Optimal. Leaf size=139 \[ -\frac{(3 A-3 B+4 C) \sin ^3(c+d x)}{3 a d}+\frac{(3 A-3 B+4 C) \sin (c+d x)}{a d}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(2 A-3 B+3 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x (2 A-3 B+3 C)}{2 a} \]
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Rubi [A] time = 0.211874, antiderivative size = 139, 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 = {3041, 2748, 2635, 8, 2633} \[ -\frac{(3 A-3 B+4 C) \sin ^3(c+d x)}{3 a d}+\frac{(3 A-3 B+4 C) \sin (c+d x)}{a d}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(2 A-3 B+3 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x (2 A-3 B+3 C)}{2 a} \]
Antiderivative was successfully verified.
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Rule 3041
Rule 2748
Rule 2635
Rule 8
Rule 2633
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+a \cos (c+d x)} \, dx &=-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac{\int \cos ^2(c+d x) (-a (2 A-3 B+3 C)+a (3 A-3 B+4 C) \cos (c+d x)) \, dx}{a^2}\\ &=-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{(2 A-3 B+3 C) \int \cos ^2(c+d x) \, dx}{a}+\frac{(3 A-3 B+4 C) \int \cos ^3(c+d x) \, dx}{a}\\ &=-\frac{(2 A-3 B+3 C) \cos (c+d x) \sin (c+d x)}{2 a d}-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{(2 A-3 B+3 C) \int 1 \, dx}{2 a}-\frac{(3 A-3 B+4 C) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a d}\\ &=-\frac{(2 A-3 B+3 C) x}{2 a}+\frac{(3 A-3 B+4 C) \sin (c+d x)}{a d}-\frac{(2 A-3 B+3 C) \cos (c+d x) \sin (c+d x)}{2 a d}-\frac{(A-B+C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{(3 A-3 B+4 C) \sin ^3(c+d x)}{3 a d}\\ \end{align*}
Mathematica [B] time = 0.79562, size = 307, normalized size = 2.21 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (-12 d x (2 A-3 B+3 C) \cos \left (c+\frac{d x}{2}\right )-12 d x (2 A-3 B+3 C) \cos \left (\frac{d x}{2}\right )+12 A \sin \left (c+\frac{d x}{2}\right )+12 A \sin \left (c+\frac{3 d x}{2}\right )+12 A \sin \left (2 c+\frac{3 d x}{2}\right )+60 A \sin \left (\frac{d x}{2}\right )-12 B \sin \left (c+\frac{d x}{2}\right )-9 B \sin \left (c+\frac{3 d x}{2}\right )-9 B \sin \left (2 c+\frac{3 d x}{2}\right )+3 B \sin \left (2 c+\frac{5 d x}{2}\right )+3 B \sin \left (3 c+\frac{5 d x}{2}\right )-60 B \sin \left (\frac{d x}{2}\right )+21 C \sin \left (c+\frac{d x}{2}\right )+18 C \sin \left (c+\frac{3 d x}{2}\right )+18 C \sin \left (2 c+\frac{3 d x}{2}\right )-2 C \sin \left (2 c+\frac{5 d x}{2}\right )-2 C \sin \left (3 c+\frac{5 d x}{2}\right )+C \sin \left (3 c+\frac{7 d x}{2}\right )+C \sin \left (4 c+\frac{7 d x}{2}\right )+69 C \sin \left (\frac{d x}{2}\right )\right )}{24 a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 420, normalized size = 3. \begin{align*}{\frac{A}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{B}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{C}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-3\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}B}{ad \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+2\,{\frac{A \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}}{ad \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+5\,{\frac{C \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}}{ad \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}-4\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}B}{ad \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+4\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}A}{ad \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+{\frac{16\,C}{3\,ad} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+1 \right ) ^{-3}}-{\frac{B}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+1 \right ) ^{-3}}+2\,{\frac{A\tan \left ( 1/2\,dx+c/2 \right ) }{ad \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}+3\,{\frac{C\tan \left ( 1/2\,dx+c/2 \right ) }{ad \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}-2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) A}{ad}}+3\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) B}{ad}}-3\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) C}{ad}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.48633, size = 540, normalized size = 3.88 \begin{align*} \frac{C{\left (\frac{\frac{9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a + \frac{3 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac{9 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{3 \, \sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - 3 \, B{\left (\frac{\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a + \frac{2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac{3 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{\sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - 3 \, A{\left (\frac{2 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac{2 \, \sin \left (d x + c\right )}{{\left (a + \frac{a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}} - \frac{\sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94933, size = 288, normalized size = 2.07 \begin{align*} -\frac{3 \,{\left (2 \, A - 3 \, B + 3 \, C\right )} d x \cos \left (d x + c\right ) + 3 \,{\left (2 \, A - 3 \, B + 3 \, C\right )} d x -{\left (2 \, C \cos \left (d x + c\right )^{3} +{\left (3 \, B - C\right )} \cos \left (d x + c\right )^{2} +{\left (6 \, A - 3 \, B + 7 \, C\right )} \cos \left (d x + c\right ) + 12 \, A - 12 \, B + 16 \, C\right )} \sin \left (d x + c\right )}{6 \,{\left (a d \cos \left (d x + c\right ) + a d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.7455, size = 1739, normalized size = 12.51 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19232, size = 279, normalized size = 2.01 \begin{align*} -\frac{\frac{3 \,{\left (d x + c\right )}{\left (2 \, A - 3 \, B + 3 \, C\right )}}{a} - \frac{6 \,{\left (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 ) + C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a} - \frac{2 \,{\left (6 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 9 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 15 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 12 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 12 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 16 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 9 \, C \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} a}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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