Optimal. Leaf size=121 \[ -\frac {(a+a \sec (e+f x))^2 \tan (e+f x)}{9 f (c-c \sec (e+f x))^5}-\frac {2 (a+a \sec (e+f x))^2 \tan (e+f x)}{63 c f (c-c \sec (e+f x))^4}-\frac {2 (a+a \sec (e+f x))^2 \tan (e+f x)}{315 c^2 f (c-c \sec (e+f x))^3} \]
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Rubi [A]
time = 0.17, antiderivative size = 121, 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 = {4036, 4035}
\begin {gather*} -\frac {2 \tan (e+f x) (a \sec (e+f x)+a)^2}{315 c^2 f (c-c \sec (e+f x))^3}-\frac {2 \tan (e+f x) (a \sec (e+f x)+a)^2}{63 c f (c-c \sec (e+f x))^4}-\frac {\tan (e+f x) (a \sec (e+f x)+a)^2}{9 f (c-c \sec (e+f x))^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 4035
Rule 4036
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^5} \, dx &=-\frac {(a+a \sec (e+f x))^2 \tan (e+f x)}{9 f (c-c \sec (e+f x))^5}+\frac {2 \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^4} \, dx}{9 c}\\ &=-\frac {(a+a \sec (e+f x))^2 \tan (e+f x)}{9 f (c-c \sec (e+f x))^5}-\frac {2 (a+a \sec (e+f x))^2 \tan (e+f x)}{63 c f (c-c \sec (e+f x))^4}+\frac {2 \int \frac {\sec (e+f x) (a+a \sec (e+f x))^2}{(c-c \sec (e+f x))^3} \, dx}{63 c^2}\\ &=-\frac {(a+a \sec (e+f x))^2 \tan (e+f x)}{9 f (c-c \sec (e+f x))^5}-\frac {2 (a+a \sec (e+f x))^2 \tan (e+f x)}{63 c f (c-c \sec (e+f x))^4}-\frac {2 (a+a \sec (e+f x))^2 \tan (e+f x)}{315 c^2 f (c-c \sec (e+f x))^3}\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 141, normalized size = 1.17 \begin {gather*} -\frac {a^2 \csc \left (\frac {e}{2}\right ) \csc ^9\left (\frac {1}{2} (e+f x)\right ) \left (3402 \sin \left (\frac {f x}{2}\right )+2520 \sin \left (e+\frac {f x}{2}\right )-1638 \sin \left (e+\frac {3 f x}{2}\right )-2310 \sin \left (2 e+\frac {3 f x}{2}\right )+1062 \sin \left (2 e+\frac {5 f x}{2}\right )+630 \sin \left (3 e+\frac {5 f x}{2}\right )-108 \sin \left (3 e+\frac {7 f x}{2}\right )-315 \sin \left (4 e+\frac {7 f x}{2}\right )+47 \sin \left (4 e+\frac {9 f x}{2}\right )\right )}{80640 c^5 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 52, normalized size = 0.43
method | result | size |
derivativedivides | \(\frac {a^{2} \left (\frac {1}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}+\frac {1}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {2}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}\right )}{4 f \,c^{5}}\) | \(52\) |
default | \(\frac {a^{2} \left (\frac {1}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}+\frac {1}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {2}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}\right )}{4 f \,c^{5}}\) | \(52\) |
risch | \(\frac {2 i a^{2} \left (315 \,{\mathrm e}^{8 i \left (f x +e \right )}-630 \,{\mathrm e}^{7 i \left (f x +e \right )}+2310 \,{\mathrm e}^{6 i \left (f x +e \right )}-2520 \,{\mathrm e}^{5 i \left (f x +e \right )}+3402 \,{\mathrm e}^{4 i \left (f x +e \right )}-1638 \,{\mathrm e}^{3 i \left (f x +e \right )}+1062 \,{\mathrm e}^{2 i \left (f x +e \right )}-108 \,{\mathrm e}^{i \left (f x +e \right )}+47\right )}{315 f \,c^{5} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{9}}\) | \(116\) |
norman | \(\frac {\frac {a^{2}}{36 c f}-\frac {8 a^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{63 c f}+\frac {139 a^{2} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{630 c f}-\frac {6 a^{2} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35 c f}+\frac {a^{2} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{20 c f}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2} c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}\) | \(131\) |
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. 293 vs.
\(2 (127) = 254\).
time = 0.31, size = 293, normalized size = 2.42 \begin {gather*} -\frac {\frac {a^{2} {\left (\frac {180 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {378 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {420 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {315 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 35\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}} + \frac {10 \, a^{2} {\left (\frac {18 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {42 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {63 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 7\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}} + \frac {7 \, a^{2} {\left (\frac {18 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {45 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 5\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{c^{5} \sin \left (f x + e\right )^{9}}}{5040 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.23, size = 150, normalized size = 1.24 \begin {gather*} \frac {47 \, a^{2} \cos \left (f x + e\right )^{5} + 127 \, a^{2} \cos \left (f x + e\right )^{4} + 101 \, a^{2} \cos \left (f x + e\right )^{3} + 11 \, a^{2} \cos \left (f x + e\right )^{2} - 8 \, a^{2} \cos \left (f x + e\right ) + 2 \, a^{2}}{315 \, {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} + 6 \, c^{5} f \cos \left (f x + e\right )^{2} - 4 \, c^{5} f \cos \left (f x + e\right ) + c^{5} 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 ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec {\left (e + f x \right )} - 1}\, dx + \int \frac {2 \sec ^{2}{\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec {\left (e + f x \right )} - 1}\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{\sec ^{5}{\left (e + f x \right )} - 5 \sec ^{4}{\left (e + f x \right )} + 10 \sec ^{3}{\left (e + f x \right )} - 10 \sec ^{2}{\left (e + f x \right )} + 5 \sec {\left (e + f x \right )} - 1}\, dx\right )}{c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.60, size = 57, normalized size = 0.47 \begin {gather*} \frac {63 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 90 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 35 \, a^{2}}{1260 \, c^{5} f \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.63, size = 67, normalized size = 0.55 \begin {gather*} \frac {a^2\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{20\,c^5\,f}-\frac {a^2\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{14\,c^5\,f}+\frac {a^2\,{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{36\,c^5\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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