }}-{\frac{{\frac{i}{8}}B}{{a}^{3}d \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{B\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{16\,{a}^{3}d}}+{\frac{{\frac{i}{16}}A\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{{a}^{3}d}} \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 = 1.39536, size = 217, normalized size = 1.94 \begin{align*} \frac{{\left (12 \,{\left (A - i \, B\right )} d x e^{\left (6 i \, d x + 6 i \, c\right )} +{\left (18 i \, A + 6 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (9 i \, A - 3 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i \, A - 2 \, B\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{96 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.2097, size = 260, normalized size = 2.32 \begin{align*} \begin{cases} \frac{\left (\left (512 i A a^{6} d^{2} e^{6 i c} - 512 B a^{6} d^{2} e^{6 i c}\right ) e^{- 6 i d x} + \left (2304 i A a^{6} d^{2} e^{8 i c} - 768 B a^{6} d^{2} e^{8 i c}\right ) e^{- 4 i d x} + \left (4608 i A a^{6} d^{2} e^{10 i c} + 1536 B a^{6} d^{2} e^{10 i c}\right ) e^{- 2 i d x}\right ) e^{- 12 i c}}{24576 a^{9} d^{3}} & \text{for}\: 24576 a^{9} d^{3} e^{12 i c} \neq 0 \\x \left (- \frac{A - i B}{8 a^{3}} + \frac{\left (A e^{6 i c} + 3 A e^{4 i c} + 3 A e^{2 i c} + A - i B e^{6 i c} - i B e^{4 i c} + i B e^{2 i c} + i B\right ) e^{- 6 i c}}{8 a^{3}}\right ) & \text{otherwise} \end{cases} + \frac{x \left (A - i B\right )}{8 a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.4069, size = 177, normalized size = 1.58 \begin{align*} -\frac{\frac{6 \,{\left (i \, A + B\right )} \log \left (\tan \left (d x + c\right ) - i\right )}{a^{3}} + \frac{6 \,{\left (-i \, A - B\right )} \log \left (i \, \tan \left (d x + c\right ) - 1\right )}{a^{3}} + \frac{-11 i \, A \tan \left (d x + c\right )^{3} - 11 \, B \tan \left (d x + c\right )^{3} - 45 \, A \tan \left (d x + c\right )^{2} + 45 i \, B \tan \left (d x + c\right )^{2} + 69 i \, A \tan \left (d x + c\right ) + 69 \, B \tan \left (d x + c\right ) + 51 \, A - 19 i \, B}{a^{3}{\left (\tan \left (d x + c\right ) - i\right )}^{3}}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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