Optimal. Leaf size=134 \[ \frac {3 \left (a^2 \sin (d+e x)+a c \cos (d+e x)\right )}{16 c^4 e (a (-\cos (d+e x))+a+c \sin (d+e x))}-\frac {\left (3 a^2+c^2\right ) \log \left (a+c \cot \left (\frac {1}{2} (d+e x)\right )\right )}{16 c^5 e}-\frac {a \sin (d+e x)+c \cos (d+e x)}{16 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^2} \]
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Rubi [A] time = 0.11, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3129, 3153, 3121, 31} \[ \frac {3 \left (a^2 \sin (d+e x)+a c \cos (d+e x)\right )}{16 c^4 e (a (-\cos (d+e x))+a+c \sin (d+e x))}-\frac {\left (3 a^2+c^2\right ) \log \left (a+c \cot \left (\frac {1}{2} (d+e x)\right )\right )}{16 c^5 e}-\frac {a \sin (d+e x)+c \cos (d+e x)}{16 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3121
Rule 3129
Rule 3153
Rubi steps
\begin {align*} \int \frac {1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^3} \, dx &=-\frac {c \cos (d+e x)+a \sin (d+e x)}{16 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))^2}+\frac {\int \frac {-4 a-2 a \cos (d+e x)+2 c \sin (d+e x)}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx}{8 c^2}\\ &=-\frac {c \cos (d+e x)+a \sin (d+e x)}{16 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))^2}+\frac {3 \left (a c \cos (d+e x)+a^2 \sin (d+e x)\right )}{16 c^4 e (a-a \cos (d+e x)+c \sin (d+e x))}+\frac {\left (3 a^2+c^2\right ) \int \frac {1}{2 a-2 a \cos (d+e x)+2 c \sin (d+e x)} \, dx}{8 c^4}\\ &=-\frac {c \cos (d+e x)+a \sin (d+e x)}{16 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))^2}+\frac {3 \left (a c \cos (d+e x)+a^2 \sin (d+e x)\right )}{16 c^4 e (a-a \cos (d+e x)+c \sin (d+e x))}-\frac {\left (3 a^2+c^2\right ) \operatorname {Subst}\left (\int \frac {1}{2 a+2 c x} \, dx,x,\cot \left (\frac {1}{2} (d+e x)\right )\right )}{8 c^4 e}\\ &=-\frac {\left (3 a^2+c^2\right ) \log \left (a+c \cot \left (\frac {1}{2} (d+e x)\right )\right )}{16 c^5 e}-\frac {c \cos (d+e x)+a \sin (d+e x)}{16 c^2 e (a-a \cos (d+e x)+c \sin (d+e x))^2}+\frac {3 \left (a c \cos (d+e x)+a^2 \sin (d+e x)\right )}{16 c^4 e (a-a \cos (d+e x)+c \sin (d+e x))}\\ \end {align*}
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Mathematica [C] time = 0.61, size = 350, normalized size = 2.61 \[ \frac {\sin \left (\frac {1}{2} (d+e x)\right ) \left (a \sin \left (\frac {1}{2} (d+e x)\right )+c \cos \left (\frac {1}{2} (d+e x)\right )\right ) \left (-6 a \left (a^2+c^2\right ) \sin ^3\left (\frac {1}{2} (d+e x)\right ) \left (a \sin \left (\frac {1}{2} (d+e x)\right )+c \cos \left (\frac {1}{2} (d+e x)\right )\right )+4 \left (3 a^2+c^2\right ) \sin ^2\left (\frac {1}{2} (d+e x)\right ) \log \left (\sin \left (\frac {1}{2} (d+e x)\right )\right ) \left (a \sin \left (\frac {1}{2} (d+e x)\right )+c \cos \left (\frac {1}{2} (d+e x)\right )\right )^2-4 \left (3 a^2+c^2\right ) \sin ^2\left (\frac {1}{2} (d+e x)\right ) \left (a \sin \left (\frac {1}{2} (d+e x)\right )+c \cos \left (\frac {1}{2} (d+e x)\right )\right )^2 \log \left (a \sin \left (\frac {1}{2} (d+e x)\right )+c \cos \left (\frac {1}{2} (d+e x)\right )\right )+c^2 (c-i a) (c+i a) \sin ^2\left (\frac {1}{2} (d+e x)\right )-c^2 \left (a \sin \left (\frac {1}{2} (d+e x)\right )+c \cos \left (\frac {1}{2} (d+e x)\right )\right )^2+3 a c \sin (d+e x) \left (a \sin \left (\frac {1}{2} (d+e x)\right )+c \cos \left (\frac {1}{2} (d+e x)\right )\right )^2\right )}{8 c^5 e (a (-\cos (d+e x))+a+c \sin (d+e x))^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.08, size = 438, normalized size = 3.27 \[ \frac {12 \, a^{2} c^{2} \cos \left (e x + d\right )^{2} - 6 \, a^{2} c^{2} - 2 \, {\left (3 \, a^{2} c^{2} - c^{4}\right )} \cos \left (e x + d\right ) + {\left (3 \, a^{4} + 4 \, a^{2} c^{2} + c^{4} + {\left (3 \, a^{4} - 2 \, a^{2} c^{2} - c^{4}\right )} \cos \left (e x + d\right )^{2} - 2 \, {\left (3 \, a^{4} + a^{2} c^{2}\right )} \cos \left (e x + d\right ) + 2 \, {\left (3 \, a^{3} c + a c^{3} - {\left (3 \, a^{3} c + a c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \log \left (a c \sin \left (e x + d\right ) + \frac {1}{2} \, a^{2} + \frac {1}{2} \, c^{2} - \frac {1}{2} \, {\left (a^{2} - c^{2}\right )} \cos \left (e x + d\right )\right ) - {\left (3 \, a^{4} + 4 \, a^{2} c^{2} + c^{4} + {\left (3 \, a^{4} - 2 \, a^{2} c^{2} - c^{4}\right )} \cos \left (e x + d\right )^{2} - 2 \, {\left (3 \, a^{4} + a^{2} c^{2}\right )} \cos \left (e x + d\right ) + 2 \, {\left (3 \, a^{3} c + a c^{3} - {\left (3 \, a^{3} c + a c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (e x + d\right ) + \frac {1}{2}\right ) - 2 \, {\left (3 \, a^{3} c - a c^{3} - 3 \, {\left (a^{3} c - a c^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{32 \, {\left (2 \, a^{2} c^{5} e \cos \left (e x + d\right ) - {\left (a^{2} c^{5} - c^{7}\right )} e \cos \left (e x + d\right )^{2} - {\left (a^{2} c^{5} + c^{7}\right )} e + 2 \, {\left (a c^{6} e \cos \left (e x + d\right ) - a c^{6} e\right )} \sin \left (e x + d\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 239, normalized size = 1.78 \[ \frac {1}{64} \, {\left (\frac {4 \, {\left (3 \, a^{2} + c^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) \right |}\right )}{c^{5}} - \frac {4 \, {\left (3 \, a^{3} + a c^{2}\right )} \log \left ({\left | a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + c \right |}\right )}{a c^{5}} + \frac {12 \, a^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 4 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} - 2 \, a c^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 18 \, a^{4} c \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 6 \, a^{2} c^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} - c^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 4 \, a^{3} c^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - a^{2} c^{3}}{{\left (a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + c \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{2} a^{2} c^{4}}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.57, size = 272, normalized size = 2.03 \[ \frac {3 a^{2}}{32 e \,c^{4} \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}+\frac {1}{16 e \,c^{2} \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}-\frac {1}{32 e \,a^{2} \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}+\frac {a^{2}}{64 e \,c^{3} \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{2}}+\frac {1}{32 e c \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{2}}+\frac {c}{64 e \,a^{2} \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{2}}-\frac {3 \ln \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right ) a^{2}}{16 e \,c^{5}}-\frac {\ln \left (c +a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{16 e \,c^{3}}-\frac {1}{64 e \,c^{3} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )^{2}}+\frac {3 \ln \left (\tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right ) a^{2}}{16 e \,c^{5}}+\frac {\ln \left (\tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{16 e \,c^{3}}+\frac {3 a}{32 e \,c^{4} \tan \left (\frac {d}{2}+\frac {e x}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 265, normalized size = 1.98 \[ -\frac {\frac {a^{2} c^{3} - \frac {4 \, a^{3} c^{2} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} - \frac {{\left (18 \, a^{4} c + 6 \, a^{2} c^{3} - c^{5}\right )} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} - \frac {2 \, {\left (6 \, a^{5} + 2 \, a^{3} c^{2} - a c^{4}\right )} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}}}{\frac {a^{2} c^{6} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + \frac {2 \, a^{3} c^{5} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}} + \frac {a^{4} c^{4} \sin \left (e x + d\right )^{4}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{4}}} + \frac {4 \, {\left (3 \, a^{2} + c^{2}\right )} \log \left (c + \frac {a \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c^{5}} - \frac {4 \, {\left (3 \, a^{2} + c^{2}\right )} \log \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c^{5}}}{64 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.47, size = 186, normalized size = 1.39 \[ \frac {\frac {2\,a\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}{c^2}-\frac {1}{2\,c}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (6\,a^4+2\,a^2\,c^2-c^4\right )}{a\,c^4}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (18\,a^4+6\,a^2\,c^2-c^4\right )}{2\,a^2\,c^3}}{e\,\left (32\,a^2\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4+64\,a\,c\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3+32\,c^2\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\right )}-\frac {\mathrm {atanh}\left (\frac {2\,a\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}{c}+1\right )\,\left (3\,a^2+c^2\right )}{8\,c^5\,e} \]
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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