Optimal. Leaf size=142 \[ \frac {3 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{16 b^4 e (a \sin (d+e x)+a+b \cos (d+e x))}-\frac {\left (3 a^2+b^2\right ) \log \left (a+b \cot \left (\frac {d}{2}+\frac {e x}{2}+\frac {\pi }{4}\right )\right )}{16 b^5 e}-\frac {a \cos (d+e x)-b \sin (d+e x)}{16 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^2} \]
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Rubi [A] time = 0.11, antiderivative size = 142, 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, 3123, 31} \[ \frac {3 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{16 b^4 e (a \sin (d+e x)+a+b \cos (d+e x))}-\frac {\left (3 a^2+b^2\right ) \log \left (a+b \cot \left (\frac {d}{2}+\frac {e x}{2}+\frac {\pi }{4}\right )\right )}{16 b^5 e}-\frac {a \cos (d+e x)-b \sin (d+e x)}{16 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3123
Rule 3129
Rule 3153
Rubi steps
\begin {align*} \int \frac {1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^3} \, dx &=-\frac {a \cos (d+e x)-b \sin (d+e x)}{16 b^2 e (a+b \cos (d+e x)+a \sin (d+e x))^2}+\frac {\int \frac {-4 a+2 b \cos (d+e x)+2 a \sin (d+e x)}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^2} \, dx}{8 b^2}\\ &=-\frac {a \cos (d+e x)-b \sin (d+e x)}{16 b^2 e (a+b \cos (d+e x)+a \sin (d+e x))^2}+\frac {3 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{16 b^4 e (a+b \cos (d+e x)+a \sin (d+e x))}+\frac {\left (3 a^2+b^2\right ) \int \frac {1}{2 a+2 b \cos (d+e x)+2 a \sin (d+e x)} \, dx}{8 b^4}\\ &=-\frac {a \cos (d+e x)-b \sin (d+e x)}{16 b^2 e (a+b \cos (d+e x)+a \sin (d+e x))^2}+\frac {3 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{16 b^4 e (a+b \cos (d+e x)+a \sin (d+e x))}-\frac {\left (3 a^2+b^2\right ) \operatorname {Subst}\left (\int \frac {1}{2 a+2 b x} \, dx,x,\cot \left (\frac {\pi }{4}+\frac {1}{2} (d+e x)\right )\right )}{8 b^4 e}\\ &=-\frac {\left (3 a^2+b^2\right ) \log \left (a+b \cot \left (\frac {d}{2}+\frac {\pi }{4}+\frac {e x}{2}\right )\right )}{16 b^5 e}-\frac {a \cos (d+e x)-b \sin (d+e x)}{16 b^2 e (a+b \cos (d+e x)+a \sin (d+e x))^2}+\frac {3 \left (a^2 \cos (d+e x)-a b \sin (d+e x)\right )}{16 b^4 e (a+b \cos (d+e x)+a \sin (d+e x))}\\ \end {align*}
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Mathematica [A] time = 2.43, size = 255, normalized size = 1.80 \[ -\frac {-\frac {b^2 \left (a^2+b^2\right )}{\left ((a-b) \sin \left (\frac {1}{2} (d+e x)\right )+(a+b) \cos \left (\frac {1}{2} (d+e x)\right )\right )^2}+\frac {6 a b \left (a^2+b^2\right ) \sin \left (\frac {1}{2} (d+e x)\right )}{(a+b) \left ((a-b) \sin \left (\frac {1}{2} (d+e x)\right )+(a+b) \cos \left (\frac {1}{2} (d+e x)\right )\right )}-2 \left (3 a^2+b^2\right ) \log \left (\sin \left (\frac {1}{2} (d+e x)\right )+\cos \left (\frac {1}{2} (d+e x)\right )\right )+2 \left (3 a^2+b^2\right ) \log \left ((a-b) \sin \left (\frac {1}{2} (d+e x)\right )+(a+b) \cos \left (\frac {1}{2} (d+e x)\right )\right )+\frac {6 a b \sin \left (\frac {1}{2} (d+e x)\right )}{\sin \left (\frac {1}{2} (d+e x)\right )+\cos \left (\frac {1}{2} (d+e x)\right )}+\frac {b^2}{\left (\sin \left (\frac {1}{2} (d+e x)\right )+\cos \left (\frac {1}{2} (d+e x)\right )\right )^2}}{32 b^5 e} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.86, size = 420, normalized size = 2.96 \[ \frac {12 \, a^{2} b^{2} \cos \left (e x + d\right )^{2} - 6 \, a^{2} b^{2} + 2 \, {\left (3 \, a^{3} b - a b^{3}\right )} \cos \left (e x + d\right ) - {\left (6 \, a^{4} + 2 \, a^{2} b^{2} - {\left (3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (e x + d\right )^{2} + 2 \, {\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (e x + d\right ) + 2 \, {\left (3 \, a^{4} + a^{2} b^{2} + {\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \log \left (2 \, a b \cos \left (e x + d\right ) + a^{2} + b^{2} + {\left (a^{2} - b^{2}\right )} \sin \left (e x + d\right )\right ) + {\left (6 \, a^{4} + 2 \, a^{2} b^{2} - {\left (3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (e x + d\right )^{2} + 2 \, {\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (e x + d\right ) + 2 \, {\left (3 \, a^{4} + a^{2} b^{2} + {\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \log \left (\sin \left (e x + d\right ) + 1\right ) - 2 \, {\left (3 \, a^{2} b^{2} - b^{4} - 3 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )}{32 \, {\left (2 \, a b^{6} e \cos \left (e x + d\right ) + 2 \, a^{2} b^{5} e - {\left (a^{2} b^{5} - b^{7}\right )} e \cos \left (e x + d\right )^{2} + 2 \, {\left (a b^{6} e \cos \left (e x + d\right ) + a^{2} b^{5} e\right )} \sin \left (e x + d\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 481, normalized size = 3.39 \[ \frac {1}{16} \, {\left (\frac {2 \, {\left (3 \, a^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} - 9 \, a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 10 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} - 6 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + b^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 9 \, a^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} - 18 \, a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 12 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} - 6 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 9 \, a^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 9 \, a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 2 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 2 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 5 \, a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + