Optimal. Leaf size=210 \[ \frac {x}{a^2 c^5}+\frac {\cot (e+f x)}{a^2 c^5 f}-\frac {\cot ^3(e+f x)}{3 a^2 c^5 f}+\frac {\cot ^5(e+f x)}{5 a^2 c^5 f}-\frac {\cot ^7(e+f x)}{7 a^2 c^5 f}+\frac {4 \cot ^9(e+f x)}{9 a^2 c^5 f}+\frac {3 \csc (e+f x)}{a^2 c^5 f}-\frac {13 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac {21 \csc ^5(e+f x)}{5 a^2 c^5 f}-\frac {15 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac {4 \csc ^9(e+f x)}{9 a^2 c^5 f} \]
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Rubi [A]
time = 0.22, antiderivative size = 210, normalized size of antiderivative = 1.00, 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 = {3989, 3971,
3554, 8, 2686, 200, 2687, 30, 276} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 200
Rule 276
Rule 2686
Rule 2687
Rule 3554
Rule 3971
Rule 3989
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 {\text {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 \text {Subst}\left (\int x^8 \, dx,x,-\cot (e+f x)\right )}{a^2 c^5 f}+\frac {3 \text {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 {\text {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 \text {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*}
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Mathematica [A]
time = 1.31, size = 383, normalized size = 1.82 \begin {gather*} \frac {\csc \left (\frac {e}{2}\right ) \sec \left (\frac {e}{2}\right ) \sec ^6(e+f x) (181440 f x \cos (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)-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)) \tan (e+f x)}{645120 a^2 c^5 f (-1+\sec (e+f x))^5 (1+\sec (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 114, normalized size = 0.54
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+128 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {1}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {8}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {29}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {64}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {99}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}}{64 f \,c^{5} a^{2}}\) | \(114\) |
default | \(\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-8 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+128 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {1}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}-\frac {8}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {29}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {64}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {99}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}}{64 f \,c^{5} a^{2}}\) | \(114\) |
risch | \(\frac {x}{a^{2} c^{5}}+\frac {2 i \left (945 \,{\mathrm e}^{11 i \left (f x +e \right )}-3150 \,{\mathrm e}^{10 i \left (f x +e \right )}+2625 \,{\mathrm e}^{9 i \left (f x +e \right )}+6300 \,{\mathrm e}^{8 i \left (f x +e \right )}-13482 \,{\mathrm e}^{7 i \left (f x +e \right )}+5292 \,{\mathrm e}^{6 i \left (f x +e \right )}+10566 \,{\mathrm e}^{5 i \left (f x +e \right )}-11736 \,{\mathrm e}^{4 i \left (f x +e \right )}+1289 \,{\mathrm e}^{3 i \left (f x +e \right )}+4866 \,{\mathrm e}^{2 i \left (f x +e \right )}-3063 \,{\mathrm e}^{i \left (f x +e \right )}+668\right )}{315 f \,c^{5} a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{9} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}\) | \(171\) |
norman | \(\frac {\frac {x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c a}+\frac {1}{576 a c f}-\frac {\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}{56 a c f}+\frac {29 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{320 a c f}-\frac {\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )}{3 a c f}+\frac {99 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 a c f}-\frac {\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )}{8 a c f}+\frac {\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )}{192 a c f}}{a \,c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}\) | \(182\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 202, normalized size = 0.96 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.78, size = 250, normalized size = 1.19 \begin {gather*} \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 {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 {\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 135, normalized size = 0.64 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.78, size = 209, normalized size = 1.00 \begin {gather*} \frac {35\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+105\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-2520\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+31185\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-6720\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+1827\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-360\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+20160\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (e+f\,x\right )}{20160\,a^2\,c^5\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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