Optimal. Leaf size=38 \[ -\frac {(a+a \sec (e+f x))^2 \tan (e+f x)}{5 f (c-c \sec (e+f x))^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {4035}
\begin {gather*} -\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{5 f (c-c \sec (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 4035
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^3} \, dx &=-\frac {(a+a \sec (e+f x))^2 \tan (e+f x)}{5 f (c-c \sec (e+f x))^3}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 25, normalized size = 0.66 \begin {gather*} \frac {a^2 \cot ^5\left (\frac {1}{2} (e+f x)\right )}{5 c^3 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 23, normalized size = 0.61
method | result | size |
derivativedivides | \(\frac {a^{2}}{5 f \,c^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}\) | \(23\) |
default | \(\frac {a^{2}}{5 f \,c^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}\) | \(23\) |
risch | \(\frac {2 i a^{2} \left (5 \,{\mathrm e}^{4 i \left (f x +e \right )}+10 \,{\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{5 f \,c^{3} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{5}}\) | \(50\) |
norman | \(\frac {\frac {a^{2}}{5 c f}-\frac {2 a^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 c f}+\frac {a^{2} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 c f}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2} c^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 205 vs.
\(2 (40) = 80\).
time = 0.28, size = 205, normalized size = 5.39 \begin {gather*} \frac {\frac {a^{2} {\left (\frac {10 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 3\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{5}}{c^{3} \sin \left (f x + e\right )^{5}} - \frac {a^{2} {\left (\frac {10 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 3\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{5}}{c^{3} \sin \left (f x + e\right )^{5}} - \frac {6 \, a^{2} {\left (\frac {5 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - 1\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{5}}{c^{3} \sin \left (f x + e\right )^{5}}}{60 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 89 vs.
\(2 (40) = 80\).
time = 2.50, size = 89, normalized size = 2.34 \begin {gather*} \frac {a^{2} \cos \left (f x + e\right )^{3} + 3 \, a^{2} \cos \left (f x + e\right )^{2} + 3 \, a^{2} \cos \left (f x + e\right ) + a^{2}}{5 \, {\left (c^{3} f \cos \left (f x + e\right )^{2} - 2 \, c^{3} f \cos \left (f x + e\right ) + c^{3} 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 {a^{2} \left (\int \frac {\sec {\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} - 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} - 1}\, dx + \int \frac {2 \sec ^{2}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} - 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} - 1}\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{\sec ^{3}{\left (e + f x \right )} - 3 \sec ^{2}{\left (e + f x \right )} + 3 \sec {\left (e + f x \right )} - 1}\, dx\right )}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.54, size = 22, normalized size = 0.58 \begin {gather*} \frac {a^{2}}{5 \, c^{3} f \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.64, size = 22, normalized size = 0.58 \begin {gather*} \frac {a^2\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{5\,c^3\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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