Optimal. Leaf size=83 \[ \frac {a \cos (d+e x)+b \sin (d+e x)}{4 b^2 e (a (-\sin (d+e x))+a+b \cos (d+e x))}-\frac {a \log \left (a+b \tan \left (\frac {d}{2}+\frac {e x}{2}+\frac {\pi }{4}\right )\right )}{4 b^3 e} \]
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Rubi [A] time = 0.05, antiderivative size = 83, 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, 12, 3122, 31} \[ \frac {a \cos (d+e x)+b \sin (d+e x)}{4 b^2 e (a (-\sin (d+e x))+a+b \cos (d+e x))}-\frac {a \log \left (a+b \tan \left (\frac {d}{2}+\frac {e x}{2}+\frac {\pi }{4}\right )\right )}{4 b^3 e} \]
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 3122
Rule 3129
Rubi steps
\begin {align*} \int \frac {1}{(2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^2} \, dx &=\frac {a \cos (d+e x)+b \sin (d+e x)}{4 b^2 e (a+b \cos (d+e x)-a \sin (d+e x))}+\frac {\int -\frac {2 a}{2 a+2 b \cos (d+e x)-2 a \sin (d+e x)} \, dx}{4 b^2}\\ &=\frac {a \cos (d+e x)+b \sin (d+e x)}{4 b^2 e (a+b \cos (d+e x)-a \sin (d+e x))}-\frac {a \int \frac {1}{2 a+2 b \cos (d+e x)-2 a \sin (d+e x)} \, dx}{2 b^2}\\ &=\frac {a \cos (d+e x)+b \sin (d+e x)}{4 b^2 e (a+b \cos (d+e x)-a \sin (d+e x))}-\frac {a \operatorname {Subst}\left (\int \frac {1}{2 a+2 b x} \, dx,x,\tan \left (\frac {\pi }{4}+\frac {1}{2} (d+e x)\right )\right )}{2 b^2 e}\\ &=-\frac {a \log \left (a+b \tan \left (\frac {d}{2}+\frac {\pi }{4}+\frac {e x}{2}\right )\right )}{4 b^3 e}+\frac {a \cos (d+e x)+b \sin (d+e x)}{4 b^2 e (a+b \cos (d+e x)-a \sin (d+e x))}\\ \end {align*}
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Mathematica [A] time = 0.62, size = 166, normalized size = 2.00 \[ \frac {\frac {b \left (a^2+b^2\right ) \sin \left (\frac {1}{2} (d+e x)\right )}{(a+b) \left ((b-a) \sin \left (\frac {1}{2} (d+e x)\right )+(a+b) \cos \left (\frac {1}{2} (d+e x)\right )\right )}-a \log \left ((b-a) \sin \left (\frac {1}{2} (d+e x)\right )+(a+b) \cos \left (\frac {1}{2} (d+e x)\right )\right )+a \log \left (\cos \left (\frac {1}{2} (d+e x)\right )-\sin \left (\frac {1}{2} (d+e x)\right )\right )+\frac {b \sin \left (\frac {1}{2} (d+e x)\right )}{\cos \left (\frac {1}{2} (d+e x)\right )-\sin \left (\frac {1}{2} (d+e x)\right )}}{4 b^3 e} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 154, normalized size = 1.86 \[ \frac {2 \, a b \cos \left (e x + d\right ) + 2 \, b^{2} \sin \left (e x + d\right ) - {\left (a b \cos \left (e x + d\right ) - a^{2} \sin \left (e x + d\right ) + a^{2}\right )} \log \left (2 \, a b \cos \left (e x + d\right ) + a^{2} + b^{2} - {\left (a^{2} - b^{2}\right )} \sin \left (e x + d\right )\right ) + {\left (a b \cos \left (e x + d\right ) - a^{2} \sin \left (e x + d\right ) + a^{2}\right )} \log \left (-\sin \left (e x + d\right ) + 1\right )}{8 \, {\left (b^{4} e \cos \left (e x + d\right ) - a b^{3} e \sin \left (e x + d\right ) + a b^{3} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 198, normalized size = 2.39 \[ -\frac {1}{4} \, {\left (\frac {2 \, {\left (a^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - a b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - a^{2}\right )}}{{\left (a b^{2} - b^{3}\right )} {\left (a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} - b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} - 2 \, a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + a + b\right )}} + \frac {a \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 2 \, b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 2 \, a - 2 \, {\left | b \right |} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 2 \, b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 2 \, a + 2 \, {\left | b \right |} \right |}}\right )}{b^{2} {\left | b \right |}}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.51, size = 178, normalized size = 2.14 \[ -\frac {a^{2}}{4 e \,b^{2} \left (a -b \right ) \left (a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-a -b \right )}-\frac {1}{4 e \left (a -b \right ) \left (a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-a -b \right )}-\frac {a \ln \left (a \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-b \tan \left (\frac {d}{2}+\frac {e x}{2}\right )-a -b \right )}{4 e \,b^{3}}-\frac {1}{4 e \,b^{2} \left (\tan \left (\frac {d}{2}+\frac {e x}{2}\right )-1\right )}+\frac {a \ln \left (\tan \left (\frac {d}{2}+\frac {e x}{2}\right )-1\right )}{4 e \,b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 182, normalized size = 2.19 \[ \frac {\frac {2 \, {\left (a^{2} - \frac {{\left (a^{2} - a b + b^{2}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}}{a^{2} b^{2} - b^{4} - \frac {2 \, {\left (a^{2} b^{2} - a b^{3}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {{\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}}} - \frac {a \log \left (a + b - \frac {{\left (a - b\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{b^{3}} + \frac {a \log \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} - 1\right )}{b^{3}}}{4 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.74, size = 126, normalized size = 1.52 \[ \frac {\frac {a^2}{b^2\,\left (a-b\right )}-\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (a^2-a\,b+b^2\right )}{b^2\,\left (a-b\right )}}{e\,\left (\left (2\,a-2\,b\right )\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2-4\,a\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+2\,a+2\,b\right )}-\frac {a\,\mathrm {atanh}\left (\frac {a-\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (2\,a-2\,b\right )}{2}}{b}\right )}{2\,b^3\,e} \]
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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