Optimal. Leaf size=206 \[ \frac{\sin (c+d x) \left (3 a^2 C-3 a b B+3 A b^2+2 b^2 C\right )}{3 b^3 d}+\frac{2 a^2 \left (A b^2-a (b B-a C)\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^4 d \sqrt{a-b} \sqrt{a+b}}+\frac{x \left (2 a^2 b B-2 a^3 C-a b^2 (2 A+C)+b^3 B\right )}{2 b^4}+\frac{(b B-a C) \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{C \sin (c+d x) \cos ^2(c+d x)}{3 b d} \]
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Rubi [A] time = 0.567419, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.122, Rules used = {3049, 3023, 2735, 2659, 205} \[ \frac{\sin (c+d x) \left (3 a^2 C-3 a b B+3 A b^2+2 b^2 C\right )}{3 b^3 d}+\frac{2 a^2 \left (A b^2-a (b B-a C)\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^4 d \sqrt{a-b} \sqrt{a+b}}+\frac{x \left (2 a^2 b B-2 a^3 C-a b^2 (2 A+C)+b^3 B\right )}{2 b^4}+\frac{(b B-a C) \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{C \sin (c+d x) \cos ^2(c+d x)}{3 b d} \]
Antiderivative was successfully verified.
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Rule 3049
Rule 3023
Rule 2735
Rule 2659
Rule 205
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx &=\frac{C \cos ^2(c+d x) \sin (c+d x)}{3 b d}+\frac{\int \frac{\cos (c+d x) \left (2 a C+b (3 A+2 C) \cos (c+d x)+3 (b B-a C) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{3 b}\\ &=\frac{(b B-a C) \cos (c+d x) \sin (c+d x)}{2 b^2 d}+\frac{C \cos ^2(c+d x) \sin (c+d x)}{3 b d}+\frac{\int \frac{3 a (b B-a C)+b (3 b B+a C) \cos (c+d x)+2 \left (3 A b^2-3 a b B+3 a^2 C+2 b^2 C\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^2}\\ &=\frac{\left (3 A b^2-3 a b B+3 a^2 C+2 b^2 C\right ) \sin (c+d x)}{3 b^3 d}+\frac{(b B-a C) \cos (c+d x) \sin (c+d x)}{2 b^2 d}+\frac{C \cos ^2(c+d x) \sin (c+d x)}{3 b d}+\frac{\int \frac{3 a b (b B-a C)+3 \left (2 a^2 b B+b^3 B-2 a^3 C-a b^2 (2 A+C)\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^3}\\ &=\frac{\left (2 a^2 b B+b^3 B-2 a^3 C-a b^2 (2 A+C)\right ) x}{2 b^4}+\frac{\left (3 A b^2-3 a b B+3 a^2 C+2 b^2 C\right ) \sin (c+d x)}{3 b^3 d}+\frac{(b B-a C) \cos (c+d x) \sin (c+d x)}{2 b^2 d}+\frac{C \cos ^2(c+d x) \sin (c+d x)}{3 b d}+\frac{\left (a^2 \left (A b^2-a (b B-a C)\right )\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{b^4}\\ &=\frac{\left (2 a^2 b B+b^3 B-2 a^3 C-a b^2 (2 A+C)\right ) x}{2 b^4}+\frac{\left (3 A b^2-3 a b B+3 a^2 C+2 b^2 C\right ) \sin (c+d x)}{3 b^3 d}+\frac{(b B-a C) \cos (c+d x) \sin (c+d x)}{2 b^2 d}+\frac{C \cos ^2(c+d x) \sin (c+d x)}{3 b d}+\frac{\left (2 a^2 \left (A b^2-a (b B-a C)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{b^4 d}\\ &=\frac{\left (2 a^2 b B+b^3 B-2 a^3 C-a b^2 (2 A+C)\right ) x}{2 b^4}+\frac{2 a^2 \left (A b^2-a (b B-a C)\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} b^4 \sqrt{a+b} d}+\frac{\left (3 A b^2-3 a b B+3 a^2 C+2 b^2 C\right ) \sin (c+d x)}{3 b^3 d}+\frac{(b B-a C) \cos (c+d x) \sin (c+d x)}{2 b^2 d}+\frac{C \cos ^2(c+d x) \sin (c+d x)}{3 b d}\\ \end{align*}
