Optimal. Leaf size=252 \[ -\frac{4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac{\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac{\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac{\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac{\cot (e+f x)}{a^3 c^6 f}-\frac{4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac{19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac{36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac{34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac{16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac{3 \csc (e+f x)}{a^3 c^6 f}+\frac{x}{a^3 c^6} \]
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Rubi [A] time = 0.299534, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {3904, 3886, 3473, 8, 2606, 194, 2607, 30, 270} \[ -\frac{4 \cot ^{11}(e+f x)}{11 a^3 c^6 f}+\frac{\cot ^9(e+f x)}{9 a^3 c^6 f}-\frac{\cot ^7(e+f x)}{7 a^3 c^6 f}+\frac{\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac{\cot (e+f x)}{a^3 c^6 f}-\frac{4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac{19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac{36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac{34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac{16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac{3 \csc (e+f x)}{a^3 c^6 f}+\frac{x}{a^3 c^6} \]
Antiderivative was successfully verified.
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Rule 3904
Rule 3886
Rule 3473
Rule 8
Rule 2606
Rule 194
Rule 2607
Rule 30
Rule 270
Rubi steps
\begin{align*} \int \frac{1}{(a+a \sec (e+f x))^3 (c-c \sec (e+f x))^6} \, dx &=\frac{\int \cot ^{12}(e+f x) (a+a \sec (e+f x))^3 \, dx}{a^6 c^6}\\ &=\frac{\int \left (a^3 \cot ^{12}(e+f x)+3 a^3 \cot ^{11}(e+f x) \csc (e+f x)+3 a^3 \cot ^{10}(e+f x) \csc ^2(e+f x)+a^3 \cot ^9(e+f x) \csc ^3(e+f x)\right ) \, dx}{a^6 c^6}\\ &=\frac{\int \cot ^{12}(e+f x) \, dx}{a^3 c^6}+\frac{\int \cot ^9(e+f x) \csc ^3(e+f x) \, dx}{a^3 c^6}+\frac{3 \int \cot ^{11}(e+f x) \csc (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{\cot ^{11}(e+f x)}{11 a^3 c^6 f}-\frac{\int \cot ^{10}(e+f x) \, dx}{a^3 c^6}-\frac{\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 \operatorname{Subst}\left (\int x^{10} \, dx,x,-\cot (e+f x)\right )}{a^3 c^6 f}-\frac{3 \operatorname{Subst}\left (\int \left (-1+x^2\right )^5 \, 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{\int \cot ^8(e+f x) \, dx}{a^3 c^6}-\frac{\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{3 \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{\cot ^7(e+f x)}{7 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{3 \csc (e+f x)}{a^3 c^6 f}-\frac{16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac{34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac{36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac{19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac{4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}-\frac{\int \cot ^6(e+f x) \, dx}{a^3 c^6}\\ &=\frac{\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac{\cot ^7(e+f x)}{7 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{3 \csc (e+f x)}{a^3 c^6 f}-\frac{16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac{34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac{36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac{19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac{4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac{\int \cot ^4(e+f x) \, dx}{a^3 c^6}\\ &=-\frac{\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac{\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac{\cot ^7(e+f x)}{7 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{3 \csc (e+f x)}{a^3 c^6 f}-\frac{16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac{34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac{36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac{19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac{4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}-\frac{\int \cot ^2(e+f x) \, dx}{a^3 c^6}\\ &=\frac{\cot (e+f x)}{a^3 c^6 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac{\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac{\cot ^7(e+f x)}{7 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{3 \csc (e+f x)}{a^3 c^6 f}-\frac{16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac{34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac{36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac{19 \csc ^9(e+f x)}{9 a^3 c^6 f}-\frac{4 \csc ^{11}(e+f x)}{11 a^3 c^6 f}+\frac{\int 1 \, dx}{a^3 c^6}\\ &=\frac{x}{a^3 c^6}+\frac{\cot (e+f x)}{a^3 c^6 f}-\frac{\cot ^3(e+f x)}{3 a^3 c^6 f}+\frac{\cot ^5(e+f x)}{5 a^3 c^6 f}-\frac{\cot ^7(e+f x)}{7 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{3 \csc (e+f x)}{a^3 c^6 f}-\frac{16 \csc ^3(e+f x)}{3 a^3 c^6 f}+\frac{34 \csc ^5(e+f x)}{5 a^3 c^6 f}-\frac{36 \csc ^7(e+f x)}{7 a^3 c^6 f}+\frac{19 \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*}
