Optimal. Leaf size=210 \[ \frac{4 \cot ^9(e+f x)}{9 a^2 c^5 f}-\frac{\cot ^7(e+f x)}{7 a^2 c^5 f}+\frac{\cot ^5(e+f x)}{5 a^2 c^5 f}-\frac{\cot ^3(e+f x)}{3 a^2 c^5 f}+\frac{\cot (e+f x)}{a^2 c^5 f}+\frac{4 \csc ^9(e+f x)}{9 a^2 c^5 f}-\frac{15 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac{21 \csc ^5(e+f x)}{5 a^2 c^5 f}-\frac{13 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac{3 \csc (e+f x)}{a^2 c^5 f}+\frac{x}{a^2 c^5} \]
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Rubi [A] time = 0.286163, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 17, 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} \[ \frac{4 \cot ^9(e+f x)}{9 a^2 c^5 f}-\frac{\cot ^7(e+f x)}{7 a^2 c^5 f}+\frac{\cot ^5(e+f x)}{5 a^2 c^5 f}-\frac{\cot ^3(e+f x)}{3 a^2 c^5 f}+\frac{\cot (e+f x)}{a^2 c^5 f}+\frac{4 \csc ^9(e+f x)}{9 a^2 c^5 f}-\frac{15 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac{21 \csc ^5(e+f x)}{5 a^2 c^5 f}-\frac{13 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac{3 \csc (e+f x)}{a^2 c^5 f}+\frac{x}{a^2 c^5} \]
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))^2 (c-c \sec (e+f x))^5} \, dx &=-\frac{\int \cot ^{10}(e+f x) (a+a \sec (e+f x))^3 \, dx}{a^5 c^5}\\ &=-\frac{\int \left (a^3 \cot ^{10}(e+f x)+3 a^3 \cot ^9(e+f x) \csc (e+f x)+3 a^3 \cot ^8(e+f x) \csc ^2(e+f x)+a^3 \cot ^7(e+f x) \csc ^3(e+f x)\right ) \, dx}{a^5 c^5}\\ &=-\frac{\int \cot ^{10}(e+f x) \, dx}{a^2 c^5}-\frac{\int \cot ^7(e+f x) \csc ^3(e+f x) \, dx}{a^2 c^5}-\frac{3 \int \cot ^9(e+f x) \csc (e+f x) \, dx}{a^2 c^5}-\frac{3 \int \cot ^8(e+f x) \csc ^2(e+f x) \, dx}{a^2 c^5}\\ &=\frac{\cot ^9(e+f x)}{9 a^2 c^5 f}+\frac{\int \cot ^8(e+f x) \, dx}{a^2 c^5}+\frac{\operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^3 \, dx,x,\csc (e+f x)\right )}{a^2 c^5 f}-\frac{3 \operatorname{Subst}\left (\int x^8 \, dx,x,-\cot (e+f x)\right )}{a^2 c^5 f}+\frac{3 \operatorname{Subst}\left (\int \left (-1+x^2\right )^4 \, dx,x,\csc (e+f x)\right )}{a^2 c^5 f}\\ &=-\frac{\cot ^7(e+f x)}{7 a^2 c^5 f}+\frac{4 \cot ^9(e+f x)}{9 a^2 c^5 f}-\frac{\int \cot ^6(e+f x) \, dx}{a^2 c^5}+\frac{\operatorname{Subst}\left (\int \left (-x^2+3 x^4-3 x^6+x^8\right ) \, dx,x,\csc (e+f x)\right )}{a^2 c^5 f}+\frac{3 \operatorname{Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,\csc (e+f x)\right )}{a^2 c^5 f}\\ &=\frac{\cot ^5(e+f x)}{5 a^2 c^5 f}-\frac{\cot ^7(e+f x)}{7 a^2 c^5 f}+\frac{4 \cot ^9(e+f x)}{9 a^2 c^5 f}+\frac{3 \csc (e+f x)}{a^2 c^5 f}-\frac{13 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac{21 \csc ^5(e+f x)}{5 a^2 c^5 f}-\frac{15 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac{4 \csc ^9(e+f x)}{9 a^2 c^5 f}+\frac{\int \cot ^4(e+f x) \, dx}{a^2 c^5}\\ &=-\frac{\cot ^3(e+f x)}{3 a^2 c^5 f}+\frac{\cot ^5(e+f x)}{5 a^2 c^5 f}-\frac{\cot ^7(e+f x)}{7 a^2 c^5 f}+\frac{4 \cot ^9(e+f x)}{9 a^2 c^5 f}+\frac{3 \csc (e+f x)}{a^2 c^5 f}-\frac{13 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac{21 \csc ^5(e+f x)}{5 a^2 c^5 f}-\frac{15 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac{4 \csc ^9(e+f x)}{9 a^2 c^5 f}-\frac{\int \cot ^2(e+f x) \, dx}{a^2 c^5}\\ &=\frac{\cot (e+f x)}{a^2 c^5 f}-\frac{\cot ^3(e+f x)}{3 a^2 c^5 f}+\frac{\cot ^5(e+f x)}{5 a^2 c^5 f}-\frac{\cot ^7(e+f x)}{7 a^2 c^5 f}+\frac{4 \cot ^9(e+f x)}{9 a^2 c^5 f}+\frac{3 \csc (e+f x)}{a^2 c^5 f}-\frac{13 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac{21 \csc ^5(e+f x)}{5 a^2 c^5 f}-\frac{15 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac{4 \csc ^9(e+f x)}{9 a^2 c^5 f}+\frac{\int 1 \, dx}{a^2 c^5}\\ &=\frac{x}{a^2 c^5}+\frac{\cot (e+f x)}{a^2 c^5 f}-\frac{\cot ^3(e+f x)}{3 a^2 c^5 f}+\frac{\cot ^5(e+f x)}{5 a^2 c^5 f}-\frac{\cot ^7(e+f x)}{7 a^2 c^5 f}+\frac{4 \cot ^9(e+f x)}{9 a^2 c^5 f}+\frac{3 \csc (e+f x)}{a^2 c^5 f}-\frac{13 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac{21 \csc ^5(e+f x)}{5 a^2 c^5 f}-\frac{15 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac{4 \csc ^9(e+f x)}{9 a^2 c^5 f}\\ \end{align*}
