Optimal. Leaf size=126 \[ \frac {x}{a^3 c}+\frac {\cot (e+f x)}{a^3 c f}-\frac {\cot ^3(e+f x)}{3 a^3 c f}+\frac {2 \cot ^5(e+f x)}{5 a^3 c f}-\frac {2 \csc (e+f x)}{a^3 c f}+\frac {4 \csc ^3(e+f x)}{3 a^3 c f}-\frac {2 \csc ^5(e+f x)}{5 a^3 c f} \]
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Rubi [A]
time = 0.14, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 12, 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 ^5(e+f x)}{5 a^3 c f}-\frac {\cot ^3(e+f x)}{3 a^3 c f}+\frac {\cot (e+f x)}{a^3 c f}-\frac {2 \csc ^5(e+f x)}{5 a^3 c f}+\frac {4 \csc ^3(e+f x)}{3 a^3 c f}-\frac {2 \csc (e+f x)}{a^3 c f}+\frac {x}{a^3 c} \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))^3 (c-c \sec (e+f x))} \, dx &=-\frac {\int \cot ^6(e+f x) (c-c \sec (e+f x))^2 \, dx}{a^3 c^3}\\ &=-\frac {\int \left (c^2 \cot ^6(e+f x)-2 c^2 \cot ^5(e+f x) \csc (e+f x)+c^2 \cot ^4(e+f x) \csc ^2(e+f x)\right ) \, dx}{a^3 c^3}\\ &=-\frac {\int \cot ^6(e+f x) \, dx}{a^3 c}-\frac {\int \cot ^4(e+f x) \csc ^2(e+f x) \, dx}{a^3 c}+\frac {2 \int \cot ^5(e+f x) \csc (e+f x) \, dx}{a^3 c}\\ &=\frac {\cot ^5(e+f x)}{5 a^3 c f}+\frac {\int \cot ^4(e+f x) \, dx}{a^3 c}-\frac {\text {Subst}\left (\int x^4 \, dx,x,-\cot (e+f x)\right )}{a^3 c f}-\frac {2 \text {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\csc (e+f x)\right )}{a^3 c f}\\ &=-\frac {\cot ^3(e+f x)}{3 a^3 c f}+\frac {2 \cot ^5(e+f x)}{5 a^3 c f}-\frac {\int \cot ^2(e+f x) \, dx}{a^3 c}-\frac {2 \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\csc (e+f x)\right )}{a^3 c f}\\ &=\frac {\cot (e+f x)}{a^3 c f}-\frac {\cot ^3(e+f x)}{3 a^3 c f}+\frac {2 \cot ^5(e+f x)}{5 a^3 c f}-\frac {2 \csc (e+f x)}{a^3 c f}+\frac {4 \csc ^3(e+f x)}{3 a^3 c f}-\frac {2 \csc ^5(e+f x)}{5 a^3 c f}+\frac {\int 1 \, dx}{a^3 c}\\ &=\frac {x}{a^3 c}+\frac {\cot (e+f x)}{a^3 c f}-\frac {\cot ^3(e+f x)}{3 a^3 c f}+\frac {2 \cot ^5(e+f x)}{5 a^3 c f}-\frac {2 \csc (e+f x)}{a^3 c f}+\frac {4 \csc ^3(e+f x)}{3 a^3 c f}-\frac {2 \csc ^5(e+f x)}{5 a^3 c f}\\ \end {align*}
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Mathematica [A]
time = 1.03, size = 197, normalized size = 1.56 \begin {gather*} -\frac {\csc \left (\frac {e}{2}\right ) \csc \left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {e}{2}\right ) \sec ^5\left (\frac {1}{2} (e+f x)\right ) (-150 f x \cos (f x)+150 f x \cos (2 e+f x)-120 f x \cos (e+2 f x)+120 f x \cos (3 e+2 f x)-30 f x \cos (2 e+3 f x)+30 f x \cos (4 e+3 f x)+80 \sin (e)+280 \sin (f x)-445 \sin (e+f x)-356 \sin (2 (e+f x))-89 \sin (3 (e+f x))+240 \sin (2 e+f x)+296 \sin (e+2 f x)+120 \sin (3 e+2 f x)+104 \sin (2 e+3 f x))}{3840 a^3 c f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 73, normalized size = 0.58
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}+\frac {5 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-11 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}+16 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f \,a^{3} c}\) | \(73\) |
default | \(\frac {-\frac {\left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}+\frac {5 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-11 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}+16 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f \,a^{3} c}\) | \(73\) |
risch | \(\frac {x}{a^{3} c}-\frac {4 i \left (15 \,{\mathrm e}^{5 i \left (f x +e \right )}+30 \,{\mathrm e}^{4 i \left (f x +e \right )}+10 \,{\mathrm e}^{3 i \left (f x +e \right )}-35 \,{\mathrm e}^{2 i \left (f x +e \right )}-37 \,{\mathrm e}^{i \left (f x +e \right )}-13\right )}{15 f \,a^{3} c \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{5} \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}\) | \(105\) |
norman | \(\frac {\frac {x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c a}+\frac {1}{8 a c f}-\frac {11 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 a c f}+\frac {5 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 a c f}-\frac {\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )}{40 a c f}}{a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 132, normalized size = 1.05 \begin {gather*} -\frac {\frac {\frac {165 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {25 \, \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}}}{a^{3} c} - \frac {240 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3} c} - \frac {15 \, {\left (\cos \left (f x + e\right ) + 1\right )}}{a^{3} c \sin \left (f x + e\right )}}{120 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.56, size = 118, normalized size = 0.94 \begin {gather*} \frac {26 \, \cos \left (f x + e\right )^{3} + 22 \, \cos \left (f x + e\right )^{2} + 15 \, {\left (f x \cos \left (f x + e\right )^{2} + 2 \, f x \cos \left (f x + e\right ) + f x\right )} \sin \left (f x + e\right ) - 17 \, \cos \left (f x + e\right ) - 16}{15 \, {\left (a^{3} c f \cos \left (f x + e\right )^{2} + 2 \, a^{3} c f \cos \left (f x + e\right ) + a^{3} c 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 ^{4}{\left (e + f x \right )} + 2 \sec ^{3}{\left (e + f x \right )} - 2 \sec {\left (e + f x \right )} - 1}\, dx}{a^{3} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.56, size = 102, normalized size = 0.81 \begin {gather*} \frac {\frac {120 \, {\left (f x + e\right )}}{a^{3} c} + \frac {15}{a^{3} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )} - \frac {3 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 25 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 165 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{15} c^{5}}}{120 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.43, size = 82, normalized size = 0.65 \begin {gather*} \frac {x}{a^3\,c}+\frac {\frac {26\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6}{15}-\frac {28\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{15}+\frac {17\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{60}-\frac {1}{40}}{a^3\,c\,f\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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