Mathematica [A] time = 2.21587, size = 499, normalized size = 1.98 \[ \frac{\csc \left (\frac{e}{2}\right ) \sec \left (\frac{e}{2}\right ) \tan (e+f x) \sec ^8(e+f x) (-86058610 \sin (e+f x)+51635166 \sin (2 (e+f x))+26599934 \sin (3 (e+f x))-39117550 \sin (4 (e+f x))+7823510 \sin (5 (e+f x))+7823510 \sin (6 (e+f x))-4694106 \sin (7 (e+f x))+782351 \sin (8 (e+f x))-55651200 \sin (2 e+f x)+47971968 \sin (e+2 f x)+14990976 \sin (3 e+2 f x)+8100992 \sin (2 e+3 f x)+24334464 \sin (4 e+3 f x)-28627840 \sin (3 e+4 f x)-19071360 \sin (5 e+4 f x)+9687680 \sin (4 e+5 f x)-147840 \sin (6 e+5 f x)+5548160 \sin (5 e+6 f x)+3991680 \sin (7 e+6 f x)-4393344 \sin (6 e+7 f x)-1330560 \sin (8 e+7 f x)+953984 \sin (7 e+8 f x)-24393600 f x \cos (2 e+f x)-14636160 f x \cos (e+2 f x)+14636160 f x \cos (3 e+2 f x)-7539840 f x \cos (2 e+3 f x)+7539840 f x \cos (4 e+3 f x)+11088000 f x \cos (3 e+4 f x)-11088000 f x \cos (5 e+4 f x)-2217600 f x \cos (4 e+5 f x)+2217600 f x \cos (6 e+5 f x)-2217600 f x \cos (5 e+6 f x)+2217600 f x \cos (7 e+6 f x)+1330560 f x \cos (6 e+7 f x)-1330560 f x \cos (8 e+7 f x)-221760 f x \cos (7 e+8 f x)+221760 f x \cos (9 e+8 f x)+17677440 \sin (e)-49287040 \sin (f x)+24393600 f x \cos (f x))}{113541120 a^3 c^6 f (\sec (e+f x)-1)^6 (\sec (e+f x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 219, normalized size = 0.9 \begin{align*} -{\frac{1}{1280\,f{a}^{3}{c}^{6}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{5}}+{\frac{5}{384\,f{a}^{3}{c}^{6}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}-{\frac{23}{128\,f{a}^{3}{c}^{6}}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) }+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{f{a}^{3}{c}^{6}}}-{\frac{1}{2816\,f{a}^{3}{c}^{6}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-11}}+{\frac{5}{1152\,f{a}^{3}{c}^{6}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-9}}-{\frac{23}{896\,f{a}^{3}{c}^{6}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-7}}+{\frac{13}{128\,f{a}^{3}{c}^{6}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-5}}-{\frac{1}{3\,f{a}^{3}{c}^{6}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-3}}+{\frac{191}{128\,f{a}^{3}{c}^{6}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.60557, size = 306, normalized size = 1.21 \begin{align*} -\frac{\frac{231 \,{\left (\frac{690 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{50 \, \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{1774080 \, \arctan \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3} c^{6}} - \frac{5 \,{\left (\frac{770 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{4554 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac{18018 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac{59136 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} + \frac{264726 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - 63\right )}{\left (\cos \left (f x + e\right ) + 1\right )}^{11}}{a^{3} c^{6} \sin \left (f x + e\right )^{11}}}{887040 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.16003, size = 799, normalized size = 3.17 \begin{align*} \frac{7453 \, \cos \left (f x + e\right )^{8} - 11964 \, \cos \left (f x + e\right )^{7} - 11866 \, \cos \left (f x + e\right )^{6} + 30542 \, \cos \left (f x + e\right )^{5} + 90 \, \cos \left (f x + e\right )^{4} - 26438 \, \cos \left (f x + e\right )^{3} + 8539 \, \cos \left (f x + e\right )^{2} + 3465 \,{\left (f x \cos \left (f x + e\right )^{7} - 3 \, f x \cos \left (f x + e\right )^{6} + f x \cos \left (f x + e\right )^{5} + 5 \, f x \cos \left (f x + e\right )^{4} - 5 \, f x \cos \left (f x + e\right )^{3} - f x \cos \left (f x + e\right )^{2} + 3 \, f x \cos \left (f x + e\right ) - f x\right )} \sin \left (f x + e\right ) + 7671 \, \cos \left (f x + e\right ) - 3712}{3465 \,{\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30013, size = 242, normalized size = 0.96 \begin{align*} \frac{\frac{887040 \,{\left (f x + e\right )}}{a^{3} c^{6}} + \frac{5 \,{\left (264726 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{10} - 59136 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} + 18018 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} - 4554 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + 770 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 63\right )}}{a^{3} c^{6} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{11}} - \frac{231 \,{\left (3 \, a^{12} c^{24} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{5} - 50 \, a^{12} c^{24} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + 690 \, a^{12} c^{24} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{a^{15} c^{30}}}{887040 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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