Optimal. Leaf size=131 \[ \frac {\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {\sec ^5(e+f x)}{9 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {2 \tan (e+f x)}{3 a^3 c^5 f}+\frac {4 \tan ^3(e+f x)}{9 a^3 c^5 f}+\frac {2 \tan ^5(e+f x)}{15 a^3 c^5 f} \]
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Rubi [A]
time = 0.13, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2815, 2751,
3852} \begin {gather*} \frac {2 \tan ^5(e+f x)}{15 a^3 c^5 f}+\frac {4 \tan ^3(e+f x)}{9 a^3 c^5 f}+\frac {2 \tan (e+f x)}{3 a^3 c^5 f}+\frac {\sec ^5(e+f x)}{9 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2751
Rule 2815
Rule 3852
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \, dx &=\frac {\int \frac {\sec ^6(e+f x)}{(c-c \sin (e+f x))^2} \, dx}{a^3 c^3}\\ &=\frac {\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {7 \int \frac {\sec ^6(e+f x)}{c-c \sin (e+f x)} \, dx}{9 a^3 c^4}\\ &=\frac {\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {\sec ^5(e+f x)}{9 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {2 \int \sec ^6(e+f x) \, dx}{3 a^3 c^5}\\ &=\frac {\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {\sec ^5(e+f x)}{9 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}-\frac {2 \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{3 a^3 c^5 f}\\ &=\frac {\sec ^5(e+f x)}{9 a^3 c^3 f (c-c \sin (e+f x))^2}+\frac {\sec ^5(e+f x)}{9 a^3 f \left (c^5-c^5 \sin (e+f x)\right )}+\frac {2 \tan (e+f x)}{3 a^3 c^5 f}+\frac {4 \tan ^3(e+f x)}{9 a^3 c^5 f}+\frac {2 \tan ^5(e+f x)}{15 a^3 c^5 f}\\ \end {align*}
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Mathematica [A]
time = 0.89, size = 213, normalized size = 1.63 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-7875 \cos (e+f x)+20480 \cos (2 (e+f x))-3325 \cos (3 (e+f x))+16384 \cos (4 (e+f x))-175 \cos (5 (e+f x))+4096 \cos (6 (e+f x))+175 \cos (7 (e+f x))+46080 \sin (e+f x)+3500 \sin (2 (e+f x))+19456 \sin (3 (e+f x))+2800 \sin (4 (e+f x))+1024 \sin (5 (e+f x))+700 \sin (6 (e+f x))-1024 \sin (7 (e+f x)))}{184320 f (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 223, normalized size = 1.70
method | result | size |
risch | \(-\frac {32 i \left (-20 i {\mathrm e}^{5 i \left (f x +e \right )}+45 \,{\mathrm e}^{6 i \left (f x +e \right )}-16 i {\mathrm e}^{3 i \left (f x +e \right )}+19 \,{\mathrm e}^{4 i \left (f x +e \right )}-4 i {\mathrm e}^{i \left (f x +e \right )}+{\mathrm e}^{2 i \left (f x +e \right )}-1\right )}{45 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{9} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5} f \,a^{3} c^{5}}\) | \(110\) |
derivativedivides | \(\frac {-\frac {4}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {5}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {49}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {49}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {35}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {49}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {51}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {99}{64 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{20 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {1}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {13}{48 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {9}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {29}{64 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{f \,c^{5} a^{3}}\) | \(223\) |
default | \(\frac {-\frac {4}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {2}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {5}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {49}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {49}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {35}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {49}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {51}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {99}{64 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{20 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {1}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {13}{48 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {9}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {29}{64 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{f \,c^{5} a^{3}}\) | \(223\) |
norman | \(\frac {\frac {4 \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {344 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a c f}-\frac {4}{9 a c f}-\frac {2 \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}+\frac {32 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{9 a c f}-\frac {4 \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {2 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{9 a c f}-\frac {76 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{9 a c f}-\frac {32 \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}+\frac {46 \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a c f}-\frac {20 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{9 a c f}+\frac {538 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{45 a c f}-\frac {64 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a c f}+\frac {236 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a c f}}{a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5} c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\) | \(330\) |
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. 662 vs.
