Optimal. Leaf size=174 \[ \frac {x \left (c^3+3 i c^2 d-3 c d^2+3 i d^3\right )}{4 a^2 (c-i d) (c+i d)^3}-\frac {d^3 \log (c \cos (e+f x)+d \sin (e+f x))}{a^2 f (c-i d) (c+i d)^3}+\frac {-3 d+i c}{4 a^2 f (c+i d)^2 (1+i \tan (e+f x))}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2} \]
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Rubi [A] time = 0.41, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3559, 3596, 3531, 3530} \[ \frac {x \left (3 i c^2 d+c^3-3 c d^2+3 i d^3\right )}{4 a^2 (c-i d) (c+i d)^3}-\frac {d^3 \log (c \cos (e+f x)+d \sin (e+f x))}{a^2 f (c-i d) (c+i d)^3}+\frac {-3 d+i c}{4 a^2 f (c+i d)^2 (1+i \tan (e+f x))}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2} \]
Antiderivative was successfully verified.
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Rule 3530
Rule 3531
Rule 3559
Rule 3596
Rubi steps
\begin {align*} \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))} \, dx &=-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2}-\frac {\int \frac {-2 a (i c-2 d)-2 i a d \tan (e+f x)}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))} \, dx}{4 a^2 (i c-d)}\\ &=\frac {i c-3 d}{4 a^2 (c+i d)^2 f (1+i \tan (e+f x))}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2}-\frac {\int \frac {-2 a^2 \left (c^2+3 i c d-4 d^2\right )-2 a^2 (c+3 i d) d \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{8 a^4 (c+i d)^2}\\ &=\frac {\left (c^3+3 i c^2 d-3 c d^2+3 i d^3\right ) x}{4 a^2 (c+i d)^2 \left (c^2+d^2\right )}+\frac {i c-3 d}{4 a^2 (c+i d)^2 f (1+i \tan (e+f x))}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2}-\frac {d^3 \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{a^2 (c+i d)^2 \left (c^2+d^2\right )}\\ &=\frac {\left (c^3+3 i c^2 d-3 c d^2+3 i d^3\right ) x}{4 a^2 (c+i d)^2 \left (c^2+d^2\right )}-\frac {d^3 \log (c \cos (e+f x)+d \sin (e+f x))}{a^2 (c+i d)^2 \left (c^2+d^2\right ) f}+\frac {i c-3 d}{4 a^2 (c+i d)^2 f (1+i \tan (e+f x))}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2}\\ \end {align*}
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Mathematica [B] time = 1.53, size = 372, normalized size = 2.14 \[ -\frac {\sec ^2(e+f x) \left (4 i c^3 f x \sin (2 (e+f x))+c^3 \sin (2 (e+f x))+4 i c^3+16 d^3 (\sin (2 (e+f x))-i \cos (2 (e+f x))) \tan ^{-1}\left (\frac {\left (d^2-c^2\right ) \sin (f x)-2 c d \cos (f x)}{\left (c^2-d^2\right ) \cos (f x)-2 c d \sin (f x)}\right )+i c^2 d \sin (2 (e+f x))-12 c^2 d f x \sin (2 (e+f x))-8 c^2 d+\cos (2 (e+f x)) \left (-8 d^3 \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right )+(c+i d)^2 (4 c f x+i c+4 i d f x+d)\right )-8 i d^3 \sin (2 (e+f x)) \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right )+c d^2 \sin (2 (e+f x))-12 i c d^2 f x \sin (2 (e+f x))+4 i c d^2+i d^3 \sin (2 (e+f x))+4 d^3 f x \sin (2 (e+f x))-8 d^3\right )}{16 a^2 f (c-i d) (c+i d)^3 (\tan (e+f x)-i)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 185, normalized size = 1.06 \[ -\frac {{\left (16 \, d^{3} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (\frac {{\left (i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d}{i \, c + d}\right ) - {\left (4 \, c^{3} + 12 i \, c^{2} d - 12 \, c d^{2} + 28 i \, d^{3}\right )} f x e^{\left (4 i \, f x + 4 i \, e\right )} - i \, c^{3} + c^{2} d - i \, c d^{2} + d^{3} + {\left (-4 i \, c^{3} + 8 \, c^{2} d - 4 i \, c d^{2} + 8 \, d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{{\left (16 \, a^{2} c^{4} + 32 i \, a^{2} c^{3} d + 32 i \, a^{2} c d^{3} - 16 \, a^{2} d^{4}\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.47, size = 296, normalized size = 1.70 \[ -\frac {2 \, {\left (\frac {d^{4} \log \left (-i \, d \tan \left (f x + e\right ) - i \, c\right )}{2 \, a^{2} c^{4} d + 4 i \, a^{2} c^{3} d^{2} + 4 i \, a^{2} c d^{4} - 2 \, a^{2} d^{5}} + \frac {{\left (c^{2} + 4 i \, c d - 7 \, d^{2}\right )} \log \left (i \, \tan \left (f x + e\right ) + 1\right )}{-16 i \, a^{2} c^{3} + 48 \, a^{2} c^{2} d + 48 i \, a^{2} c d^{2} - 16 \, a^{2} d^{3}} - \frac {\log \left (\tan \left (f x + e\right ) + i\right )}{-16 i \, a^{2} c - 16 \, a^{2} d} - \frac {3 \, c^{2} \tan \left (f x + e\right )^{2} + 12 i \, c d \tan \left (f x + e\right )^{2} - 21 \, d^{2} \tan \left (f x + e\right )^{2} - 10 i \, c^{2} \tan \left (f x + e\right ) + 40 \, c d \tan \left (f x + e\right ) + 54 i \, d^{2} \tan \left (f x + e\right ) - 11 \, c^{2} - 36 i \, c d + 37 \, d^{2}}{{\left (-32 i \, a^{2} c^{3} + 96 \, a^{2} c^{2} d + 96 i \, a^{2} c d^{2} - 32 \, a^{2} d^{3}\right )} {\left (\tan \left (f x + e\right ) - i\right )}^{2}}\right )}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.34, size = 339, normalized size = 1.95 \[ -\frac {i \ln \left (\tan \left (f x +e \right )+i\right )}{f \,a^{2} \left (8 i d -8 c \right )}+\frac {d^{3} \ln \left (c +d \tan \left (f x +e \right )\right )}{f \,a^{2} \left (i d -c \right ) \left (i d +c \right )^{3}}+\frac {i c d}{f \,a^{2} \left (i d +c \right )^{3} \left (\tan \left (f x +e \right )-i\right )}+\frac {c^{2}}{4 f \,a^{2} \left (i d +c \right )^{3} \left (\tan \left (f x +e \right )-i\right )}-\frac {3 d^{2}}{4 f \,a^{2} \left (i d +c \right )^{3} \left (\tan \left (f x +e \right )-i\right )}-\frac {i c^{2}}{4 f \,a^{2} \left (i d +c \right )^{3} \left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {i d^{2}}{4 f \,a^{2} \left (i d +c \right )^{3} \left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {c d}{2 f \,a^{2} \left (i d +c \right )^{3} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {i \ln \left (\tan \left (f x +e \right )-i\right ) c^{2}}{8 f \,a^{2} \left (i d +c \right )^{3}}+\frac {7 i \ln \left (\tan \left (f x +e \right )-i\right ) d^{2}}{8 f \,a^{2} \left (i d +c \right )^{3}}+\frac {\ln \left (\tan \left (f x +e \right )-i\right ) c d}{2 f \,a^{2} \left (i d +c \right )^{3}} \]
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 = 9.23, size = 1384, normalized size = 7.95 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.53, size = 614, normalized size = 3.53 \[ \frac {x \left (- c^{2} - 4 i c d + 7 d^{2}\right )}{- 4 a^{2} c^{3} - 12 i a^{2} c^{2} d + 12 a^{2} c d^{2} + 4 i a^{2} d^{3}} + \begin {cases} \frac {\left (4 i a^{2} c^{2} f e^{2 i e} - 8 a^{2} c d f e^{2 i e} - 4 i a^{2} d^{2} f e^{2 i e}\right ) e^{- 4 i f x} + \left (16 i a^{2} c^{2} f e^{4 i e} - 48 a^{2} c d f e^{4 i e} - 32 i a^{2} d^{2} f e^{4 i e}\right ) e^{- 2 i f x}}{64 a^{4} c^{3} f^{2} e^{6 i e} + 192 i a^{4} c^{2} d f^{2} e^{6 i e} - 192 a^{4} c d^{2} f^{2} e^{6 i e} - 64 i a^{4} d^{3} f^{2} e^{6 i e}} & \text {for}\: 64 a^{4} c^{3} f^{2} e^{6 i e} + 192 i a^{4} c^{2} d f^{2} e^{6 i e} - 192 a^{4} c d^{2} f^{2} e^{6 i e} - 64 i a^{4} d^{3} f^{2} e^{6 i e} \neq 0 \\x \left (- \frac {i c^{2} - 4 c d - 7 i d^{2}}{4 i a^{2} c^{3} - 12 a^{2} c^{2} d - 12 i a^{2} c d^{2} + 4 a^{2} d^{3}} + \frac {- c^{2} e^{4 i e} - 2 c^{2} e^{2 i e} - c^{2} - 4 i c d e^{4 i e} - 6 i c d e^{2 i e} - 2 i c d + 7 d^{2} e^{4 i e} + 4 d^{2} e^{2 i e} + d^{2}}{- 4 a^{2} c^{3} e^{4 i e} - 12 i a^{2} c^{2} d e^{4 i e} + 12 a^{2} c d^{2} e^{4 i e} + 4 i a^{2} d^{3} e^{4 i e}}\right ) & \text {otherwise} \end {cases} - \frac {d^{3} \log {\left (\frac {- i c + d}{- i c e^{2 i e} - d e^{2 i e}} + e^{2 i f x} \right )}}{a^{2} f \left (c - i d\right ) \left (c + i d\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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