Optimal. Leaf size=153 \[ -\frac{4 (A-B) \sin ^3(c+d x)}{3 a d}+\frac{4 (A-B) \sin (c+d x)}{a d}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(4 A-5 B) \sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac{3 (4 A-5 B) \sin (c+d x) \cos (c+d x)}{8 a d}-\frac{3 x (4 A-5 B)}{8 a} \]
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Rubi [A] time = 0.206986, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {2977, 2748, 2633, 2635, 8} \[ -\frac{4 (A-B) \sin ^3(c+d x)}{3 a d}+\frac{4 (A-B) \sin (c+d x)}{a d}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(4 A-5 B) \sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac{3 (4 A-5 B) \sin (c+d x) \cos (c+d x)}{8 a d}-\frac{3 x (4 A-5 B)}{8 a} \]
Antiderivative was successfully verified.
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Rule 2977
Rule 2748
Rule 2633
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx &=\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac{\int \cos ^3(c+d x) (4 a (A-B)-a (4 A-5 B) \cos (c+d x)) \, dx}{a^2}\\ &=\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{(4 A-5 B) \int \cos ^4(c+d x) \, dx}{a}+\frac{(4 (A-B)) \int \cos ^3(c+d x) \, dx}{a}\\ &=-\frac{(4 A-5 B) \cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{(3 (4 A-5 B)) \int \cos ^2(c+d x) \, dx}{4 a}-\frac{(4 (A-B)) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a d}\\ &=\frac{4 (A-B) \sin (c+d x)}{a d}-\frac{3 (4 A-5 B) \cos (c+d x) \sin (c+d x)}{8 a d}-\frac{(4 A-5 B) \cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{4 (A-B) \sin ^3(c+d x)}{3 a d}-\frac{(3 (4 A-5 B)) \int 1 \, dx}{8 a}\\ &=-\frac{3 (4 A-5 B) x}{8 a}+\frac{4 (A-B) \sin (c+d x)}{a d}-\frac{3 (4 A-5 B) \cos (c+d x) \sin (c+d x)}{8 a d}-\frac{(4 A-5 B) \cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac{(A-B) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{4 (A-B) \sin ^3(c+d x)}{3 a d}\\ \end{align*}
Mathematica [B] time = 0.607227, size = 311, normalized size = 2.03 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (-72 d x (4 A-5 B) \cos \left (c+\frac{d x}{2}\right )-72 d x (4 A-5 B) \cos \left (\frac{d x}{2}\right )+168 A \sin \left (c+\frac{d x}{2}\right )+144 A \sin \left (c+\frac{3 d x}{2}\right )+144 A \sin \left (2 c+\frac{3 d x}{2}\right )-16 A \sin \left (2 c+\frac{5 d x}{2}\right )-16 A \sin \left (3 c+\frac{5 d x}{2}\right )+8 A \sin \left (3 c+\frac{7 d x}{2}\right )+8 A \sin \left (4 c+\frac{7 d x}{2}\right )+552 A \sin \left (\frac{d x}{2}\right )-168 B \sin \left (c+\frac{d x}{2}\right )-120 B \sin \left (c+\frac{3 d x}{2}\right )-120 B \sin \left (2 c+\frac{3 d x}{2}\right )+40 B \sin \left (2 c+\frac{5 d x}{2}\right )+40 B \sin \left (3 c+\frac{5 d x}{2}\right )-5 B \sin \left (3 c+\frac{7 d x}{2}\right )-5 B \sin \left (4 c+\frac{7 d x}{2}\right )+3 B \sin \left (4 c+\frac{9 d x}{2}\right )+3 B \sin \left (5 c+\frac{9 d x}{2}\right )-552 B \sin \left (\frac{d x}{2}\right )\right )}{192 a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.069, size = 351, normalized size = 2.3 \begin{align*}{\frac{A}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{B}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{25\,B}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}+5\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}A}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{4}}}-{\frac{115\,B}{12\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}+{\frac{31\,A}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-{\frac{109\,B}{12\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}+{\frac{25\,A}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-{\frac{7\,B}{4\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}+3\,{\frac{A\tan \left ( 1/2\,dx+c/2 \right ) }{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{4}}}-3\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) A}{da}}+{\frac{15\,B}{4\,da}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.53736, size = 532, normalized size = 3.48 \begin{align*} -\frac{B{\left (\frac{\frac{21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{109 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{115 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{75 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a + \frac{4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac{45 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{12 \, \sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - 4 \, A{\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 )}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37627, size = 301, normalized size = 1.97 \begin{align*} -\frac{9 \,{\left (4 \, A - 5 \, B\right )} d x \cos \left (d x + c\right ) + 9 \,{\left (4 \, A - 5 \, B\right )} d x -{\left (6 \, B \cos \left (d x + c\right )^{4} + 2 \,{\left (4 \, A - B\right )} \cos \left (d x + c\right )^{3} -{\left (4 \, A - 13 \, B\right )} \cos \left (d x + c\right )^{2} +{\left (28 \, A - 19 \, B\right )} \cos \left (d x + c\right ) + 64 \, A - 64 \, B\right )} \sin \left (d x + c\right )}{24 \,{\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 = 13.353, size = 1794, normalized size = 11.73 \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 = 1.18388, size = 244, normalized size = 1.59 \begin{align*} -\frac{\frac{9 \,{\left (d x + c\right )}{\left (4 \, A - 5 \, B\right )}}{a} - \frac{24 \,{\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 )\right )}}{a} - \frac{2 \,{\left (60 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 75 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 124 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 115 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 100 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 109 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 36 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 21 \, 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 )}^{4} a}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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