Optimal. Leaf size=123 \[ \frac {\log \left (\tan \left (\frac {1}{2} (d+e x)\right )+1\right )}{4 a^3 e}+\frac {3 (\cos (d+e x)-\sin (d+e x))}{16 e \left (a^3 \sin (d+e x)+a^3 \cos (d+e x)+a^3\right )}-\frac {a \cos (d+e x)-a \sin (d+e x)}{16 e \left (a^2 \sin (d+e x)+a^2 \cos (d+e x)+a^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 123, 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, 3124, 31} \[ \frac {\log \left (\tan \left (\frac {1}{2} (d+e x)\right )+1\right )}{4 a^3 e}+\frac {3 (\cos (d+e x)-\sin (d+e x))}{16 e \left (a^3 \sin (d+e x)+a^3 \cos (d+e x)+a^3\right )}-\frac {a \cos (d+e x)-a \sin (d+e x)}{16 e \left (a^2 \sin (d+e x)+a^2 \cos (d+e x)+a^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 3124
Rule 3129
Rule 3153
Rubi steps
\begin {align*} \int \frac {1}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^3} \, dx &=-\frac {a \cos (d+e x)-a \sin (d+e x)}{16 e \left (a^2+a^2 \cos (d+e x)+a^2 \sin (d+e x)\right )^2}+\frac {\int \frac {-4 a+2 a \cos (d+e x)+2 a \sin (d+e x)}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^2} \, dx}{8 a^2}\\ &=-\frac {a \cos (d+e x)-a \sin (d+e x)}{16 e \left (a^2+a^2 \cos (d+e x)+a^2 \sin (d+e x)\right )^2}+\frac {3 (\cos (d+e x)-\sin (d+e x))}{16 e \left (a^3+a^3 \cos (d+e x)+a^3 \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^2}\\ &=-\frac {a \cos (d+e x)-a \sin (d+e x)}{16 e \left (a^2+a^2 \cos (d+e x)+a^2 \sin (d+e x)\right )^2}+\frac {3 (\cos (d+e x)-\sin (d+e x))}{16 e \left (a^3+a^3 \cos (d+e x)+a^3 \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^2 e}\\ &=\frac {\log \left (1+\tan \left (\frac {1}{2} (d+e x)\right )\right )}{4 a^3 e}-\frac {a \cos (d+e x)-a \sin (d+e x)}{16 e \left (a^2+a^2 \cos (d+e x)+a^2 \sin (d+e x)\right )^2}+\frac {3 (\cos (d+e x)-\sin (d+e x))}{16 e \left (a^3+a^3 \cos (d+e x)+a^3 \sin (d+e x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.57, size = 135, normalized size = 1.10 \[ \frac {\sec ^2\left (\frac {1}{2} (d+e x)\right )+2 \left (-3 \tan \left (\frac {1}{2} (d+e x)\right )-8 \log \left (\cos \left (\frac {1}{2} (d+e x)\right )\right )-\frac {6 \sin \left (\frac {1}{2} (d+e x)\right )}{\sin \left (\frac {1}{2} (d+e x)\right )+\cos \left (\frac {1}{2} (d+e x)\right )}-\frac {1}{\left (\sin \left (\frac {1}{2} (d+e x)\right )+\cos \left (\frac {1}{2} (d+e x)\right )\right )^2}+8 \log \left (\sin \left (\frac {1}{2} (d+e x)\right )+\cos \left (\frac {1}{2} (d+e x)\right )\right )\right )}{64 a^3 e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.76, size = 143, normalized size = 1.16 \[ \frac {6 \, \cos \left (e x + d\right )^{2} - 4 \, {\left ({\left (\cos \left (e x + d\right ) + 1\right )} \sin \left (e x + d\right ) + \cos \left (e x + d\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (e x + d\right ) + \frac {1}{2}\right ) + 4 \, {\left ({\left (\cos \left (e x + d\right ) + 1\right )} \sin \left (e x + d\right ) + \cos \left (e x + d\right ) + 1\right )} \log \left (\sin \left (e x + d\right ) + 1\right ) + 2 \, \cos \left (e x + d\right ) - 2 \, \sin \left (e x + d\right ) - 3}{32 \, {\left (a^{3} e \cos \left (e x + d\right ) + a^{3} e + {\left (a^{3} e \cos \left (e x + d\right ) + a^{3} e\right )} \sin \left (e x + d\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 107, normalized size = 0.87 \[ \frac {1}{64} \, {\left (\frac {16 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 1 \right |}\right )}{a^{3}} - \frac {4 \, {\left (6 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 8 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 3\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 