Mathematica [A] time = 0.650606, size = 179, normalized size = 0.87 \[ \frac{-6 (c+d x) \left (-2 a^2 b B+2 a^3 C+a b^2 (2 A+C)-b^3 B\right )+3 b \sin (c+d x) \left (4 a^2 C-4 a b B+4 A b^2+3 b^2 C\right )-\frac{24 a^2 \left (a (a C-b B)+A b^2\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}}+3 b^2 (b B-a C) \sin (2 (c+d x))+b^3 C \sin (3 (c+d x))}{12 b^4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 814, normalized size = 4. \begin{align*} \text{result too large to display} \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 = 2.00078, size = 1288, normalized size = 6.25 \begin{align*} \left [-\frac{3 \,{\left (2 \, C a^{5} - 2 \, B a^{4} b +{\left (2 \, A - C\right )} a^{3} b^{2} + B a^{2} b^{3} -{\left (2 \, A + C\right )} a b^{4} + B b^{5}\right )} d x + 3 \,{\left (C a^{4} - B a^{3} b + A a^{2} b^{2}\right )} \sqrt{-a^{2} + b^{2}} \log \left (\frac{2 \, a b \cos \left (d x + c\right ) +{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt{-a^{2} + b^{2}}{\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) -{\left (6 \, C a^{4} b - 6 \, B a^{3} b^{2} + 2 \,{\left (3 \, A - C\right )} a^{2} b^{3} + 6 \, B a b^{4} - 2 \,{\left (3 \, A + 2 \, C\right )} b^{5} + 2 \,{\left (C a^{2} b^{3} - C b^{5}\right )} \cos \left (d x + c\right )^{2} - 3 \,{\left (C a^{3} b^{2} - B a^{2} b^{3} - C a b^{4} + B b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \,{\left (a^{2} b^{4} - b^{6}\right )} d}, -\frac{3 \,{\left (2 \, C a^{5} - 2 \, B a^{4} b +{\left (2 \, A - C\right )} a^{3} b^{2} + B a^{2} b^{3} -{\left (2 \, A + C\right )} a b^{4} + B b^{5}\right )} d x - 6 \,{\left (C a^{4} - B a^{3} b + A a^{2} b^{2}\right )} \sqrt{a^{2} - b^{2}} \arctan \left (-\frac{a \cos \left (d x + c\right ) + b}{\sqrt{a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) -{\left (6 \, C a^{4} b - 6 \, B a^{3} b^{2} + 2 \,{\left (3 \, A - C\right )} a^{2} b^{3} + 6 \, B a b^{4} - 2 \,{\left (3 \, A + 2 \, C\right )} b^{5} + 2 \,{\left (C a^{2} b^{3} - C b^{5}\right )} \cos \left (d x + c\right )^{2} - 3 \,{\left (C a^{3} b^{2} - B a^{2} b^{3} - C a b^{4} + B b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \,{\left (a^{2} b^{4} - b^{6}\right )} d}\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 [B] time = 1.30182, size = 572, normalized size = 2.78 \begin{align*} -\frac{\frac{3 \,{\left (2 \, C a^{3} - 2 \, B a^{2} b + 2 \, A a b^{2} + C a b^{2} - B b^{3}\right )}{\left (d x + c\right )}}{b^{4}} + \frac{12 \,{\left (C a^{4} - B a^{3} b + A a^{2} b^{2}\right )}{\left (\pi \left \lfloor \frac{d x + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\sqrt{a^{2} - b^{2}}}\right )\right )}}{\sqrt{a^{2} - b^{2}} b^{4}} - \frac{2 \,{\left (6 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 6 \, B a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, B b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 12 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 12 \, B a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 12 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 4 \, C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, C a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 6 \, B a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \, C a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, B b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, C b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{3} b^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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