Optimal. Leaf size=38 \[ \frac {\csc (e+f x)}{a^2 c^2 f}-\frac {\csc ^3(e+f x)}{3 a^2 c^2 f} \]
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Rubi [A]
time = 0.07, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4043, 2686}
\begin {gather*} \frac {\csc (e+f x)}{a^2 c^2 f}-\frac {\csc ^3(e+f x)}{3 a^2 c^2 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2686
Rule 4043
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^2} \, dx &=\frac {\int \cot ^3(e+f x) \csc (e+f x) \, dx}{a^2 c^2}\\ &=-\frac {\text {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{a^2 c^2 f}\\ &=\frac {\csc (e+f x)}{a^2 c^2 f}-\frac {\csc ^3(e+f x)}{3 a^2 c^2 f}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 33, normalized size = 0.87 \begin {gather*} \frac {\frac {\csc (e+f x)}{f}-\frac {\csc ^3(e+f x)}{3 f}}{a^2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 66, normalized size = 1.74
method | result | size |
default | \(\frac {-\frac {\cos ^{4}\left (f x +e \right )}{3 \sin \left (f x +e \right )^{3}}+\frac {\cos ^{4}\left (f x +e \right )}{3 \sin \left (f x +e \right )}+\frac {\left (2+\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )}{3}}{a^{2} c^{2} f}\) | \(66\) |
risch | \(\frac {2 i \left (3 \,{\mathrm e}^{5 i \left (f x +e \right )}-2 \,{\mathrm e}^{3 i \left (f x +e \right )}+3 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{3 f \,a^{2} c^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{3}}\) | \(73\) |
norman | \(\frac {-\frac {1}{24 a c f}+\frac {3 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 a c f}+\frac {3 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 a c f}-\frac {\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )}{24 a c f}}{a c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}\) | \(97\) |
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 = 0.28, size = 33, normalized size = 0.87 \begin {gather*} \frac {3 \, \sin \left (f x + e\right )^{2} - 1}{3 \, a^{2} c^{2} f \sin \left (f x + e\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.39, size = 53, normalized size = 1.39 \begin {gather*} \frac {3 \, \cos \left (f x + e\right )^{2} - 2}{3 \, {\left (a^{2} c^{2} f \cos \left (f x + e\right )^{2} - a^{2} c^{2} f\right )} \sin \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\sec {\left (e + f x \right )}}{\sec ^{4}{\left (e + f x \right )} - 2 \sec ^{2}{\left (e + f x \right )} + 1}\, dx}{a^{2} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.54, size = 31, normalized size = 0.82 \begin {gather*} \frac {3 \, \sin \left (f x + e\right )^{2} - 1}{3 \, a^{2} c^{2} f \sin \left (f x + e\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.57, size = 28, normalized size = 0.74 \begin {gather*} \frac {{\sin \left (e+f\,x\right )}^2-\frac {1}{3}}{a^2\,c^2\,f\,{\sin \left (e+f\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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