Optimal. Leaf size=303 \[ \frac{\sin (c+d x) \left (2 a^2 b B-3 a^3 C-a b^2 (A-2 C)-b^3 B\right )}{b^3 d \left (a^2-b^2\right )}-\frac{2 a \left (a^2 A b^2-4 a^2 b^2 C-2 a^3 b B+3 a^4 C+3 a b^3 B-2 A b^4\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \cos (c+d x) \left (3 a^2 C-2 a b B+2 A b^2-b^2 C\right )}{2 b^2 d \left (a^2-b^2\right )}+\frac{x \left (6 a^2 C-4 a b B+2 A b^2+b^2 C\right )}{2 b^4} \]
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Rubi [A] time = 1.11738, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used = {3047, 3049, 3023, 2735, 2659, 205} \[ \frac{\sin (c+d x) \left (2 a^2 b B-3 a^3 C-a b^2 (A-2 C)-b^3 B\right )}{b^3 d \left (a^2-b^2\right )}-\frac{2 a \left (a^2 A b^2-4 a^2 b^2 C-2 a^3 b B+3 a^4 C+3 a b^3 B-2 A b^4\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \cos (c+d x) \left (3 a^2 C-2 a b B+2 A b^2-b^2 C\right )}{2 b^2 d \left (a^2-b^2\right )}+\frac{x \left (6 a^2 C-4 a b B+2 A b^2+b^2 C\right )}{2 b^4} \]
Antiderivative was successfully verified.
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Rule 3047
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))^2} \, dx &=-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac{\int \frac{\cos (c+d x) \left (2 \left (A b^2-a (b B-a C)\right )+b (b B-a (A+C)) \cos (c+d x)-\left (2 A b^2-2 a b B+3 a^2 C-b^2 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{b \left (a^2-b^2\right )}\\ &=\frac{\left (2 A b^2-2 a b B+3 a^2 C-b^2 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac{\int \frac{-a \left (2 A b^2-2 a b B+3 a^2 C-b^2 C\right )+b \left (2 A b^2-2 a b B+a^2 C+b^2 C\right ) \cos (c+d x)-2 \left (2 a^2 b B-b^3 B-a b^2 (A-2 C)-3 a^3 C\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )}\\ &=\frac{\left (2 a^2 b B-b^3 B-a b^2 (A-2 C)-3 a^3 C\right ) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (2 A b^2-2 a b B+3 a^2 C-b^2 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac{\int \frac{-a b \left (2 A b^2-2 a b B+3 a^2 C-b^2 C\right )-\left (a^2-b^2\right ) \left (2 A b^2-4 a b B+6 a^2 C+b^2 C\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )}\\ &=\frac{\left (2 A b^2-4 a b B+6 a^2 C+b^2 C\right ) x}{2 b^4}+\frac{\left (2 a^2 b B-b^3 B-a b^2 (A-2 C)-3 a^3 C\right ) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (2 A b^2-2 a b B+3 a^2 C-b^2 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac{\left (a \left (2 A b^4+2 a^3 b B-3 a b^3 B-a^2 b^2 (A-4 C)-3 a^4 C\right )\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{b^4 \left (a^2-b^2\right )}\\ &=\frac{\left (2 A b^2-4 a b B+6 a^2 C+b^2 C\right ) x}{2 b^4}+\frac{\left (2 a^2 b B-b^3 B-a b^2 (A-2 C)-3 a^3 C\right ) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (2 A b^2-2 a b B+3 a^2 C-b^2 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac{\left (2 a \left (2 A b^4+2 a^3 b B-3 a b^3 B-a^2 b^2 (A-4 C)-3 a^4 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 \left (a^2-b^2\right ) d}\\ &=\frac{\left (2 A b^2-4 a b B+6 a^2 C+b^2 C\right ) x}{2 b^4}-\frac{2 a \left (a^2 A b^2-2 A b^4-2 a^3 b B+3 a b^3 B+3 a^4 C-4 a^2 b^2 C\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{(a-b)^{3/2} b^4 (a+b)^{3/2} d}+\frac{\left (2 a^2 b B-b^3 B-a b^2 (A-2 C)-3 a^3 C\right ) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d}+\frac{\left (2 A b^2-2 a b B+3 a^2 C-b^2 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x))}\\ \end{align*}
Mathematica [A] time = 1.3755, size = 208, normalized size = 0.69 \[ \frac{2 (c+d x) \left (6 a^2 C-4 a b B+2 A b^2+b^2 C\right )-\frac{4 a^2 b \sin (c+d x) \left (a (a C-b B)+A b^2\right )}{(a-b) (a+b) (a+b \cos (c+d x))}-\frac{8 a \left (a^2 b^2 (A-4 C)-2 a^3 b B+3 a^4 C+3 a b^3 B-2 A b^4\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\left (b^2-a^2\right )^{3/2}}+4 b (b B-2 a C) \sin (c+d x)+b^2 C \sin (2 (c+d x))}{4 b^4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.043, size = 845, normalized size = 2.8 \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 = 2.54595, size = 2381, normalized size = 7.86 \begin{align*} \text{result too large to display} \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 = 1.24799, size = 508, normalized size = 1.68 \begin{align*} \frac{\frac{4 \,{\left (3 \, C a^{5} - 2 \, B a^{4} b + A a^{3} b^{2} - 4 \, C a^{3} b^{2} + 3 \, B a^{2} b^{3} - 2 \, A a b^{4}\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 )}}{{\left (a^{2} b^{4} - b^{6}\right )} \sqrt{a^{2} - b^{2}}} - \frac{4 \,{\left (C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - B a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + A a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (a^{2} b^{3} - b^{5}\right )}{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a + b\right )}} + \frac{{\left (6 \, C a^{2} - 4 \, B a b + 2 \, A b^{2} + C b^{2}\right )}{\left (d x + c\right )}}{b^{4}} - \frac{2 \,{\left (4 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 4 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - C b \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 )}^{2} b^{3}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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