Optimal. Leaf size=166 \[ \frac {x}{a^2 c^4}+\frac {\cot (e+f x)}{a^2 c^4 f}-\frac {\cot ^3(e+f x)}{3 a^2 c^4 f}+\frac {\cot ^5(e+f x)}{5 a^2 c^4 f}-\frac {2 \cot ^7(e+f x)}{7 a^2 c^4 f}+\frac {2 \csc (e+f x)}{a^2 c^4 f}-\frac {2 \csc ^3(e+f x)}{a^2 c^4 f}+\frac {6 \csc ^5(e+f x)}{5 a^2 c^4 f}-\frac {2 \csc ^7(e+f x)}{7 a^2 c^4 f} \]
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Rubi [A]
time = 0.16, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3989, 3971,
3554, 8, 2686, 200, 2687, 30} \begin {gather*} -\frac {2 \cot ^7(e+f x)}{7 a^2 c^4 f}+\frac {\cot ^5(e+f x)}{5 a^2 c^4 f}-\frac {\cot ^3(e+f x)}{3 a^2 c^4 f}+\frac {\cot (e+f x)}{a^2 c^4 f}-\frac {2 \csc ^7(e+f x)}{7 a^2 c^4 f}+\frac {6 \csc ^5(e+f x)}{5 a^2 c^4 f}-\frac {2 \csc ^3(e+f x)}{a^2 c^4 f}+\frac {2 \csc (e+f x)}{a^2 c^4 f}+\frac {x}{a^2 c^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 200
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))^4} \, dx &=\frac {\int \cot ^8(e+f x) (a+a \sec (e+f x))^2 \, dx}{a^4 c^4}\\ &=\frac {\int \left (a^2 \cot ^8(e+f x)+2 a^2 \cot ^7(e+f x) \csc (e+f x)+a^2 \cot ^6(e+f x) \csc ^2(e+f x)\right ) \, dx}{a^4 c^4}\\ &=\frac {\int \cot ^8(e+f x) \, dx}{a^2 c^4}+\frac {\int \cot ^6(e+f x) \csc ^2(e+f x) \, dx}{a^2 c^4}+\frac {2 \int \cot ^7(e+f x) \csc (e+f x) \, dx}{a^2 c^4}\\ &=-\frac {\cot ^7(e+f x)}{7 a^2 c^4 f}-\frac {\int \cot ^6(e+f x) \, dx}{a^2 c^4}+\frac {\text {Subst}\left (\int x^6 \, dx,x,-\cot (e+f x)\right )}{a^2 c^4 f}-\frac {2 \text {Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\csc (e+f x)\right )}{a^2 c^4 f}\\ &=\frac {\cot ^5(e+f x)}{5 a^2 c^4 f}-\frac {2 \cot ^7(e+f x)}{7 a^2 c^4 f}+\frac {\int \cot ^4(e+f x) \, dx}{a^2 c^4}-\frac {2 \text {Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\csc (e+f x)\right )}{a^2 c^4 f}\\ &=-\frac {\cot ^3(e+f x)}{3 a^2 c^4 f}+\frac {\cot ^5(e+f x)}{5 a^2 c^4 f}-\frac {2 \cot ^7(e+f x)}{7 a^2 c^4 f}+\frac {2 \csc (e+f x)}{a^2 c^4 f}-\frac {2 \csc ^3(e+f x)}{a^2 c^4 f}+\frac {6 \csc ^5(e+f x)}{5 a^2 c^4 f}-\frac {2 \csc ^7(e+f x)}{7 a^2 c^4 f}-\frac {\int \cot ^2(e+f x) \, dx}{a^2 c^4}\\ &=\frac {\cot (e+f x)}{a^2 c^4 f}-\frac {\cot ^3(e+f x)}{3 a^2 c^4 f}+\frac {\cot ^5(e+f x)}{5 a^2 c^4 f}-\frac {2 \cot ^7(e+f x)}{7 a^2 c^4 f}+\frac {2 \csc (e+f x)}{a^2 c^4 f}-\frac {2 \csc ^3(e+f x)}{a^2 c^4 f}+\frac {6 \csc ^5(e+f x)}{5 a^2 c^4 f}-\frac {2 \csc ^7(e+f x)}{7 a^2 c^4 f}+\frac {\int 1 \, dx}{a^2 c^4}\\ &=\frac {x}{a^2 c^4}+\frac {\cot (e+f x)}{a^2 c^4 f}-\frac {\cot ^3(e+f x)}{3 a^2 c^4 f}+\frac {\cot ^5(e+f x)}{5 a^2 c^4 f}-\frac {2 \cot ^7(e+f x)}{7 a^2 c^4 f}+\frac {2 \csc (e+f x)}{a^2 c^4 f}-\frac {2 \csc ^3(e+f x)}{a^2 c^4 f}+\frac {6 \csc ^5(e+f x)}{5 a^2 c^4 f}-\frac {2 \csc ^7(e+f x)}{7 a^2 c^4 f}\\ \end {align*}
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Mathematica [A]
time = 1.36, size = 315, normalized size = 1.90 \begin {gather*} \frac {\csc \left (\frac {e}{2}\right ) \csc ^7\left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {e}{2}\right ) \sec ^3\left (\frac {1}{2} (e+f x)\right ) (5880 f x \cos (f x)-5880 f x \cos (2 e+f x)-3360 f x \cos (e+2 f x)+3360 f x \cos (3 e+2 f x)-1260 f x \cos (2 e+3 f x)+1260 f x \cos (4 e+3 f x)+1680 f x \cos (3 e+4 f x)-1680 f x \cos (5 e+4 f x)-420 f x \cos (4 e+5 f x)+420 f x \cos (6 e+5 f x)+4032 \sin (e)-9632 \sin (f x)-16002 \sin (e+f x)+9144 \sin (2 (e+f x))+3429 \sin (3 (e+f x))-4572 \sin (4 (e+f x))+1143 \sin (5 (e+f x))-11760 \sin (2 e+f x)+8864 \sin (e+2 f x)+3360 \sin (3 e+2 f x)+2064 \sin (2 e+3 f x)+2520 \sin (4 e+3 f x)-4432 \sin (3 e+4 f x)-1680 \sin (5 e+4 f x)+1528 \sin (4 e+5 f x))}{860160 a^2 c^4 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 101, normalized size = 0.61
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+64 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {7}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {22}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {42}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}}{32 f \,c^{4} a^{2}}\) | \(101\) |
