Optimal. Leaf size=168 \[ -\frac {19 (a \cos (d+e x)-a \sin (d+e x))}{96 e \left (a^5 \sin (d+e x)+a^5 \cos (d+e x)+a^5\right )}-\frac {\log \left (\tan \left (\frac {1}{2} (d+e x)\right )+1\right )}{4 a^4 e}+\frac {5 (\cos (d+e x)-\sin (d+e x))}{96 e \left (a^2 \sin (d+e x)+a^2 \cos (d+e x)+a^2\right )^2}-\frac {\cos (d+e x)-\sin (d+e x)}{48 a e (a \sin (d+e x)+a \cos (d+e x)+a)^3} \]
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Rubi [A] time = 0.19, antiderivative size = 168, 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, 3124, 31} \[ -\frac {\log \left (\tan \left (\frac {1}{2} (d+e x)\right )+1\right )}{4 a^4 e}-\frac {19 (a \cos (d+e x)-a \sin (d+e x))}{96 e \left (a^5 \sin (d+e x)+a^5 \cos (d+e x)+a^5\right )}+\frac {5 (\cos (d+e x)-\sin (d+e x))}{96 e \left (a^2 \sin (d+e x)+a^2 \cos (d+e x)+a^2\right )^2}-\frac {\cos (d+e x)-\sin (d+e x)}{48 a e (a \sin (d+e x)+a \cos (d+e x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3124
Rule 3129
Rule 3153
Rule 3156
Rubi steps
\begin {align*} \int \frac {1}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^4} \, dx &=-\frac {\cos (d+e x)-\sin (d+e x)}{48 a e (a+a \cos (d+e x)+a \sin (d+e x))^3}+\frac {\int \frac {-6 a+4 a \cos (d+e x)+4 a \sin (d+e x)}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^3} \, dx}{12 a^2}\\ &=-\frac {\cos (d+e x)-\sin (d+e x)}{48 a e (a+a \cos (d+e x)+a \sin (d+e x))^3}+\frac {5 (\cos (d+e x)-\sin (d+e x))}{96 e \left (a^2+a^2 \cos (d+e x)+a^2 \sin (d+e x)\right )^2}+\frac {\int \frac {56 a^2-20 a^2 \cos (d+e x)-20 a^2 \sin (d+e x)}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^2} \, dx}{96 a^4}\\ &=-\frac {\cos (d+e x)-\sin (d+e x)}{48 a e (a+a \cos (d+e x)+a \sin (d+e x))^3}+\frac {5 (\cos (d+e x)-\sin (d+e x))}{96 e \left (a^2+a^2 \cos (d+e x)+a^2 \sin (d+e x)\right )^2}-\frac {19 \left (a^3 \cos (d+e x)-a^3 \sin (d+e x)\right )}{96 e \left (a^7+a^7 \cos (d+e x)+a^7 \sin (d+e x)\right )}-\frac {\int \frac {1}{2 a+2 a \cos (d+e x)+2 a \sin (d+e x)} \, dx}{2 a^3}\\ &=-\frac {\cos (d+e x)-\sin (d+e x)}{48 a e (a+a \cos (d+e x)+a \sin (d+e x))^3}+\frac {5 (\cos (d+e x)-\sin (d+e x))}{96 e \left (a^2+a^2 \cos (d+e x)+a^2 \sin (d+e x)\right )^2}-\frac {19 \left (a^3 \cos (d+e x)-a^3 \sin (d+e x)\right )}{96 e \left (a^7+a^7 \cos (d+e x)+a^7 \sin (d+e x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {1}{4 a+4 a x} \, dx,x,\tan \left (\frac {1}{2} (d+e x)\right )\right )}{a^3 e}\\ &=-\frac {\log \left (1+\tan \left (\frac {1}{2} (d+e x)\right )\right )}{4 a^4 e}-\frac {\cos (d+e x)-\sin (d+e x)}{48 a e (a+a \cos (d+e x)+a \sin (d+e x))^3}+\frac {5 (\cos (d+e x)-\sin (d+e x))}{96 e \left (a^2+a^2 \cos (d+e x)+a^2 \sin (d+e x)\right )^2}-\frac {19 \left (a^3 \cos (d+e x)-a^3 \sin (d+e x)\right )}{96 e \left (a^7+a^7 \cos (d+e x)+a^7 \sin (d+e x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.98, size = 247, normalized size = 1.47 \[ \frac {19 \tan \left (\frac {1}{2} (d+e x)\right )}{192 a^4 e}-\frac {\sec ^2\left (\frac {1}{2} (d+e x)\right )}{64 a^4 e}+\frac {\log \left (\cos \left (\frac {1}{2} (d+e x)\right )\right )}{4 a^4 e}+\frac {19 \sin \left (\frac {1}{2} (d+e x)\right )}{96 a^4 e \left (\sin \left (\frac {1}{2} (d+e x)\right )+\cos \left (\frac {1}{2} (d+e x)\right )\right )}+\frac {5}{192 a^4 e \left (\sin \left (\frac {1}{2} (d+e x)\right )+\cos \left (\frac {1}{2} (d+e x)\right )\right )^2}+\frac {\sin \left (\frac {1}{2} (d+e x)\right )}{96 a^4 e \left (\sin \left (\frac {1}{2} (d+e x)\right )+\cos \left (\frac {1}{2} (d+e x)\right )\right )^3}+\frac {\tan \left (\frac {1}{2} (d+e x)\right ) \sec ^2\left (\frac {1}{2} (d+e x)\right )}{384 a^4 e}-\frac {\log \left (\sin \left (\frac {1}{2} (d+e x)\right )+\cos \left (\frac {1}{2} (d+e x)\right )\right )}{4 a^4 e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 237, normalized size = 1.41 \[ \frac {38 \, \cos \left (e x + d\right )^{3} + 66 \, \cos \left (e x + d\right )^{2} + 24 \, {\left (\cos \left (e x + d\right )^{3} - {\left (\cos \left (e x + d\right )^{2} + 3 \, \cos \left (e x + d\right ) + 2\right )} \sin \left (e x + d\right ) - 3 \, \cos \left (e x + d\right ) - 2\right )} \log \left (\frac {1}{2} \, \cos \left (e x + d\right ) + \frac {1}{2}\right ) - 24 \, {\left (\cos \left (e x + d\right )^{3} - {\left (\cos \left (e x + d\right )^{2} + 3 \, \cos \left (e x + d\right ) + 2\right )} \sin \left (e x + d\right ) - 3 \, \cos \left (e x + d\right ) - 2\right )} \log \left (\sin \left (e x + d\right ) + 1\right ) + {\left (38 \, \cos \left (e x + d\right )^{2} - 35\right )} \sin \left (e x + d\right ) - 3 \, \cos \left (e x + d\right ) - 33}{192 \, {\left (a^{4} e \cos \left (e x + d\right )^{3} - 3 \, a^{4} e \cos \left (e x + d\right ) - 2 \, a^{4} e - {\left (a^{4} e \cos \left (e x + d\right )^{2} + 3 \, a^{4} e \cos \left (e x + d\right ) + 2 \, a^{4} e\right )} \sin \left (e x + d\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 139, normalized size = 0.83 \[ -\frac {1}{384} \, {\left (\frac {96 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 1 \right |}\right )}{a^{4}} - \frac {4 \, {\left (44 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 105 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 87 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 24\right )}}{a^{4} {\left (\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 1\right )}^{3}} - \frac {a^{8} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} - 6 \, a^{8} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 39 \, a^{8} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )}{a^{12}}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 140, normalized size = 0.83 \[ \frac {\tan ^{3}\left (\frac {d}{2}+\frac {e x}{2}\right )}{384 a^{4} e}-\frac {\tan ^{2}\left (\frac {d}{2}+\frac {e x}{2}\right )}{64 a^{4} e}+\frac {13 \tan \left (\frac {d}{2}+\frac {e x}{2}\right )}{128 a^{4} e}-\frac {1}{48 a^{4} e \left (1+\tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{3}}+\frac {3}{32 a^{4} e \left (1+\tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{2}}-\frac {9}{32 a^{4} e \left (1+\tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}-\frac {\ln \left (1+\tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{4 a^{4} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 208, normalized size = 1.24 \[ -\frac {\frac {4 \, {\left (\frac {45 \, \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {27 \, \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + 20\right )}}{a^{4} + \frac {3 \, a^{4} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {3 \, a^{4} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + \frac {a^{4} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}}} - \frac {\frac {39 \, \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} - \frac {6 \, \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + \frac {\sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}}}{a^{4}} + \frac {96 \, \log \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + 1\right )}{a^{4}}}{384 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.44, size = 161, normalized size = 0.96 \[ \frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3}{384\,a^4\,e}-\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2}{64\,a^4\,e}-\frac {\ln \left (\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+1\right )}{4\,a^4\,e}+\frac {13\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}{128\,a^4\,e}-\frac {9\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2+15\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+\frac {20}{3}}{e\,\left (32\,a^4\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3+96\,a^4\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2+96\,a^4\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+32\,a^4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 23.78, size = 792, normalized size = 4.71 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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