Optimal. Leaf size=207 \[ -\frac{a \left (5 a^2+3 c^2\right ) \log \left (a+c \tan \left (\frac{1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac{c \left (15 a^2+4 c^2\right ) \cos (d+e x)-a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a \cos (d+e x)+a+c \sin (d+e x))}+\frac{5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a \cos (d+e x)+a+c \sin (d+e x))^2}-\frac{c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a \cos (d+e x)+a+c \sin (d+e x))^3} \]
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Rubi [A] time = 0.248415, antiderivative size = 207, normalized size of antiderivative = 1., 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} \[ -\frac{a \left (5 a^2+3 c^2\right ) \log \left (a+c \tan \left (\frac{1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac{c \left (15 a^2+4 c^2\right ) \cos (d+e x)-a \left (15 a^2+4 c^2\right ) \sin (d+e x)}{96 c^6 e (a \cos (d+e x)+a+c \sin (d+e x))}+\frac{5 \left (a c \cos (d+e x)-a^2 \sin (d+e x)\right )}{96 c^4 e (a \cos (d+e x)+a+c \sin (d+e x))^2}-\frac{c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a \cos (d+e x)+a+c \sin (d+e x))^3} \]
Antiderivative was successfully verified.
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Rule 3129
Rule 3156
Rule 3153
Rule 3124
Rule 31
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}{4 a+4 c x} \, dx,x,\tan \left (\frac{1}{2} (d+e x)\right )\right )}{8 c^6 e}\\ &=-\frac{a \left (5 a^2+3 c^2\right ) \log \left (a+c \tan \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*}
Mathematica [B] time = 1.6272, size = 492, normalized size = 2.38 \[ \frac{\cos \left (\frac{1}{2} (d+e x)\right ) \left (a \cos \left (\frac{1}{2} (d+e x)\right )+c \sin \left (\frac{1}{2} (d+e x)\right )\right ) \left (\frac{c \left (255 a^4 c^2 \sin (d+e x)+72 a^4 c^2 \sin (2 (d+e x))-37 a^4 c^2 \sin (3 (d+e x))+129 a^2 c^4 \sin (d+e x)+36 a^2 c^4 \sin (2 (d+e x))-27 a^2 c^4 \sin (3 (d+e x))-35 a^3 c^3 \cos (3 (d+e x))+3 a c \left (25 a^2 c^2+25 a^4-4 c^4\right ) \cos (d+e x)-6 \left (15 a^3 c^3+25 a^5 c+4 a c^5\right ) \cos (2 (d+e x))+130 a^3 c^3-75 a^5 c \cos (3 (d+e x))+150 a^5 c+150 a^6 \sin (d+e x)+120 a^6 \sin (2 (d+e x))+30 a^6 \sin (3 (d+e x))-4 a c^5 \cos (3 (d+e x))+24 a c^5+12 c^6 \sin (d+e x)-4 c^6 \sin (3 (d+e x))\right )}{a}+192 \left (5 a^3+3 a c^2\right ) \cos ^3\left (\frac{1}{2} (d+e x)\right ) \log \left (\cos \left (\frac{1}{2} (d+e x)\right )\right ) \left (a \cos \left (\frac{1}{2} (d+e x)\right )+c \sin \left (\frac{1}{2} (d+e x)\right )\right )^3-192 \left (5 a^3+3 a c^2\right ) \cos ^3\left (\frac{1}{2} (d+e x)\right ) \left (a \cos \left (\frac{1}{2} (d+e x)\right )+c \sin \left (\frac{1}{2} (d+e x)\right )\right )^3 \log \left (a \cos \left (\frac{1}{2} (d+e x)\right )+c \sin \left (\frac{1}{2} (d+e x)\right )\right )\right )}{384 c^7 e (a \cos (d+e x)+a+c \sin (d+e x))^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.222, size = 378, normalized size = 1.8 \begin{align*}{\frac{1}{384\,{c}^{4}e} \left ( \tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{3}}-{\frac{a}{64\,{c}^{5}e} \left ( \tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{2}}+{\frac{5\,{a}^{2}}{64\,e{c}^{6}}\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) }+{\frac{3}{128\,{c}^{4}e}\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) }+{\frac{3\,{a}^{5}}{128\,e{c}^{7}} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-2}}+{\frac{3\,{a}^{3}}{64\,{c}^{5}e} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-2}}+{\frac{3\,a}{128\,{c}^{3}e} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-2}}-{\frac{{a}^{6}}{384\,e{c}^{7}} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-3}}-{\frac{{a}^{4}}{128\,{c}^{5}e} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-3}}-{\frac{{a}^{2}}{128\,{c}^{3}e} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-3}}-{\frac{1}{384\,ce} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-3}}-{\frac{5\,{a}^{3}}{32\,e{c}^{7}}\ln \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) }-{\frac{3\,a}{32\,{c}^{5}e}\ln \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) }-{\frac{15\,{a}^{4}}{128\,e{c}^{7}} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-1}}-{\frac{9\,{a}^{2}}{64\,{c}^{5}e} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-1}}-{\frac{3}{128\,{c}^{3}e} \left ( a+c\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14598, size = 414, normalized size = 2. \begin{align*} -\frac{\frac{37 \, a^{6} + 39 \, a^{4} c^{2} + 3 \, a^{2} c^{4} + c^{6} + \frac{9 \,{\left (9 \, a^{5} c + 10 \, a^{3} c^{3} + a c^{5}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac{9 \,{\left (5 \, a^{4} c^{2} + 6 \, a^{2} c^{4} + c^{6}\right )} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}}}{a^{3} c^{7} + \frac{3 \, a^{2} c^{8} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac{3 \, a c^{9} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + \frac{c^{10} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}}} + \frac{\frac{6 \, a c \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} - \frac{c^{2} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}} - \frac{3 \,{\left (10 \, a^{2} + 3 \, c^{2}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}}{c^{6}} + \frac{12 \,{\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left (a + \frac{c \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c^{7}}}{384 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.70019, size = 1782, normalized size = 8.61 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14161, size = 410, normalized size = 1.98 \begin{align*} -\frac{1}{384} \,{\left (\frac{12 \,{\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left ({\left | c \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + a \right |}\right )}{c^{7}} - \frac{110 \, a^{3} c^{3} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{3} + 66 \, a c^{5} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{3} + 285 \, a^{4} c^{2} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} + 144 \, a^{2} c^{4} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} - 9 \, c^{6} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} + 249 \, a^{5} c \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + 108 \, a^{3} c^{3} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) - 9 \, a c^{5} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + 73 \, a^{6} + 27 \, a^{4} c^{2} - 3 \, a^{2} c^{4} - c^{6}}{{\left (c \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + a\right )}^{3} c^{7}} - \frac{c^{8} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{3} - 6 \, a c^{7} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} + 30 \, a^{2} c^{6} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + 9 \, c^{8} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )}{c^{12}}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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