Optimal. Leaf size=162 \[ -\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}-\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {17 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \csc ^7(e+f x)}{a^3 c^6 f}+\frac {22 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {\csc (e+f x)}{a^3 c^6 f} \]
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Rubi [A] time = 0.26, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {3958, 2606, 194, 2607, 30, 270, 14} \[ -\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}-\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {17 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \csc ^7(e+f x)}{a^3 c^6 f}+\frac {22 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {\csc (e+f x)}{a^3 c^6 f} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 194
Rule 270
Rule 2606
Rule 2607
Rule 3958
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx &=\frac {\int \left (a^3 \cot ^{11}(e+f x) \csc (e+f x)+3 a^3 \cot ^{10}(e+f x) \csc ^2(e+f x)+3 a^3 \cot ^9(e+f x) \csc ^3(e+f x)+a^3 \cot ^8(e+f x) \csc ^4(e+f x)\right ) \, dx}{a^6 c^6}\\ &=\frac {\int \cot ^{11}(e+f x) \csc (e+f x) \, dx}{a^3 c^6}+\frac {\int \cot ^8(e+f x) \csc ^4(e+f x) \, dx}{a^3 c^6}+\frac {3 \int \cot ^{10}(e+f x) \csc ^2(e+f x) \, dx}{a^3 c^6}+\frac {3 \int \cot ^9(e+f x) \csc ^3(e+f x) \, dx}{a^3 c^6}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-1+x^2\right )^5 \, dx,x,\csc (e+f x)\right )}{a^3 c^6 f}+\frac {\operatorname {Subst}\left (\int x^8 \left (1+x^2\right ) \, dx,x,-\cot (e+f x)\right )}{a^3 c^6 f}+\frac {3 \operatorname {Subst}\left (\int x^{10} \, dx,x,-\cot (e+f x)\right )}{a^3 c^6 f}-\frac {3 \operatorname {Subst}\left (\int x^2 \left (-1+x^2\right )^4 \, dx,x,\csc (e+f x)\right )}{a^3 c^6 f}\\ &=-\frac {3 \cot ^{11}(e+f x)}{11 a^3 c^6 f}-\frac {\operatorname {Subst}\left (\int \left (-1+5 x^2-10 x^4+10 x^6-5 x^8+x^{10}\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c^6 f}+\frac {\operatorname {Subst}\left (\int \left (x^8+x^{10}\right ) \, dx,x,-\cot (e+f x)\right )}{a^3 c^6 f}-\frac {3 \operatorname {Subst}\left (\int \left (x^2-4 x^4+6 x^6-4 x^8+x^{10}\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c^6 f}\\ &=-\frac {\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac {\csc (e+f x)}{a^3 c^6 f}-\frac {8 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac {22 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac {4 \csc ^7(e+f x)}{a^3 c^6 f}+\frac {17 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac {4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}\\ \end {align*}
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Mathematica [A] time = 2.32, size = 289, normalized size = 1.78 \[ \frac {\csc (e) (-3440690 \sin (e+f x)+2064414 \sin (2 (e+f x))+1063486 \sin (3 (e+f x))-1563950 \sin (4 (e+f x))+312790 \sin (5 (e+f x))+312790 \sin (6 (e+f x))-187674 \sin (7 (e+f x))+31279 \sin (8 (e+f x))-1499520 \sin (2 e+f x)+1051776 \sin (e+2 f x)+4224 \sin (3 e+2 f x)-85376 \sin (2 e+3 f x)+629376 \sin (4 e+3 f x)-483200 \sin (3 e+4 f x)-316800 \sin (5 e+4 f x)+392320 \sin (4 e+5 f x)-232320 \sin (6 e+5 f x)-30080 \sin (5 e+6 f x)+190080 \sin (7 e+6 f x)-32640 \sin (6 e+7 f x)-63360 \sin (8 e+7 f x)+16000 \sin (7 e+8 f x)+1119360 \sin (e)-260480 \sin (f x)) \tan (e+f x) \sec ^8(e+f x)}{8110080 a^3 c^6 f (\sec (e+f x)-1)^6 (\sec (e+f x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 217, normalized size = 1.34 \[ \frac {125 \, \cos \left (f x + e\right )^{8} + 120 \, \cos \left (f x + e\right )^{7} - 680 \, \cos \left (f x + e\right )^{6} + 400 \, \cos \left (f x + e\right )^{5} + 720 \, \cos \left (f x + e\right )^{4} - 832 \, \cos \left (f x + e\right )^{3} - 64 \, \cos \left (f x + e\right )^{2} + 384 \, \cos \left (f x + e\right ) - 128}{495 \, {\left (a^{3} c^{6} f \cos \left (f x + e\right )^{7} - 3 \, a^{3} c^{6} f \cos \left (f x + e\right )^{6} + a^{3} c^{6} f \cos \left (f x + e\right )^{5} + 5 \, a^{3} c^{6} f \cos \left (f x + e\right )^{4} - 5 \, a^{3} c^{6} f \cos \left (f x + e\right )^{3} - a^{3} c^{6} f \cos \left (f x + e\right )^{2} + 3 \, a^{3} c^{6} f \cos \left (f x + e\right ) - a^{3} c^{6} f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 164, normalized size = 1.01 \[ \frac {\frac {27720 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 11550 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 5544 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 1980 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 440 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 45}{a^{3} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11}} + \frac {33 \, {\left (3 \, a^{12} c^{24} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 40 \, a^{12} c^{24} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 420 \, a^{12} c^{24} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15} c^{30}}}{126720 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.00, size = 128, normalized size = 0.79 \[ \frac {\frac {\left (\tan ^{5}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{5}-\frac {8 \left (\tan ^{3}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3}+28 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )-\frac {1}{11 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{11}}-\frac {4}{\tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{7}}+\frac {8}{9 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{9}}-\frac {70}{3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{3}}+\frac {56}{5 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )^{5}}+\frac {56}{\tan \left (\frac {e}{2}+\frac {f x}{2}\right )}}{256 f \,a^{3} c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 200, normalized size = 1.23 \[ \frac {\frac {33 \, {\left (\frac {420 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {40 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}\right )}}{a^{3} c^{6}} + \frac {{\left (\frac {440 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {1980 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {5544 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {11550 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac {27720 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - 45\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{a^{3} c^{6} \sin \left (f x + e\right )^{11}}}{126720 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.11, size = 120, normalized size = 0.74 \[ -\frac {\frac {605\,\cos \left (e+f\,x\right )}{8}+\frac {1023\,\cos \left (2\,e+2\,f\,x\right )}{16}-\frac {349\,\cos \left (3\,e+3\,f\,x\right )}{8}-\frac {325\,\cos \left (4\,e+4\,f\,x\right )}{32}+\frac {305\,\cos \left (5\,e+5\,f\,x\right )}{8}-\frac {215\,\cos \left (6\,e+6\,f\,x\right )}{16}+\frac {15\,\cos \left (7\,e+7\,f\,x\right )}{8}+\frac {125\,\cos \left (8\,e+8\,f\,x\right )}{128}-\frac {8745}{128}}{126720\,a^3\,c^6\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}} \]
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 \[ \frac {\int \frac {\sec {\left (e + f x \right )}}{\sec ^{9}{\left (e + f x \right )} - 3 \sec ^{8}{\left (e + f x \right )} + 8 \sec ^{6}{\left (e + f x \right )} - 6 \sec ^{5}{\left (e + f x \right )} - 6 \sec ^{4}{\left (e + f x \right )} + 8 \sec ^{3}{\left (e + f x \right )} - 3 \sec {\left (e + f x \right )} + 1}\, dx}{a^{3} c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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