Optimal. Leaf size=28 \[ \frac {c^2 \tan (e+f x)}{f (a+i a \tan (e+f x))^2} \]
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Rubi [A]
time = 0.07, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3603, 3568, 34}
\begin {gather*} \frac {c^2 \tan (e+f x)}{f (a+i a \tan (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 34
Rule 3568
Rule 3603
Rubi steps
\begin {align*} \int \frac {(c-i c \tan (e+f x))^2}{(a+i a \tan (e+f x))^2} \, dx &=\left (a^2 c^2\right ) \int \frac {\sec ^4(e+f x)}{(a+i a \tan (e+f x))^4} \, dx\\ &=-\frac {\left (i c^2\right ) \text {Subst}\left (\int \frac {a-x}{(a+x)^3} \, dx,x,i a \tan (e+f x)\right )}{a f}\\ &=\frac {c^2 \tan (e+f x)}{f (a+i a \tan (e+f x))^2}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 34, normalized size = 1.21 \begin {gather*} \frac {c^2 (i \cos (4 (e+f x))+\sin (4 (e+f x)))}{4 a^2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 39, normalized size = 1.39
method | result | size |
risch | \(\frac {i c^{2} {\mathrm e}^{-4 i \left (f x +e \right )}}{4 f \,a^{2}}\) | \(22\) |
derivativedivides | \(\frac {c^{2} \left (-\frac {1}{\tan \left (f x +e \right )-i}-\frac {i}{\left (\tan \left (f x +e \right )-i\right )^{2}}\right )}{f \,a^{2}}\) | \(39\) |
default | \(\frac {c^{2} \left (-\frac {1}{\tan \left (f x +e \right )-i}-\frac {i}{\left (\tan \left (f x +e \right )-i\right )^{2}}\right )}{f \,a^{2}}\) | \(39\) |
norman | \(\frac {\frac {c^{2} \tan \left (f x +e \right )}{a f}-\frac {c^{2} \left (\tan ^{3}\left (f x +e \right )\right )}{a f}-\frac {2 i c^{2} \left (\tan ^{2}\left (f x +e \right )\right )}{a f}}{a \left (1+\tan ^{2}\left (f x +e \right )\right )^{2}}\) | \(73\) |
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 = 1.03, size = 21, normalized size = 0.75 \begin {gather*} \frac {i \, c^{2} e^{\left (-4 i \, f x - 4 i \, e\right )}}{4 \, a^{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. 51 vs. \(2 (24) = 48\).
time = 0.13, size = 51, normalized size = 1.82 \begin {gather*} \begin {cases} \frac {i c^{2} e^{- 4 i e} e^{- 4 i f x}}{4 a^{2} f} & \text {for}\: a^{2} f e^{4 i e} \neq 0 \\\frac {c^{2} x e^{- 4 i e}}{a^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.59, size = 54, normalized size = 1.93 \begin {gather*} -\frac {2 \, {\left (c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{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 = 4.60, size = 28, normalized size = 1.00 \begin {gather*} -\frac {c^2\,\mathrm {tan}\left (e+f\,x\right )}{a^2\,f\,{\left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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