Optimal. Leaf size=25 \[ -\frac {i a}{2 f (c-i c \tan (e+f x))^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3603, 3568, 32}
\begin {gather*} -\frac {i a}{2 f (c-i c \tan (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 3568
Rule 3603
Rubi steps
\begin {align*} \int \frac {a+i a \tan (e+f x)}{(c-i c \tan (e+f x))^2} \, dx &=(a c) \int \frac {\sec ^2(e+f x)}{(c-i c \tan (e+f x))^3} \, dx\\ &=\frac {(i a) \text {Subst}\left (\int \frac {1}{(c+x)^3} \, dx,x,-i c \tan (e+f x)\right )}{f}\\ &=-\frac {i a}{2 f (c-i c \tan (e+f x))^2}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(51\) vs. \(2(25)=50\).
time = 0.40, size = 51, normalized size = 2.04 \begin {gather*} \frac {a (3 \cos (e+f x)-i \sin (e+f x)) (-i \cos (3 (e+f x))+\sin (3 (e+f x)))}{8 c^2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 22, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {i a}{2 f \,c^{2} \left (\tan \left (f x +e \right )+i\right )^{2}}\) | \(22\) |
default | \(\frac {i a}{2 f \,c^{2} \left (\tan \left (f x +e \right )+i\right )^{2}}\) | \(22\) |
risch | \(-\frac {i a \,{\mathrm e}^{4 i \left (f x +e \right )}}{8 c^{2} f}-\frac {i a \,{\mathrm e}^{2 i \left (f x +e \right )}}{4 c^{2} f}\) | \(40\) |
norman | \(\frac {\frac {a \tan \left (f x +e \right )}{c f}-\frac {i a}{2 c f}+\frac {i a \left (\tan ^{2}\left (f x +e \right )\right )}{2 c f}}{c \left (1+\tan ^{2}\left (f x +e \right )\right )^{2}}\) | \(60\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.04, size = 35, normalized size = 1.40 \begin {gather*} \frac {-i \, a e^{\left (4 i \, f x + 4 i \, e\right )} - 2 i \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{8 \, c^{2} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 88 vs. \(2 (20) = 40\).
time = 0.14, size = 88, normalized size = 3.52 \begin {gather*} \begin {cases} \frac {- 4 i a c^{2} f e^{4 i e} e^{4 i f x} - 8 i a c^{2} f e^{2 i e} e^{2 i f x}}{32 c^{4} f^{2}} & \text {for}\: c^{4} f^{2} \neq 0 \\\frac {x \left (a e^{4 i e} + a e^{2 i e}\right )}{2 c^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 65 vs. \(2 (20) = 40\).
time = 0.51, size = 65, normalized size = 2.60 \begin {gather*} -\frac {2 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + i \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{c^{2} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.61, size = 21, normalized size = 0.84 \begin {gather*} \frac {a\,1{}\mathrm {i}}{2\,c^2\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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