Optimal. Leaf size=61 \[ \frac{2 \tan ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (d+e x)\right )+c}{\sqrt{a^2-b^2-c^2}}\right )}{e \sqrt{a^2-b^2-c^2}} \]
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Rubi [A] time = 0.0837496, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {3124, 618, 204} \[ \frac{2 \tan ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (d+e x)\right )+c}{\sqrt{a^2-b^2-c^2}}\right )}{e \sqrt{a^2-b^2-c^2}} \]
Antiderivative was successfully verified.
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Rule 3124
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{a+b \cos (d+e x)+c \sin (d+e x)} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{a+b+2 c x+(a-b) x^2} \, dx,x,\tan \left (\frac{1}{2} (d+e x)\right )\right )}{e}\\ &=-\frac{4 \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2-c^2\right )-x^2} \, dx,x,2 c+2 (a-b) \tan \left (\frac{1}{2} (d+e x)\right )\right )}{e}\\ &=\frac{2 \tan ^{-1}\left (\frac{c+(a-b) \tan \left (\frac{1}{2} (d+e x)\right )}{\sqrt{a^2-b^2-c^2}}\right )}{\sqrt{a^2-b^2-c^2} e}\\ \end{align*}
Mathematica [A] time = 0.117129, size = 57, normalized size = 0.93 \[ -\frac{2 \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (d+e x)\right )+c}{\sqrt{-a^2+b^2+c^2}}\right )}{e \sqrt{-a^2+b^2+c^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.079, size = 61, normalized size = 1. \begin{align*} 2\,{\frac{1}{e\sqrt{{a}^{2}-{b}^{2}-{c}^{2}}}\arctan \left ( 1/2\,{\frac{2\, \left ( a-b \right ) \tan \left ( d/2+1/2\,ex \right ) +2\,c}{\sqrt{{a}^{2}-{b}^{2}-{c}^{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.33889, size = 953, normalized size = 15.62 \begin{align*} \left [-\frac{\sqrt{-a^{2} + b^{2} + c^{2}} \log \left (-\frac{a^{2} b^{2} - 2 \, b^{4} - c^{4} -{\left (a^{2} + 3 \, b^{2}\right )} c^{2} -{\left (2 \, a^{2} b^{2} - b^{4} - 2 \, a^{2} c^{2} + c^{4}\right )} \cos \left (e x + d\right )^{2} - 2 \,{\left (a b^{3} + a b c^{2}\right )} \cos \left (e x + d\right ) - 2 \,{\left (a b^{2} c + a c^{3} -{\left (b c^{3} -{\left (2 \, a^{2} b - b^{3}\right )} c\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right ) + 2 \,{\left (2 \, a b c \cos \left (e x + d\right )^{2} - a b c +{\left (b^{2} c + c^{3}\right )} \cos \left (e x + d\right ) -{\left (b^{3} + b c^{2} +{\left (a b^{2} - a c^{2}\right )} \cos \left (e x + d\right )\right )} \sin \left (e x + d\right )\right )} \sqrt{-a^{2} + b^{2} + c^{2}}}{2 \, a b \cos \left (e x + d\right ) +{\left (b^{2} - c^{2}\right )} \cos \left (e x + d\right )^{2} + a^{2} + c^{2} + 2 \,{\left (b c \cos \left (e x + d\right ) + a c\right )} \sin \left (e x + d\right )}\right )}{2 \,{\left (a^{2} - b^{2} - c^{2}\right )} e}, \frac{\arctan \left (-\frac{{\left (a b \cos \left (e x + d\right ) + a c \sin \left (e x + d\right ) + b^{2} + c^{2}\right )} \sqrt{a^{2} - b^{2} - c^{2}}}{{\left (c^{3} -{\left (a^{2} - b^{2}\right )} c\right )} \cos \left (e x + d\right ) +{\left (a^{2} b - b^{3} - b c^{2}\right )} \sin \left (e x + d\right )}\right )}{\sqrt{a^{2} - b^{2} - c^{2}} e}\right ] \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 = 1.11689, size = 123, normalized size = 2.02 \begin{align*} -\frac{2 \,{\left (\pi \left \lfloor \frac{x e + d}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac{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 ) + c}{\sqrt{a^{2} - b^{2} - c^{2}}}\right )\right )} e^{\left (-1\right )}}{\sqrt{a^{2} - b^{2} - c^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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