Mathematica [A] time = 1.23536, size = 383, normalized size = 1.82 \[ \frac{\csc \left (\frac{e}{2}\right ) \sec \left (\frac{e}{2}\right ) \tan (e+f x) \sec ^6(e+f x) (-675036 \sin (e+f x)+506277 \sin (2 (e+f x))+37502 \sin (3 (e+f x))-225012 \sin (4 (e+f x))+112506 \sin (5 (e+f x))-18751 \sin (6 (e+f x))-431424 \sin (2 e+f x)+375552 \sin (e+2 f x)+201600 \sin (3 e+2 f x)-41248 \sin (2 e+3 f x)+84000 \sin (4 e+3 f x)-155712 \sin (3 e+4 f x)-100800 \sin (5 e+4 f x)+98016 \sin (4 e+5 f x)+30240 \sin (6 e+5 f x)-21376 \sin (5 e+6 f x)-181440 f x \cos (2 e+f x)-136080 f x \cos (e+2 f x)+136080 f x \cos (3 e+2 f x)-10080 f x \cos (2 e+3 f x)+10080 f x \cos (4 e+3 f x)+60480 f x \cos (3 e+4 f x)-60480 f x \cos (5 e+4 f x)-30240 f x \cos (4 e+5 f x)+30240 f x \cos (6 e+5 f x)+5040 f x \cos (5 e+6 f x)-5040 f x \cos (7 e+6 f x)+169344 \sin (e)-338112 \sin (f x)+181440 f x \cos (f x))}{645120 a^2 c^5 f (\sec (e+f x)-1)^5 (\sec (e+f x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.074, size = 175, normalized size = 0.8 \begin{align*}{\frac{1}{192\,f{a}^{2}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}-{\frac{1}{8\,f{a}^{2}{c}^{5}}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) }+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{f{a}^{2}{c}^{5}}}+{\frac{1}{576\,f{a}^{2}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-9}}-{\frac{1}{56\,f{a}^{2}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-7}}+{\frac{29}{320\,f{a}^{2}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-5}}-{\frac{1}{3\,f{a}^{2}{c}^{5}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{-3}}+{\frac{99}{64\,f{a}^{2}{c}^{5}} \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 = 1.55535, size = 251, normalized size = 1.2 \begin{align*} -\frac{\frac{105 \,{\left (\frac{24 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2} c^{5}} - \frac{40320 \, \arctan \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2} c^{5}} + \frac{{\left (\frac{360 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{1827 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac{6720 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac{31185 \, \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}}{a^{2} c^{5} \sin \left (f x + e\right )^{9}}}{20160 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.1041, size = 598, normalized size = 2.85 \begin{align*} \frac{668 \, \cos \left (f x + e\right )^{6} - 1059 \, \cos \left (f x + e\right )^{5} - 573 \, \cos \left (f x + e\right )^{4} + 1813 \, \cos \left (f x + e\right )^{3} - 393 \, \cos \left (f x + e\right )^{2} + 315 \,{\left (f x \cos \left (f x + e\right )^{5} - 3 \, f x \cos \left (f x + e\right )^{4} + 2 \, f x \cos \left (f x + e\right )^{3} + 2 \, 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 ) - 789 \, \cos \left (f x + e\right ) + 368}{315 \,{\left (a^{2} c^{5} f \cos \left (f x + e\right )^{5} - 3 \, a^{2} c^{5} f \cos \left (f x + e\right )^{4} + 2 \, a^{2} c^{5} f \cos \left (f x + e\right )^{3} + 2 \, a^{2} c^{5} f \cos \left (f x + e\right )^{2} - 3 \, a^{2} c^{5} f \cos \left (f x + e\right ) + a^{2} c^{5} 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] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{1}{\sec ^{7}{\left (e + f x \right )} - 3 \sec ^{6}{\left (e + f x \right )} + \sec ^{5}{\left (e + f x \right )} + 5 \sec ^{4}{\left (e + f x \right )} - 5 \sec ^{3}{\left (e + f x \right )} - \sec ^{2}{\left (e + f x \right )} + 3 \sec{\left (e + f x \right )} - 1}\, dx}{a^{2} c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.40587, size = 193, normalized size = 0.92 \begin{align*} \frac{\frac{20160 \,{\left (f x + e\right )}}{a^{2} c^{5}} + \frac{31185 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{8} - 6720 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{6} + 1827 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 360 \, \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 35}{a^{2} c^{5} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{9}} + \frac{105 \,{\left (a^{4} c^{10} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} - 24 \, a^{4} c^{10} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right )}}{a^{6} c^{15}}}{20160 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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