b^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 3 \, a^{5} - 4 \, a^{3} b^{2} - a b^{4}\right )}}{{\left (a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} - b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 2 \, a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + a + b\right )}^{2}} + \frac {{\left (3 \, a^{2} + b^{2}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 2 \, b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 2 \, a - 2 \, {\left | b \right |} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 2 \, b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 2 \, a + 2 \, {\left | b \right |} \right |}}\right )}{b^{4} {\left | b \right |}}\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 = 639, normalized size = 4.50 \[ -\frac {3 \ln \left (a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )+a +b \right ) a^{3}}{16 e \,b^{5} \left (a -b \right )}+\frac {3 \ln \left (a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )+a +b \right ) a^{2}}{16 e \,b^{4} \left (a -b \right )}-\frac {\ln \left (a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )+a +b \right ) a}{16 e \,b^{3} \left (a -b \right )}+\frac {\ln \left (a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )+a +b \right )}{16 e \,b^{2} \left (a -b \right )}+\frac {a^{4}}{16 e \,b^{3} \left (a -b \right )^{2} \left (a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )+a +b \right )^{2}}+\frac {a^{2}}{8 e b \left (a -b \right )^{2} \left (a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )+a +b \right )^{2}}+\frac {b}{16 e \left (a -b \right )^{2} \left (a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )+a +b \right )^{2}}+\frac {3 a^{4}}{16 e \,b^{4} \left (a -b \right )^{2} \left (a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )+a +b \right )}-\frac {a^{3}}{4 e \,b^{3} \left (a -b \right )^{2} \left (a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )+a +b \right )}+\frac {a^{2}}{8 e \,b^{2} \left (a -b \right )^{2} \left (a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )+a +b \right )}-\frac {a}{4 e b \left (a -b \right )^{2} \left (a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )+a +b \right )}-\frac {1}{16 e \left (a -b \right )^{2} \left (a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )+a +b \right )}-\frac {1}{16 e \,b^{3} \left (1+\tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{2}}+\frac {3 a}{16 e \,b^{4} \left (1+\tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}+\frac {1}{16 e \,b^{3} \left (1+\tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}+\frac {3 \ln \left (1+\tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right ) a^{2}}{16 e \,b^{5}}+\frac {\ln \left (1+\tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{16 e \,b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 493, normalized size = 3.47 \[ \frac {\frac {2 \, {\left (3 \, a^{5} - 4 \, a^{3} b^{2} - a b^{4} + \frac {{\left (9 \, a^{5} - 9 \, a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 5 \, a b^{4} + b^{5}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {{\left (9 \, a^{5} - 18 \, a^{4} b + 12 \, a^{3} b^{2} - 6 \, a^{2} b^{3} + a b^{4}\right )} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + \frac {{\left (3 \, a^{5} - 9 \, a^{4} b + 10 \, a^{3} b^{2} - 6 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}}\right )}}{a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8} + \frac {4 \, {\left (a^{4} b^{4} - a^{3} b^{5} - a^{2} b^{6} + a b^{7}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {2 \, {\left (3 \, a^{4} b^{4} - 6 \, a^{3} b^{5} + 2 \, a^{2} b^{6} + 2 \, a b^{7} - b^{8}\right )} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + \frac {4 \, {\left (a^{4} b^{4} - 3 \, a^{3} b^{5} + 3 \, a^{2} b^{6} - a b^{7}\right )} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}} + \frac {{\left (a^{4} b^{4} - 4 \, a^{3} b^{5} + 6 \, a^{2} b^{6} - 4 \, a b^{7} + b^{8}\right )} \sin \left (e x + d\right )^{4}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{4}}} - \frac {{\left (3 \, a^{2} + b^{2}\right )} \log \left (-a - b - \frac {{\left (a - b\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{b^{5}} + \frac {{\left (3 \, a^{2} + b^{2}\right )} \log \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + 1\right )}{b^{5}}}{16 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.45, size = 360, normalized size = 2.54 \[ -\frac {\frac {-3\,a^5+4\,a^3\,b^2+a\,b^4}{2\,b^4\,{\left (a-b\right )}^2}+\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (-9\,a^5+9\,a^4\,b+2\,a^3\,b^2-2\,a^2\,b^3+5\,a\,b^4-b^5\right )}{2\,b^4\,{\left (a-b\right )}^2}+\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (-3\,a^4+6\,a^3\,b-4\,a^2\,b^2+2\,a\,b^3+b^4\right )}{2\,b^4\,\left (a-b\right )}-\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (9\,a^5-18\,a^4\,b+12\,a^3\,b^2-6\,a^2\,b^3+a\,b^4\right )}{2\,b^4\,{\left (a-b\right )}^2}}{e\,\left (8\,a\,b+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (24\,a^2-8\,b^2\right )-{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (16\,a\,b-16\,a^2\right )+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (4\,a^2-8\,a\,b+4\,b^2\right )+4\,a^2+4\,b^2+\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (16\,a^2+16\,b\,a\right )\right )}-\frac {\mathrm {atanh}\left (\frac {2\,a+\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (2\,a-2\,b\right )}{2\,b}\right )\,\left (3\,a^2+b^2\right )}{8\,b^5\,e} \]
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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