Optimal. Leaf size=298 \[ \frac{\sin (c+d x) \left (-a^2 b (2 A-C)+a^3 B-2 a b^2 B+3 A b^3\right )}{a^3 d \left (a^2-b^2\right )}-\frac{\sin (c+d x) \cos (c+d x) \left (a^2 (-(A-2 C))-2 a b B+3 A b^2\right )}{2 a^2 d \left (a^2-b^2\right )}-\frac{2 b \left (4 a^2 A b^2-a^2 b^2 C-3 a^3 b B+2 a^4 C+2 a b^3 B-3 A b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac{x \left (a^2 (A+2 C)-4 a b B+6 A b^2\right )}{2 a^4} \]
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Rubi [A] time = 1.23513, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used = {4100, 4104, 3919, 3831, 2659, 208} \[ \frac{\sin (c+d x) \left (-a^2 b (2 A-C)+a^3 B-2 a b^2 B+3 A b^3\right )}{a^3 d \left (a^2-b^2\right )}-\frac{\sin (c+d x) \cos (c+d x) \left (a^2 (-(A-2 C))-2 a b B+3 A b^2\right )}{2 a^2 d \left (a^2-b^2\right )}-\frac{2 b \left (4 a^2 A b^2-a^2 b^2 C-3 a^3 b B+2 a^4 C+2 a b^3 B-3 A b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}+\frac{x \left (a^2 (A+2 C)-4 a b B+6 A b^2\right )}{2 a^4} \]
Antiderivative was successfully verified.
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Rule 4100
Rule 4104
Rule 3919
Rule 3831
Rule 2659
Rule 208
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx &=\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac{\int \frac{\cos ^2(c+d x) \left (3 A b^2-2 a b B-a^2 (A-2 C)+a (A b-a B+b C) \sec (c+d x)-2 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{a \left (a^2-b^2\right )}\\ &=-\frac{\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac{\int \frac{\cos (c+d x) \left (2 \left (3 A b^3+a^3 B-2 a b^2 B-a^2 b (2 A-C)\right )+a \left (A b^2-2 a b B+a^2 (A+2 C)\right ) \sec (c+d x)-b \left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )}\\ &=\frac{\left (3 A b^3+a^3 B-2 a b^2 B-a^2 b (2 A-C)\right ) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac{\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac{\int \frac{-\left (a^2-b^2\right ) \left (6 A b^2-4 a b B+a^2 (A+2 C)\right )+a b \left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^3 \left (a^2-b^2\right )}\\ &=\frac{\left (6 A b^2-4 a b B+a^2 (A+2 C)\right ) x}{2 a^4}+\frac{\left (3 A b^3+a^3 B-2 a b^2 B-a^2 b (2 A-C)\right ) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac{\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac{\left (b \left (3 A b^4+3 a^3 b B-2 a b^3 B-a^2 b^2 (4 A-C)-2 a^4 C\right )\right ) \int \frac{\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^4 \left (a^2-b^2\right )}\\ &=\frac{\left (6 A b^2-4 a b B+a^2 (A+2 C)\right ) x}{2 a^4}+\frac{\left (3 A b^3+a^3 B-2 a b^2 B-a^2 b (2 A-C)\right ) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac{\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac{\left (3 A b^4+3 a^3 b B-2 a b^3 B-a^2 b^2 (4 A-C)-2 a^4 C\right ) \int \frac{1}{1+\frac{a \cos (c+d x)}{b}} \, dx}{a^4 \left (a^2-b^2\right )}\\ &=\frac{\left (6 A b^2-4 a b B+a^2 (A+2 C)\right ) x}{2 a^4}+\frac{\left (3 A b^3+a^3 B-2 a b^2 B-a^2 b (2 A-C)\right ) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac{\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac{\left (2 \left (3 A b^4+3 a^3 b B-2 a b^3 B-a^2 b^2 (4 A-C)-2 a^4 C\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{a}{b}+\left (1-\frac{a}{b}\right ) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^4 \left (a^2-b^2\right ) d}\\ &=\frac{\left (6 A b^2-4 a b B+a^2 (A+2 C)\right ) x}{2 a^4}-\frac{2 b \left (4 a^2 A b^2-3 A b^4-3 a^3 b B+2 a b^3 B+2 a^4 C-a^2 b^2 C\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^4 (a-b)^{3/2} (a+b)^{3/2} d}+\frac{\left (3 A b^3+a^3 B-2 a b^2 B-a^2 b (2 A-C)\right ) \sin (c+d x)}{a^3 \left (a^2-b^2\right ) d}-\frac{\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) d}+\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 1.40944, size = 206, normalized size = 0.69 \[ \frac{2 (c+d x) \left (a^2 (A+2 C)-4 a b B+6 A b^2\right )-\frac{8 b \left (a^2 b^2 (C-4 A)+3 a^3 b B-2 a^4 C-2 a b^3 B+3 A b^4\right ) \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+a^2 A \sin (2 (c+d x))+\frac{4 a b^2 \sin (c+d x) \left (a (a C-b B)+A b^2\right )}{(a-b) (a+b) (a \cos (c+d x)+b)}+4 a (a B-2 A b) \sin (c+d x)}{4 a^4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.138, size = 857, normalized size = 2.9 \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 = 0.819258, size = 2398, normalized size = 8.05 \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 = 1.27657, size = 513, normalized size = 1.72 \begin{align*} -\frac{\frac{4 \,{\left (2 \, C a^{4} b - 3 \, B a^{3} b^{2} + 4 \, A a^{2} b^{3} - C a^{2} b^{3} + 2 \, B a b^{4} - 3 \, A b^{5}\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^{6} - a^{4} b^{2}\right )} \sqrt{-a^{2} + b^{2}}} + \frac{4 \,{\left (C a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - B a b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + A b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (a^{5} - a^{3} b^{2}\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 (A a^{2} + 2 \, C a^{2} - 4 \, B a b + 6 \, A b^{2}\right )}{\left (d x + c\right )}}{a^{4}} + \frac{2 \,{\left (A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 4 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 4 \, A 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} a^{3}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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