Optimal. Leaf size=76 \[ -\frac{\left (b^2-2 c (a+c)\right ) \tanh ^{-1}\left (\frac{b+2 c \cos (x)}{\sqrt{b^2-4 a c}}\right )}{c^2 \sqrt{b^2-4 a c}}-\frac{b \log \left (a+b \cos (x)+c \cos ^2(x)\right )}{2 c^2}+\frac{\cos (x)}{c} \]
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Rubi [A] time = 0.129778, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {3259, 1657, 634, 618, 206, 628} \[ -\frac{\left (b^2-2 c (a+c)\right ) \tanh ^{-1}\left (\frac{b+2 c \cos (x)}{\sqrt{b^2-4 a c}}\right )}{c^2 \sqrt{b^2-4 a c}}-\frac{b \log \left (a+b \cos (x)+c \cos ^2(x)\right )}{2 c^2}+\frac{\cos (x)}{c} \]
Antiderivative was successfully verified.
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Rule 3259
Rule 1657
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{\sin ^3(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1-x^2}{a+b x+c x^2} \, dx,x,\cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{1}{c}+\frac{a+c+b x}{c \left (a+b x+c x^2\right )}\right ) \, dx,x,\cos (x)\right )\\ &=\frac{\cos (x)}{c}-\frac{\operatorname{Subst}\left (\int \frac{a+c+b x}{a+b x+c x^2} \, dx,x,\cos (x)\right )}{c}\\ &=\frac{\cos (x)}{c}-\frac{b \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,\cos (x)\right )}{2 c^2}+\frac{\left (b^2-2 c (a+c)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,\cos (x)\right )}{2 c^2}\\ &=\frac{\cos (x)}{c}-\frac{b \log \left (a+b \cos (x)+c \cos ^2(x)\right )}{2 c^2}-\frac{\left (b^2-2 c (a+c)\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c \cos (x)\right )}{c^2}\\ &=-\frac{\left (b^2-2 c (a+c)\right ) \tanh ^{-1}\left (\frac{b+2 c \cos (x)}{\sqrt{b^2-4 a c}}\right )}{c^2 \sqrt{b^2-4 a c}}+\frac{\cos (x)}{c}-\frac{b \log \left (a+b \cos (x)+c \cos ^2(x)\right )}{2 c^2}\\ \end{align*}
Mathematica [A] time = 0.256545, size = 131, normalized size = 1.72 \[ \frac{2 c \cos (x) \sqrt{b^2-4 a c}+\left (-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2\right ) \log \left (\sqrt{b^2-4 a c}-b-2 c \cos (x)\right )-\left (b \sqrt{b^2-4 a c}-2 c (a+c)+b^2\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c \cos (x)\right )}{2 c^2 \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 141, normalized size = 1.9 \begin{align*}{\frac{\cos \left ( x \right ) }{c}}-{\frac{b\ln \left ( a+b\cos \left ( x \right ) +c \left ( \cos \left ( x \right ) \right ) ^{2} \right ) }{2\,{c}^{2}}}-2\,{\frac{a}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{b+2\,c\cos \left ( x \right ) }{\sqrt{4\,ac-{b}^{2}}}} \right ) }-2\,{\frac{1}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{b+2\,c\cos \left ( x \right ) }{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{2}}{{c}^{2}}\arctan \left ({(b+2\,c\cos \left ( x \right ) ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \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 [A] time = 1.96448, size = 641, normalized size = 8.43 \begin{align*} \left [-\frac{{\left (b^{2} - 2 \, a c - 2 \, c^{2}\right )} \sqrt{b^{2} - 4 \, a c} \log \left (-\frac{2 \, c^{2} \cos \left (x\right )^{2} + 2 \, b c \cos \left (x\right ) + b^{2} - 2 \, a c + \sqrt{b^{2} - 4 \, a c}{\left (2 \, c \cos \left (x\right ) + b\right )}}{c \cos \left (x\right )^{2} + b \cos \left (x\right ) + a}\right ) - 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} \cos \left (x\right ) +{\left (b^{3} - 4 \, a b c\right )} \log \left (c \cos \left (x\right )^{2} + b \cos \left (x\right ) + a\right )}{2 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, -\frac{2 \,{\left (b^{2} - 2 \, a c - 2 \, c^{2}\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c \cos \left (x\right ) + b\right )}}{b^{2} - 4 \, a c}\right ) - 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} \cos \left (x\right ) +{\left (b^{3} - 4 \, a b c\right )} \log \left (c \cos \left (x\right )^{2} + b \cos \left (x\right ) + a\right )}{2 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\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 = 1.13827, size = 103, normalized size = 1.36 \begin{align*} \frac{\cos \left (x\right )}{c} - \frac{b \log \left (c \cos \left (x\right )^{2} + b \cos \left (x\right ) + a\right )}{2 \, c^{2}} + \frac{{\left (b^{2} - 2 \, a c - 2 \, c^{2}\right )} \arctan \left (\frac{2 \, c \cos \left (x\right ) + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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