\(2 (130) = 260\).
time = 0.33, size = 662, normalized size = 5.05 \begin {gather*} \frac {2 \, {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {80 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {190 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {50 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {269 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {96 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {516 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {354 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {69 \, \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac {240 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + \frac {30 \, \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} - \frac {90 \, \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}} + \frac {45 \, \sin \left (f x + e\right )^{13}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{13}} + 10\right )}}{45 \, {\left (a^{3} c^{5} - \frac {4 \, a^{3} c^{5} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {a^{3} c^{5} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {16 \, a^{3} c^{5} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {19 \, a^{3} c^{5} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {20 \, a^{3} c^{5} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {45 \, a^{3} c^{5} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {45 \, a^{3} c^{5} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {20 \, a^{3} c^{5} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac {19 \, a^{3} c^{5} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - \frac {16 \, a^{3} c^{5} \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} - \frac {a^{3} c^{5} \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}} + \frac {4 \, a^{3} c^{5} \sin \left (f x + e\right )^{13}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{13}} - \frac {a^{3} c^{5} \sin \left (f x + e\right )^{14}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{14}}\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 144, normalized size = 1.10 \begin {gather*} -\frac {32 \, \cos \left (f x + e\right )^{6} - 16 \, \cos \left (f x + e\right )^{4} - 4 \, \cos \left (f x + e\right )^{2} - {\left (16 \, \cos \left (f x + e\right )^{6} - 24 \, \cos \left (f x + e\right )^{4} - 10 \, \cos \left (f x + e\right )^{2} - 7\right )} \sin \left (f x + e\right ) - 2}{45 \, {\left (a^{3} c^{5} f \cos \left (f x + e\right )^{7} + 2 \, a^{3} c^{5} f \cos \left (f x + e\right )^{5} \sin \left (f x + e\right ) - 2 \, a^{3} c^{5} f \cos \left (f x + e\right )^{5}\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. 4335 vs.
\(2 (117) = 234\).
time = 50.44, size = 4335, normalized size = 33.09 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 217, normalized size = 1.66 \begin {gather*} -\frac {\frac {3 \, {\left (435 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 1470 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2060 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1330 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 353\right )}}{a^{3} c^{5} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}} + \frac {4455 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 26460 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 78120 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 137340 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 157374 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 118356 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 57744 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 16596 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2339}{a^{3} c^{5} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}}}{2880 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.20, size = 190, normalized size = 1.45 \begin {gather*} -\frac {\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {65\,\cos \left (\frac {5\,e}{2}+\frac {5\,f\,x}{2}\right )}{32}-\frac {225\,\cos \left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}\right )}{32}-5\,\cos \left (\frac {7\,e}{2}+\frac {7\,f\,x}{2}\right )+\cos \left (\frac {9\,e}{2}+\frac {9\,f\,x}{2}\right )-\frac {37\,\cos \left (\frac {11\,e}{2}+\frac {11\,f\,x}{2}\right )}{32}+\frac {5\,\cos \left (\frac {13\,e}{2}+\frac {13\,f\,x}{2}\right )}{32}-\frac {89\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}{4}+11\,\sin \left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}\right )-\frac {63\,\sin \left (\frac {5\,e}{2}+\frac {5\,f\,x}{2}\right )}{8}+\frac {25\,\sin \left (\frac {7\,e}{2}+\frac {7\,f\,x}{2}\right )}{8}-\frac {5\,\sin \left (\frac {9\,e}{2}+\frac {9\,f\,x}{2}\right )}{8}+\frac {3\,\sin \left (\frac {11\,e}{2}+\frac {11\,f\,x}{2}\right )}{8}+\frac {\sin \left (\frac {13\,e}{2}+\frac {13\,f\,x}{2}\right )}{4}\right )}{2880\,a^3\,c^5\,f\,{\cos \left (\frac {e}{2}-\frac {\pi }{4}+\frac {f\,x}{2}\right )}^5\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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