Optimal. Leaf size=141 \[ \frac {\cot ^7(e+f x)}{7 a^2 c^5 f}+\frac {4 \cot ^9(e+f x)}{9 a^2 c^5 f}+\frac {\csc (e+f x)}{a^2 c^5 f}-\frac {7 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac {3 \csc ^5(e+f x)}{a^2 c^5 f}-\frac {13 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac {4 \csc ^9(e+f x)}{9 a^2 c^5 f} \]
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Rubi [A]
time = 0.19, antiderivative size = 141, 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 = {4043, 2686,
200, 2687, 30, 276, 14} \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 {4 \csc ^9(e+f x)}{9 a^2 c^5 f}-\frac {13 \csc ^7(e+f x)}{7 a^2 c^5 f}+\frac {3 \csc ^5(e+f x)}{a^2 c^5 f}-\frac {7 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac {\csc (e+f x)}{a^2 c^5 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 200
Rule 276
Rule 2686
Rule 2687
Rule 4043
Rubi steps
\begin {align*} \int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c-c \sec (e+f x))^5} \, dx &=-\frac {\int \left (a^3 \cot ^9(e+f x) \csc (e+f x)+3 a^3 \cot ^8(e+f x) \csc ^2(e+f x)+3 a^3 \cot ^7(e+f x) \csc ^3(e+f x)+a^3 \cot ^6(e+f x) \csc ^4(e+f x)\right ) \, dx}{a^5 c^5}\\ &=-\frac {\int \cot ^9(e+f x) \csc (e+f x) \, dx}{a^2 c^5}-\frac {\int \cot ^6(e+f x) \csc ^4(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 {3 \int \cot ^7(e+f x) \csc ^3(e+f x) \, dx}{a^2 c^5}\\ &=\frac {\text {Subst}\left (\int \left (-1+x^2\right )^4 \, dx,x,\csc (e+f x)\right )}{a^2 c^5 f}-\frac {\text {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (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 x^2 \left (-1+x^2\right )^3 \, dx,x,\csc (e+f x)\right )}{a^2 c^5 f}\\ &=\frac {\cot ^9(e+f x)}{3 a^2 c^5 f}+\frac {\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 {\text {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (e+f x)\right )}{a^2 c^5 f}+\frac {3 \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 {\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 {\csc (e+f x)}{a^2 c^5 f}-\frac {7 \csc ^3(e+f x)}{3 a^2 c^5 f}+\frac {3 \csc ^5(e+f x)}{a^2 c^5 f}-\frac {13 \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 = 1.25, size = 211, normalized size = 1.50 \begin {gather*} -\frac {\csc (e) \csc ^6\left (\frac {1}{2} (e+f x)\right ) \csc ^3(e+f x) (-9408 \sin (e)+9792 \sin (f x)+36252 \sin (e+f x)-27189 \sin (2 (e+f x))-2014 \sin (3 (e+f x))+12084 \sin (4 (e+f x))-6042 \sin (5 (e+f x))+1007 \sin (6 (e+f x))+12096 \sin (2 e+f x)-14400 \sin (e+2 f x)-2016 \sin (3 e+2 f x)+7520 \sin (2 e+3 f x)-8736 \sin (4 e+3 f x)+1248 \sin (3 e+4 f x)+6048 \sin (5 e+4 f x)-1632 \sin (4 e+5 f x)-2016 \sin (6 e+5 f x)+608 \sin (5 e+6 f x))}{516096 a^2 c^5 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 102, normalized size = 0.72
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {20}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {3}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}+\frac {1}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}+\frac {15}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}-\frac {6}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}}{64 f \,c^{5} a^{2}}\) | \(102\) |
default | \(\frac {-\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}+6 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-\frac {20}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {3}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}+\frac {1}{9 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}+\frac {15}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}-\frac {6}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}}{64 f \,c^{5} a^{2}}\) | \(102\) |
risch | \(\frac {2 i \left (63 \,{\mathrm e}^{11 i \left (f x +e \right )}-189 \,{\mathrm e}^{10 i \left (f x +e \right )}+273 \,{\mathrm e}^{9 i \left (f x +e \right )}+63 \,{\mathrm e}^{8 i \left (f x +e \right )}-378 \,{\mathrm e}^{7 i \left (f x +e \right )}+294 \,{\mathrm e}^{6 i \left (f x +e \right )}+306 \,{\mathrm e}^{5 i \left (f x +e \right )}-450 \,{\mathrm e}^{4 i \left (f x +e \right )}+235 \,{\mathrm e}^{3 i \left (f x +e \right )}+39 \,{\mathrm e}^{2 i \left (f x +e \right )}-51 \,{\mathrm e}^{i \left (f x +e \right )}+19\right )}{63 f \,c^{5} a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{9} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}\) | \(162\) |
norman | \(\frac {\frac {1}{576 a c f}-\frac {3 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{224 a c f}+\frac {3 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 a c f}-\frac {5 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{48 a c f}+\frac {15 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 a c f}+\frac {3 \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32 a c f}-\frac {\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )}{192 a c f}}{a \,c^{4} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{9}}\) | \(163\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 175, normalized size = 1.24 \begin {gather*} \frac {\frac {21 \, {\left (\frac {18 \, \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 {{\left (\frac {54 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {189 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {420 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {945 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - 7\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{9}}{a^{2} c^{5} \sin \left (f x + e\right )^{9}}}{4032 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.11, size = 175, normalized size = 1.24 \begin {gather*} \frac {19 \, \cos \left (f x + e\right )^{6} + 6 \, \cos \left (f x + e\right )^{5} - 66 \, \cos \left (f x + e\right )^{4} + 56 \, \cos \left (f x + e\right )^{3} + 24 \, \cos \left (f x + e\right )^{2} - 48 \, \cos \left (f x + e\right ) + 16}{63 \, {\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 {\sec {\left (e + f x \right )}}{\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 = 0.57, size = 122, normalized size = 0.87 \begin {gather*} \frac {\frac {945 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 420 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 189 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 54 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7}{a^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9}} - \frac {21 \, {\left (a^{4} c^{10} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 18 \, a^{4} c^{10} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6} c^{15}}}{4032 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.24, size = 102, normalized size = 0.72 \begin {gather*} \frac {-21\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+378\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+945\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-420\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+189\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-54\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+7}{4032\,a^2\,c^5\,f\,{\mathrm {tan}\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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