Optimal. Leaf size=136 \[ -\frac{\cos (x) \left (b^2-c (a+2 c)\right )}{c^3}+\frac{b \left (b^2-2 c (a+c)\right ) \log \left (a+b \cos (x)+c \cos ^2(x)\right )}{2 c^4}+\frac{\left (-2 b^2 c (2 a+c)+2 c^2 (a+c)^2+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c \cos (x)}{\sqrt{b^2-4 a c}}\right )}{c^4 \sqrt{b^2-4 a c}}+\frac{b \cos ^2(x)}{2 c^2}-\frac{\cos ^3(x)}{3 c} \]
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Rubi [A] time = 0.23179, antiderivative size = 136, 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} \[ -\frac{\cos (x) \left (b^2-c (a+2 c)\right )}{c^3}+\frac{b \left (b^2-2 c (a+c)\right ) \log \left (a+b \cos (x)+c \cos ^2(x)\right )}{2 c^4}+\frac{\left (-2 b^2 c (2 a+c)+2 c^2 (a+c)^2+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c \cos (x)}{\sqrt{b^2-4 a c}}\right )}{c^4 \sqrt{b^2-4 a c}}+\frac{b \cos ^2(x)}{2 c^2}-\frac{\cos ^3(x)}{3 c} \]
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 ^5(x)}{a+b \cos (x)+c \cos ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^2}{a+b x+c x^2} \, dx,x,\cos (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{b^2-c (a+2 c)}{c^3}-\frac{b x}{c^2}+\frac{x^2}{c}-\frac{-a^2 c-c^3+a \left (b^2-2 c^2\right )+b \left (b^2-2 c (a+c)\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx,x,\cos (x)\right )\\ &=-\frac{\left (b^2-c (a+2 c)\right ) \cos (x)}{c^3}+\frac{b \cos ^2(x)}{2 c^2}-\frac{\cos ^3(x)}{3 c}+\frac{\operatorname{Subst}\left (\int \frac{-a^2 c-c^3+a \left (b^2-2 c^2\right )+b \left (b^2-2 c (a+c)\right ) x}{a+b x+c x^2} \, dx,x,\cos (x)\right )}{c^3}\\ &=-\frac{\left (b^2-c (a+2 c)\right ) \cos (x)}{c^3}+\frac{b \cos ^2(x)}{2 c^2}-\frac{\cos ^3(x)}{3 c}+\frac{\left (b \left (b^2-2 c (a+c)\right )\right ) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,\cos (x)\right )}{2 c^4}-\frac{\left (b^4+2 c^2 (a+c)^2-2 b^2 c (2 a+c)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,\cos (x)\right )}{2 c^4}\\ &=-\frac{\left (b^2-c (a+2 c)\right ) \cos (x)}{c^3}+\frac{b \cos ^2(x)}{2 c^2}-\frac{\cos ^3(x)}{3 c}+\frac{b \left (b^2-2 c (a+c)\right ) \log \left (a+b \cos (x)+c \cos ^2(x)\right )}{2 c^4}+\frac{\left (b^4+2 c^2 (a+c)^2-2 b^2 c (2 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^4}\\ &=\frac{\left (b^4+2 c^2 (a+c)^2-2 b^2 c (2 a+c)\right ) \tanh ^{-1}\left (\frac{b+2 c \cos (x)}{\sqrt{b^2-4 a c}}\right )}{c^4 \sqrt{b^2-4 a c}}-\frac{\left (b^2-c (a+2 c)\right ) \cos (x)}{c^3}+\frac{b \cos ^2(x)}{2 c^2}-\frac{\cos ^3(x)}{3 c}+\frac{b \left (b^2-2 c (a+c)\right ) \log \left (a+b \cos (x)+c \cos ^2(x)\right )}{2 c^4}\\ \end{align*}
Mathematica [A] time = 0.534214, size = 239, normalized size = 1.76 \[ \frac{\frac{6 \left (b^3 \sqrt{b^2-4 a c}+2 b^2 c (2 a+c)-2 b c (a+c) \sqrt{b^2-4 a c}-2 c^2 (a+c)^2-b^4\right ) \log \left (\sqrt{b^2-4 a c}-b-2 c \cos (x)\right )}{\sqrt{b^2-4 a c}}+\frac{6 \left (b^3 \sqrt{b^2-4 a c}-2 b^2 c (2 a+c)-2 b c (a+c) \sqrt{b^2-4 a c}+2 c^2 (a+c)^2+b^4\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c \cos (x)\right )}{\sqrt{b^2-4 a c}}+3 c \cos (x) \left (c (4 a+7 c)-4 b^2\right )+3 b c^2 \cos (2 x)+c^3 (-\cos (3 x))}{12 c^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.027, size = 344, normalized size = 2.5 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{3}}{3\,c}}+{\frac{b \left ( \cos \left ( x \right ) \right ) ^{2}}{2\,{c}^{2}}}+{\frac{\cos \left ( x \right ) a}{{c}^{2}}}-{\frac{\cos \left ( x \right ){b}^{2}}{{c}^{3}}}+2\,{\frac{\cos \left ( x \right ) }{c}}-{\frac{\ln \left ( a+b\cos \left ( x \right ) +c \left ( \cos \left ( x \right ) \right ) ^{2} \right ) ab}{{c}^{3}}}+{\frac{\ln \left ( a+b\cos \left ( x \right ) +c \left ( \cos \left ( x \right ) \right ) ^{2} \right ){b}^{3}}{2\,{c}^{4}}}-{\frac{b\ln \left ( a+b\cos \left ( x \right ) +c \left ( \cos \left ( x \right ) \right ) ^{2} \right ) }{{c}^{2}}}-2\,{\frac{{a}^{2}}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{b+2\,c\cos \left ( x \right ) }{\sqrt{4\,ac-{b}^{2}}}} \right ) }+4\,{\frac{a{b}^{2}}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{b+2\,c\cos \left ( x \right ) }{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\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}^{4}}{{c}^{4}}\arctan \left ({(b+2\,c\cos \left ( x \right ) ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+2\,{\frac{{b}^{2}}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{b+2\,c\cos \left ( x \right ) }{\sqrt{4\,ac-{b}^{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 [A] time = 2.59324, size = 1125, normalized size = 8.27 \begin{align*} \left [-\frac{2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \cos \left (x\right )^{3} - 3 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} \cos \left (x\right )^{2} - 3 \,{\left (b^{4} - 4 \, a b^{2} c + 4 \, a c^{3} + 2 \, c^{4} + 2 \,{\left (a^{2} - b^{2}\right )} 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 ) + 6 \,{\left (b^{4} c - 5 \, a b^{2} c^{2} + 8 \, a c^{4} + 2 \,{\left (2 \, a^{2} - b^{2}\right )} c^{3}\right )} \cos \left (x\right ) - 3 \,{\left (b^{5} - 6 \, a b^{3} c + 8 \, a b c^{3} + 2 \,{\left (4 \, a^{2} b - b^{3}\right )} c^{2}\right )} \log \left (c \cos \left (x\right )^{2} + b \cos \left (x\right ) + a\right )}{6 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}, -\frac{2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \cos \left (x\right )^{3} - 3 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} \cos \left (x\right )^{2} - 6 \,{\left (b^{4} - 4 \, a b^{2} c + 4 \, a c^{3} + 2 \, c^{4} + 2 \,{\left (a^{2} - b^{2}\right )} 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 ) + 6 \,{\left (b^{4} c - 5 \, a b^{2} c^{2} + 8 \, a c^{4} + 2 \,{\left (2 \, a^{2} - b^{2}\right )} c^{3}\right )} \cos \left (x\right ) - 3 \,{\left (b^{5} - 6 \, a b^{3} c + 8 \, a b c^{3} + 2 \,{\left (4 \, a^{2} b - b^{3}\right )} c^{2}\right )} \log \left (c \cos \left (x\right )^{2} + b \cos \left (x\right ) + a\right )}{6 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\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 = 1.14366, size = 207, normalized size = 1.52 \begin{align*} -\frac{2 \, c^{2} \cos \left (x\right )^{3} - 3 \, b c \cos \left (x\right )^{2} + 6 \, b^{2} \cos \left (x\right ) - 6 \, a c \cos \left (x\right ) - 12 \, c^{2} \cos \left (x\right )}{6 \, c^{3}} + \frac{{\left (b^{3} - 2 \, a b c - 2 \, b c^{2}\right )} \log \left (c \cos \left (x\right )^{2} + b \cos \left (x\right ) + a\right )}{2 \, c^{4}} - \frac{{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2} - 2 \, b^{2} c^{2} + 4 \, a c^{3} + 2 \, c^{4}\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^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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