1\right )}^{2}} + \frac {a^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} - 6 \, a^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )}{a^{6}}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.45, size = 100, normalized size = 0.81 \[ \frac {\tan ^{2}\left (\frac {d}{2}+\frac {e x}{2}\right )}{64 a^{3} e}-\frac {3 \tan \left (\frac {d}{2}+\frac {e x}{2}\right )}{32 a^{3} e}-\frac {1}{16 a^{3} e \left (1+\tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )^{2}}+\frac {1}{4 a^{3} e \left (1+\tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}+\frac {\ln \left (1+\tan \left (\frac {d}{2}+\frac {e x}{2}\right )\right )}{4 a^{3} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 146, normalized size = 1.19 \[ \frac {\frac {4 \, {\left (\frac {4 \, \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + 3\right )}}{a^{3} + \frac {2 \, a^{3} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {a^{3} \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}}} - \frac {\frac {6 \, \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} - \frac {\sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}}}{a^{3}} + \frac {16 \, \log \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + 1\right )}{a^{3}}}{64 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.45, size = 90, normalized size = 0.73 \[ \frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2}{64\,a^3\,e}+\frac {\ln \left (\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+1\right )}{4\,a^3\,e}-\frac {3\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}{32\,a^3\,e}+\frac {\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}{4}+\frac {3}{16}}{a^3\,e\,{\left (\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 6.49, size = 423, normalized size = 3.44 \[ \begin {cases} \frac {16 \log {\left (\tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {d}{2} + \frac {e x}{2} \right )}}{64 a^{3} e \tan ^{2}{\left (\frac {d}{2} + \frac {e x}{2} \right )} + 128 a^{3} e \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 64 a^{3} e} + \frac {32 \log {\left (\tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 1 \right )} \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )}}{64 a^{3} e \tan ^{2}{\left (\frac {d}{2} + \frac {e x}{2} \right )} + 128 a^{3} e \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 64 a^{3} e} + \frac {16 \log {\left (\tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 1 \right )}}{64 a^{3} e \tan ^{2}{\left (\frac {d}{2} + \frac {e x}{2} \right )} + 128 a^{3} e \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 64 a^{3} e} + \frac {\tan ^{4}{\left (\frac {d}{2} + \frac {e x}{2} \right )}}{64 a^{3} e \tan ^{2}{\left (\frac {d}{2} + \frac {e x}{2} \right )} + 128 a^{3} e \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 64 a^{3} e} - \frac {4 \tan ^{3}{\left (\frac {d}{2} + \frac {e x}{2} \right )}}{64 a^{3} e \tan ^{2}{\left (\frac {d}{2} + \frac {e x}{2} \right )} + 128 a^{3} e \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 64 a^{3} e} + \frac {32 \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )}}{64 a^{3} e \tan ^{2}{\left (\frac {d}{2} + \frac {e x}{2} \right )} + 128 a^{3} e \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 64 a^{3} e} + \frac {23}{64 a^{3} e \tan ^{2}{\left (\frac {d}{2} + \frac {e x}{2} \right )} + 128 a^{3} e \tan {\left (\frac {d}{2} + \frac {e x}{2} \right )} + 64 a^{3} e} & \text {for}\: e \neq 0 \\\frac {x}{\left (2 a \sin {\relax (d )} + 2 a \cos {\relax (d )} + 2 a\right )^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________