Optimal. Leaf size=207 \[ -\frac{a \left (15 a^2+4 c^2\right ) \sin (d+e x)+c \left (15 a^2+4 c^2\right ) \cos (d+e x)}{96 c^6 e (a (-\cos (d+e x))+a+c \sin (d+e x))}+\frac{5 \left (a^2 \sin (d+e x)+a c \cos (d+e x)\right )}{96 c^4 e (a (-\cos (d+e x))+a+c \sin (d+e x))^2}+\frac{a \left (5 a^2+3 c^2\right ) \log \left (a+c \cot \left (\frac{1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac{a \sin (d+e x)+c \cos (d+e x)}{48 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.24045, 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, 3121, 31} \[ -\frac{a \left (15 a^2+4 c^2\right ) \sin (d+e x)+c \left (15 a^2+4 c^2\right ) \cos (d+e x)}{96 c^6 e (a (-\cos (d+e x))+a+c \sin (d+e x))}+\frac{5 \left (a^2 \sin (d+e x)+a c \cos (d+e x)\right )}{96 c^4 e (a (-\cos (d+e x))+a+c \sin (d+e x))^2}+\frac{a \left (5 a^2+3 c^2\right ) \log \left (a+c \cot \left (\frac{1}{2} (d+e x)\right )\right )}{32 c^7 e}-\frac{a \sin (d+e x)+c \cos (d+e x)}{48 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3129
Rule 3156
Rule 3153
Rule 3121
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}{2 a+2 c x} \, dx,x,\cot \left (\frac{1}{2} (d+e x)\right )\right )}{16 c^6 e}\\ &=\frac{a \left (5 a^2+3 c^2\right ) \log \left (a+c \cot \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 = 1.08889, size = 494, normalized size = 2.39 \[ \frac{\sin \left (\frac{1}{2} (d+e x)\right ) \left (a \sin \left (\frac{1}{2} (d+e x)\right )+c \cos \left (\frac{1}{2} (d+e x)\right )\right ) \left (75 a^3 c^3 \sin (d+e x)-156 a^3 c^3 \sin (2 (d+e x))+79 a^3 c^3 \sin (3 (d+e x))-255 a^4 c^2 \cos (d+e x)+174 a^4 c^2 \cos (2 (d+e x))-49 a^4 c^2 \cos (3 (d+e x))-42 a^2 c^4 \cos (d+e x)+18 a^2 c^4 \cos (3 (d+e x))-192 \left (5 a^3+3 a c^2\right ) \sin ^3\left (\frac{1}{2} (d+e x)\right ) \log \left (\sin \left (\frac{1}{2} (d+e x)\right )\right ) \left (a \sin \left (\frac{1}{2} (d+e x)\right )+c \cos \left (\frac{1}{2} (d+e x)\right )\right )^3+192 \left (5 a^3+3 a c^2\right ) \sin ^3\left (\frac{1}{2} (d+e x)\right ) \left (a \sin \left (\frac{1}{2} (d+e x)\right )+c \cos \left (\frac{1}{2} (d+e x)\right )\right )^3 \log \left (a \sin \left (\frac{1}{2} (d+e x)\right )+c \cos \left (\frac{1}{2} (d+e x)\right )\right )+130 a^4 c^2+24 a^2 c^4+75 a^5 c \sin (d+e x)-60 a^5 c \sin (2 (d+e x))+15 a^5 c \sin (3 (d+e x))-225 a^6 \cos (d+e x)+90 a^6 \cos (2 (d+e x))-15 a^6 \cos (3 (d+e x))+150 a^6-12 a c^5 \sin (d+e x)-12 a c^5 \sin (2 (d+e x))+20 a c^5 \sin (3 (d+e x))-24 c^6 \cos (d+e x)+8 c^6 \cos (3 (d+e x))\right )}{384 c^7 e (a (-\cos (d+e x))+a+c \sin (d+e x))^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.229, size = 416, normalized size = 2. \begin{align*} -{\frac{{a}^{3}}{64\,{c}^{5}e} \left ( c+a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-2}}-{\frac{3\,a}{128\,{c}^{3}e} \left ( c+a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-2}}+{\frac{c}{128\,{a}^{3}e} \left ( c+a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-2}}-{\frac{5\,{a}^{3}}{64\,e{c}^{6}} \left ( c+a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-1}}-{\frac{9\,a}{128\,{c}^{4}e} \left ( c+a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-1}}-{\frac{1}{128\,{a}^{3}e} \left ( c+a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-1}}-{\frac{{a}^{3}}{384\,{c}^{4}e} \left ( c+a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-3}}-{\frac{a}{128\,{c}^{2}e} \left ( c+a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-3}}-{\frac{1}{128\,ae} \left ( c+a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-3}}-{\frac{{c}^{2}}{384\,{a}^{3}e} \left ( c+a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-3}}+{\frac{5\,{a}^{3}}{32\,e{c}^{7}}\ln \left ( c+a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) }+{\frac{3\,a}{32\,{c}^{5}e}\ln \left ( c+a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) }-{\frac{1}{384\,{c}^{4}e} \left ( \tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-3}}-{\frac{5\,{a}^{2}}{64\,e{c}^{6}} \left ( \tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-1}}-{\frac{3}{128\,{c}^{4}e} \left ( \tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-1}}+{\frac{a}{64\,{c}^{5}e} \left ( \tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) ^{-2}}-{\frac{5\,{a}^{3}}{32\,e{c}^{7}}\ln \left ( \tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) }-{\frac{3\,a}{32\,{c}^{5}e}\ln \left ( \tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.12117, size = 516, normalized size = 2.49 \begin{align*} -\frac{\frac{a^{3} c^{5} - \frac{3 \, a^{4} c^{4} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac{3 \,{\left (5 \, a^{5} c^{3} + 3 \, a^{3} c^{5}\right )} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + \frac{{\left (110 \, a^{6} c^{2} + 66 \, a^{4} c^{4} + 3 \, a^{2} c^{6} + c^{8}\right )} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}} + \frac{3 \,{\left (50 \, a^{7} c + 30 \, a^{5} c^{3} + a c^{7}\right )} \sin \left (e x + d\right )^{4}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{4}} + \frac{3 \,{\left (20 \, a^{8} + 12 \, a^{6} c^{2} + a^{2} c^{6}\right )} \sin \left (e x + d\right )^{5}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{5}}}{\frac{a^{3} c^{9} \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}} + \frac{3 \, a^{4} c^{8} \sin \left (e x + d\right )^{4}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{4}} + \frac{3 \, a^{5} c^{7} \sin \left (e x + d\right )^{5}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{5}} + \frac{a^{6} c^{6} \sin \left (e x + d\right )^{6}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{6}}} - \frac{12 \,{\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left (c + \frac{a \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )}{c^{7}} + \frac{12 \,{\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left (\frac{\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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.64159, size = 1783, normalized size = 8.61 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20794, size = 490, normalized size = 2.37 \begin{align*} -\frac{1}{384} \,{\left (\frac{12 \,{\left (5 \, a^{3} + 3 \, a c^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) \right |}\right )}{c^{7}} - \frac{12 \,{\left (5 \, a^{4} + 3 \, a^{2} c^{2}\right )} \log \left ({\left | a \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + c \right |}\right )}{a c^{7}} + \frac{60 \, a^{8} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{5} + 36 \, a^{6} c^{2} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{5} + 3 \, a^{2} c^{6} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{5} + 150 \, a^{7} c \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{4} + 90 \, a^{5} c^{3} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{4} + 3 \, a c^{7} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{4} + 110 \, a^{6} c^{2} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{3} + 66 \, a^{4} c^{4} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{3} + 3 \, a^{2} c^{6} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{3} + c^{8} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{3} + 15 \, a^{5} c^{3} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} + 9 \, a^{3} c^{5} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} - 3 \, a^{4} c^{4} \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + a^{3} c^{5}}{{\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 )}^{3} a^{3} c^{6}}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]