Optimal. Leaf size=83 \[ \frac{a \cos (d+e x)+b \sin (d+e x)}{4 b^2 e (a (-\sin (d+e x))+a+b \cos (d+e x))}-\frac{a \log \left (a+b \tan \left (\frac{d}{2}+\frac{e x}{2}+\frac{\pi }{4}\right )\right )}{4 b^3 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0529481, antiderivative size = 83, normalized size of antiderivative = 1., 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.
[In]
[Out]
Rule 3129
Rule 12
Rule 3122
Rule 31
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*}
Mathematica [A] time = 0.556618, size = 166, normalized size = 2. \[ \frac{\frac{b \left (a^2+b^2\right ) \sin \left (\frac{1}{2} (d+e x)\right )}{(a+b) \left ((b-a) \sin \left (\frac{1}{2} (d+e x)\right )+(a+b) \cos \left (\frac{1}{2} (d+e x)\right )\right )}-a \log \left ((b-a) \sin \left (\frac{1}{2} (d+e x)\right )+(a+b) \cos \left (\frac{1}{2} (d+e x)\right )\right )+a \log \left (\cos \left (\frac{1}{2} (d+e x)\right )-\sin \left (\frac{1}{2} (d+e x)\right )\right )+\frac{b \sin \left (\frac{1}{2} (d+e x)\right )}{\cos \left (\frac{1}{2} (d+e x)\right )-\sin \left (\frac{1}{2} (d+e x)\right )}}{4 b^3 e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.148, size = 178, normalized size = 2.1 \begin{align*} -{\frac{1}{4\,{b}^{2}e} \left ( \tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) -1 \right ) ^{-1}}+{\frac{a}{4\,{b}^{3}e}\ln \left ( \tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) -1 \right ) }-{\frac{{a}^{2}}{4\,{b}^{2}e \left ( a-b \right ) } \left ( a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) -b\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) -a-b \right ) ^{-1}}-{\frac{1}{4\,e \left ( a-b \right ) } \left ( a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) -b\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) -a-b \right ) ^{-1}}-{\frac{a}{4\,{b}^{3}e}\ln \left ( a\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) -b\tan \left ({\frac{d}{2}}+{\frac{ex}{2}} \right ) -a-b \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.00558, size = 246, normalized size = 2.96 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.27783, size = 377, normalized size = 4.54 \begin{align*} \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 )}} \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 [B] time = 1.16489, size = 258, normalized size = 3.11 \begin{align*} -\frac{1}{4} \,{\left (\frac{{\left (a^{2} - a b\right )} \log \left ({\left | a \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) - b \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) - a - b \right |}\right )}{a b^{3} - b^{4}} + \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 ({\left | \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) - 1 \right |}\right )}{b^{3}}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]