default | \(\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+64 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {1}{7 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}+\frac {7}{5 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}-\frac {22}{3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}+\frac {42}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}}{32 f \,c^{4} a^{2}}\) | \(101\) |
risch | \(\frac {x}{a^{2} c^{4}}+\frac {2 i \left (210 \,{\mathrm e}^{9 i \left (f x +e \right )}-315 \,{\mathrm e}^{8 i \left (f x +e \right )}-420 \,{\mathrm e}^{7 i \left (f x +e \right )}+1470 \,{\mathrm e}^{6 i \left (f x +e \right )}-504 \,{\mathrm e}^{5 i \left (f x +e \right )}-1204 \,{\mathrm e}^{4 i \left (f x +e \right )}+1108 \,{\mathrm e}^{3 i \left (f x +e \right )}+258 \,{\mathrm e}^{2 i \left (f x +e \right )}-554 \,{\mathrm e}^{i \left (f x +e \right )}+191\right )}{105 f \,c^{4} a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{7} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}\) | \(149\) |
norman | \(\frac {\frac {x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c a}-\frac {1}{224 a c f}+\frac {7 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{160 a c f}-\frac {11 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{48 a c f}+\frac {21 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16 a c f}-\frac {7 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32 a c f}+\frac {\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )}{96 a c f}}{a \,c^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{7}}\) | \(160\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 181, normalized size = 1.09 \begin {gather*} -\frac {\frac {35 \, {\left (\frac {21 \, \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^{4}} - \frac {6720 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2} c^{4}} - \frac {{\left (\frac {147 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {770 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {4410 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 15\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{7}}{a^{2} c^{4} \sin \left (f x + e\right )^{7}}}{3360 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.03, size = 179, normalized size = 1.08 \begin {gather*} \frac {191 \, \cos \left (f x + e\right )^{5} - 172 \, \cos \left (f x + e\right )^{4} - 253 \, \cos \left (f x + e\right )^{3} + 258 \, \cos \left (f x + e\right )^{2} + 105 \, {\left (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 ) - f x\right )} \sin \left (f x + e\right ) + 87 \, \cos \left (f x + e\right ) - 96}{105 \, {\left (a^{2} c^{4} f \cos \left (f x + e\right )^{4} - 2 \, a^{2} c^{4} f \cos \left (f x + e\right )^{3} + 2 \, a^{2} c^{4} f \cos \left (f x + e\right ) - a^{2} c^{4} 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 ^{6}{\left (e + f x \right )} - 2 \sec ^{5}{\left (e + f x \right )} - \sec ^{4}{\left (e + f x \right )} + 4 \sec ^{3}{\left (e + f x \right )} - \sec ^{2}{\left (e + f x \right )} - 2 \sec {\left (e + f x \right )} + 1}\, dx}{a^{2} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 122, normalized size = 0.73 \begin {gather*} \frac {\frac {3360 \, {\left (f x + e\right )}}{a^{2} c^{4}} + \frac {4410 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 770 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 147 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 15}{a^{2} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7}} + \frac {35 \, {\left (a^{4} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 21 \, a^{4} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6} c^{12}}}{3360 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.64, size = 185, normalized size = 1.11 \begin {gather*} \frac {35\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-15\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-735\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+4410\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-770\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+147\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+3360\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (e+f\,x\right )}{3360\,a^2\,c^4\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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