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 = 0.10, 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 = {3958, 2606} \[ \frac {\csc (e+f x)}{a^2 c^2 f}-\frac {\csc ^3(e+f x)}{3 a^2 c^2 f} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 3958
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 {\operatorname {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 \[ \frac {\frac {\csc (e+f x)}{f}-\frac {\csc ^3(e+f x)}{3 f}}{a^2 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 50, normalized size = 1.32 \[ \frac {3 \, \cos \left (f x + e\right )^{2} - 2}{3 \, {\left (a^{2} c^{2} f \cos \left (f x + e\right )^{2} - a^{2} c^{2} f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 33, normalized size = 0.87 \[ \frac {3 \, \sin \left (f x + e\right )^{2} - 1}{3 \, a^{2} c^{2} f \sin \left (f x + e\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (f x +e \right )}{\left (a +a \sec \left (f x +e \right )\right )^{2} \left (c -c \sec \left (f x +e \right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 31, normalized size = 0.82 \[ \frac {3 \, \sin \left (f x + e\right )^{2} - 1}{3 \, a^{2} c^{2} f \sin \left (f x + e\right )^{3}} \]
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 \[ \frac {{\sin \left (e+f\,x\right )}^2-\frac {1}{3}}{a^2\,c^2\,f\,{\sin \left (e+f\,x\right )}^3} \]
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 \[ \frac {\int \frac {\sec {\left (e + f x \right )}}{\sec ^{4}{\left (e + f x \right )} - 2 \sec ^{2}{\left (e + f x \right )} + 1}\, dx}{a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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