Optimal. Leaf size=16 \[ \frac {\tan (e+f x)}{a c f} \]
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Rubi [A]
time = 0.05, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2815, 3852, 8}
\begin {gather*} \frac {\tan (e+f x)}{a c f} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2815
Rule 3852
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))} \, dx &=\frac {\int \sec ^2(e+f x) \, dx}{a c}\\ &=-\frac {\text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{a c f}\\ &=\frac {\tan (e+f x)}{a c f}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {\tan (e+f x)}{a c f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 17, normalized size = 1.06
method | result | size |
default | \(\frac {\tan \left (f x +e \right )}{a c f}\) | \(17\) |
risch | \(\frac {2 i}{\left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) a c f}\) | \(41\) |
norman | \(-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a c f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}\) | \(47\) |
derivativedivides | error in RationalFunction: argument is not a rational function\ | N/A |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 17, normalized size = 1.06 \begin {gather*} \frac {\tan \left (f x + e\right )}{a c f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 26, normalized size = 1.62 \begin {gather*} \frac {\sin \left (f x + e\right )}{a c f \cos \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs.
\(2 (10) = 20\).
time = 0.60, size = 49, normalized size = 3.06 \begin {gather*} \begin {cases} - \frac {2 \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a c f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - a c f} & \text {for}\: f \neq 0 \\\frac {x}{\left (a \sin {\left (e \right )} + a\right ) \left (- c \sin {\left (e \right )} + c\right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 17, normalized size = 1.06 \begin {gather*} \frac {\tan \left (f x + e\right )}{a c f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.85, size = 35, normalized size = 2.19 \begin {gather*} -\frac {2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{a\,c\,f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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