Optimal. Leaf size=207 \[ \frac {5 \left (a^2 \sin (d+e x)+a c \cos (d+e x)\right )}{96 c^4 e (a (-\cos (d+e x))+a+c \sin (d+e x))^2}+\frac {a \left (5 a^2+3 c^2\right ) \log \left (a+c \cot \left (\frac {1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac {a \left (15 a^2+4 c^2\right ) \sin (d+e x)+c \left (15 a^2+4 c^2\right ) \cos (d+e x)}{96 c^6 e (a (-\cos (d+e x))+a+c \sin (d+e x))}-\frac {a \sin (d+e x)+c \cos (d+e x)}{48 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^3} \]
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Rubi [A] time = 0.24, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3129, 3156, 3153, 3121, 31} \[ -\frac {a \left (15 a^2+4 c^2\right ) \sin (d+e x)+c \left (15 a^2+4 c^2\right ) \cos (d+e x)}{96 c^6 e (a (-\cos (d+e x))+a+c \sin (d+e x))}+\frac {5 \left (a^2 \sin (d+e x)+a c \cos (d+e x)\right )}{96 c^4 e (a (-\cos (d+e x))+a+c \sin (d+e x))^2}+\frac {a \left (5 a^2+3 c^2\right ) \log \left (a+c \cot \left (\frac {1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac {a \sin (d+e x)+c \cos (d+e x)}{48 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^3} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3121
Rule 3129
Rule 3153
Rule 3156
Rubi steps
\begin {align*} \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx &=-\frac {c \cos (d+e x)+a \sin (d+e x)}{48 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))^3}+\frac {\int \frac {-6 a-4 a \cos (d+e x)+4 c \sin (d+e x)}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^3} \, dx}{12 c^2}\\ &=-\frac {c \cos (d+e x)+a \sin (d+e x)}{48 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))^3}+\frac {5 \left (a c \cos (d+e x)+a^2 \sin (d+e x)\right )}{96 c^4 e (a-a \cos (d+e x)+c \sin (d+e x))^2}+\frac {\int \frac {8 \left (5 a^2+2 c^2\right )+20 a^2 \cos (d+e x)-20 a c \sin (d+e x)}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx}{96 c^4}\\ &=-\frac {c \cos (d+e x)+a \sin (d+e x)}{48 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))^3}+\frac {5 \left (a c \cos (d+e x)+a^2 \sin (d+e x)\right )}{96 c^4 e (a-a \cos (d+e x)+c \sin (d+e x))^2}-\frac {c \left (15 a^2+4 c^2\right ) \cos (d+e x)+a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a-a \cos (d+e x)+c \sin (d+e x))}-\frac {\left (a \left (5 a^2+3 c^2\right )\right ) \int \frac {1}{2 a-2 a \cos (d+e x)+2 c \sin (d+e x)} \, dx}{16 c^6}\\ &=-\frac {c \cos (d+e x)+a \sin (d+e x)}{48 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))^3}+\frac {5 \left (a c \cos (d+e x)+a^2 \sin (d+e x)\right )}{96 c^4 e (a-a \cos (d+e x)+c \sin (d+e x))^2}-\frac {c \left (15 a^2+4 c^2\right ) \cos (d+e x)+a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a-a \cos (d+e x)+c \sin (d+e x))}+\frac {\left (a \left (5 a^2+3 c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 a+2 c x} \, dx,x,\cot \left (\frac {1}{2} (d+e x)\right )\right )}{16 c^6 e}\\ &=\frac {a \left (5 a^2+3 c^2\right ) \log \left (a+c \cot \left (\frac {1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac {c \cos (d+e x)+a \sin (d+e x)}{48 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))^3}+\frac {5 \left (a c \cos (d+e x)+a^2 \sin (d+e x)\right )}{96 c^4 e (a-a \cos (d+e x)+c \sin (d+e x))^2}-\frac {c \left (15 a^2+4 c^2\right ) \cos (d+e x)+a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a-a \cos (d+e x)+c \sin (d+e x))}\\ \end {align*}
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Mathematica [B] time = 1.17, size = 494, normalized size = 2.39 \[ \frac {\sin \left (\frac {1}{2} (d+e x)\right ) \left (a \sin \left (\frac {1}{2} (d+e x)\right )+c \cos \left (\frac {1}{2} (d+e x)\right )\right ) \left (-225 a^6 \cos (d+e x)+90 a^6 \cos (2 (d+e x))-15 a^6 \cos (3 (d+e x))+150 a^6+75 a^5 c \sin (d+e x)-60 a^5 c \sin (2 (d+e x))+15 a^5 c \sin (3 (d+e x))-255 a^4 c^2 \cos (d+e x)+174 a^4 c^2 \cos (2 (d+e x))-49 a^4 c^2 \cos (3 (d+e x))+130 a^4 c^2+75 a^3 c^3 \sin (d+e x)-156 a^3 c^3 \sin (2 (d+e x))+79 a^3 c^3 \sin (3 (d+e x))-192 \left (5 a^3+3 a c^2\right ) \sin ^3\left (\frac {1}{2} (d+e x)\right ) \log \left (\sin \left (\frac {1}{2} (d+e x)\right )\right ) \left (a \sin \left (\frac {1}{2} (d+e x)\right )+c \cos \left (\frac {1}{2} (d+e x)\right )\right )^3+192 \left (5 a^3+3 a c^2\right ) \sin ^3\left (\frac {1}{2} (d+e x)\right ) \left (a \sin \left (\frac {1}{2} (d+e x)\right )+c \cos \left (\frac {1}{2} (d+e x)\right )\right )^3 \log \left (a \sin \left (\frac {1}{2} (d+e x)\right )+c \cos \left (\frac {1}{2} (d+e x)\right )\right )-42 a^2 c^4 \cos (d+e x)+18 a^2 c^4 \cos (3 (d+e x))+24 a^2 c^4-12 a c^5 \sin (d+e x)-12 a c^5 \sin (2 (d+e x))+20 a c^5 \sin (3 (d+e x))-24 c^6 \cos (d+e x)+8 c^6 \cos (3 (d+e x))\right )}{384 c^7 e (a (-\cos (d+e x))+a+c \sin (d+e x))^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.47, size = 796, normalized size = 3.85 \[ \frac {60 \, a^{4} c^{2} + 6 \, a^{2} c^{4} + 2 \, {\left (45 \, a^{4} c^{2} - 3 \, a^{2} c^{4} - 4 \, c^{6}\right )} \cos \left (e x + d\right )^{3} - 12 \, {\left (10 \, a^{4} c^{2} + a^{2} c^{4}\right )} \cos \left (e x + d\right )^{2} - 6 \, {\left (5 \, a^{4} c^{2} - 2 \, a^{2} c^{4} - 2 \, c^{6}\right )} \cos \left (e x + d\right ) - 3 \, {\left (5 \, a^{6} + 18 \, a^{4} c^{2} + 9 \, a^{2} c^{4} - {\left (5 \, a^{6} - 12 \, a^{4} c^{2} - 9 \, a^{2} c^{4}\right )} \cos \left (e x + d\right )^{3} + 3 \, {\left (5 \, a^{6} - 2 \, a^{4} c^{2} - 3 \, a^{2} c^{4}\right )} \cos \left (e x + d\right )^{2} - 3 \, {\left (5 \, a^{6} + 8 \, a^{4} c^{2} + 3 \, a^{2} c^{4}\right )} \cos \left (e x + d\right ) + {\left (15 \, a^{5} c + 14 \, a^{3} c^{3} + 3 \, a c^{5} + {\left (15 \, a^{5} c + 4 \, a^{3} c^{3} - 3 \, a c^{5}\right )} \cos \left (e x + d\right )^{2} - 6 \, {\left (5 \, a^{5} c + 3 \, a^{3} c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \log \left (a c \sin \left (e x + d\right ) + \frac {1}{2} \, a^{2} + \frac {1}{2} \, c^{2} - \frac {1}{2} \, {\left (a^{2} - c^{2}\right )} \cos \left (e x + d\right )\right ) + 3 \, {\left (5 \, a^{6} + 18 \, a^{4} c^{2} + 9 \, a^{2} c^{4} - {\left (5 \, a^{6} - 12 \, a^{4} c^{2} - 9 \, a^{2} c^{4}\right )} \cos \left (e x + d\right )^{3} + 3 \, {\left (5 \, a^{6} - 2 \, a^{4} c^{2} - 3 \, a^{2} c^{4}\right )} \cos \left (e x + d\right )^{2} - 3 \, {\left (5 \, a^{6} + 8 \, a^{4} c^{2} + 3 \, a^{2} c^{4}\right )} \cos \left (e x + d\right ) + {\left (15 \, a^{5} c + 14 \, a^{3} c^{3} + 3 \, a c^{5} + {\left (15 \, a^{5} c + 4 \, a^{3} c^{3} - 3 \, a c^{5}\right )} \cos \left (e x + d\right )^{2} - 6 \, {\left (5 \, a^{5} c + 3 \, a^{3} c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (e x + d\right ) + \frac {1}{2}\right ) + 2 \, {\left (15 \, a^{5} c + 14 \, a^{3} c^{3} + 6 \, a c^{5} + {\left (15 \, a^{5} c - 41 \, a^{3} c^{3} - 12 \, a c^{5}\right )} \cos \left (e x + d\right )^{2} - 3 \, {\left (10 \, a^{5} c - 9 \, a^{3} c^{3} - a c^{5}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{192 \, {\left ({\left (a^{3} c^{7} - 3 \, a c^{9}\right )} e \cos \left (e x + d\right )^{3} - 3 \, {\left (a^{3} c^{7} - a c^{9}\right )} e \cos \left (e x + d\right )^{2} + 3 \, {\left (a^{3} c^{7} + a c^{9}\right )} e \cos \left (e x + d\right ) - {\left (a^{3} c^{7} + 3 \, a c^{9}\right )} e + {\left (6 \, a^{2} c^{8} e \cos \left (e x + d\right ) - {\left (3 \, a^{2} c^{8} - c^{10}\right )} e \cos \left (e x + d\right )^{2} - {\left (3 \, a^{2} c^{8} + c^{10}\right )} e\right )} \sin \left (e x + d\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 363, normalized size = 1.75 \[ -\frac {1}{384} \, {\left (\frac {12 \, {\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) \right |}\right )}{c^{7}} - \frac {12 \, {\left (5 \, a^{4} + 3 \, a^{2} c^{2}\right )} \log \left ({\left | a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + c \right |}\right )}{a c^{7}} + \frac {60 \, a^{8} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 36 \, a^{6} c^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 3 \, a^{2} c^{6} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 150 \, a^{7} c \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} + 90 \, a^{5} c^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} + 3 \, a c^{7} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} + 110 \, a^{6} c^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 66 \, a^{4} c^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 3 \, a^{2} c^{6} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + c^{8} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 15 \, a^{5} c^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 9 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} - 3 \, a^{4} c^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + a^{3} c^{5}}{{\left (a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + c \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{3} a^{3} c^{6}}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.59, size = 416, normalized size = 2.01 \[ -\frac {a^{3}}{64 e \,c^{5} \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{2}}-\frac {3 a}{128 e \,c^{3} \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{2}}+\frac {c}{128 e \,a^{3} \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{2}}-\frac {5 a^{3}}{64 e \,c^{6} \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}-\frac {9 a}{128 e \,c^{4} \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}-\frac {1}{128 e \,a^{3} \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}-\frac {a^{3}}{384 e \,c^{4} \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{3}}-\frac {a}{128 e \,c^{2} \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{3}}-\frac {1}{128 e a \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{3}}-\frac {c^{2}}{384 e \,a^{3} \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{3}}+\frac {5 a^{3} \ln \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{32 e \,c^{7}}+\frac {3 a \ln \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{32 e \,c^{5}}-\frac {1}{384 e \,c^{4} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )^{3}}-\frac {5 a^{2}}{64 e \,c^{6} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )}-\frac {3}{128 e \,c^{4} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )}+\frac {a}{64 e \,c^{5} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )^{2}}-\frac {5 a^{3} \ln \left (\tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{32 e \,c^{7}}-\frac {3 a \ln \left (\tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{32 e \,c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 382, normalized size = 1.85 \[ -\frac {\frac {a^{3} c^{5} - \frac {3 \, a^{4} c^{4} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {3 \, {\left (5 \, a^{5} c^{3} + 3 \, a^{3} c^{5}\right )} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + \frac {{\left (110 \, a^{6} c^{2} + 66 \, a^{4} c^{4} + 3 \, a^{2} c^{6} + c^{8}\right )} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}} + \frac {3 \, {\left (50 \, a^{7} c + 30 \, a^{5} c^{3} + a c^{7}\right )} \sin \left (e x + d\right )^{4}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{4}} + \frac {3 \, {\left (20 \, a^{8} + 12 \, a^{6} c^{2} + a^{2} c^{6}\right )} \sin \left (e x + d\right )^{5}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{5}}}{\frac {a^{3} c^{9} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}} + \frac {3 \, a^{4} c^{8} \sin \left (e x + d\right )^{4}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{4}} + \frac {3 \, a^{5} c^{7} \sin \left (e x + d\right )^{5}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{5}} + \frac {a^{6} c^{6} \sin \left (e x + d\right )^{6}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{6}}} - \frac {12 \, {\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left (c + \frac {a \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c^{7}} + \frac {12 \, {\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c^{7}}}{384 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.05, size = 301, normalized size = 1.45 \[ \frac {a\,\mathrm {atanh}\left (\frac {a\,\left (c+2\,a\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\right )\,\left (5\,a^2+3\,c^2\right )}{c\,\left (5\,a^3+3\,a\,c^2\right )}\right )\,\left (5\,a^2+3\,c^2\right )}{16\,c^7\,e}-\frac {\frac {1}{3\,c}-\frac {a\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}{c^2}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (5\,a^2+3\,c^2\right )}{c^3}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (110\,a^6+66\,a^4\,c^2+3\,a^2\,c^4+c^6\right )}{3\,a^3\,c^4}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\left (20\,a^6+12\,a^4\,c^2+c^6\right )}{a\,c^6}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (50\,a^6+30\,a^4\,c^2+c^6\right )}{a^2\,c^5}}{e\,\left (128\,a^3\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6+384\,a^2\,c\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5+384\,a\,c^2\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4+128\,c^3\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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