Optimal. Leaf size=122 \[ \frac {(3 B-4 C) \sin ^3(c+d x)}{3 a d}-\frac {(3 B-4 C) \sin (c+d x)}{a d}+\frac {(B-C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}+\frac {3 (B-C) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac {3 x (B-C)}{2 a} \]
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Rubi [A] time = 0.26, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3029, 2977, 2748, 2635, 8, 2633} \[ \frac {(3 B-4 C) \sin ^3(c+d x)}{3 a d}-\frac {(3 B-4 C) \sin (c+d x)}{a d}+\frac {(B-C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}+\frac {3 (B-C) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac {3 x (B-C)}{2 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2748
Rule 2977
Rule 3029
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx &=\int \frac {\cos ^3(c+d x) (B+C \cos (c+d x))}{a+a \cos (c+d x)} \, dx\\ &=\frac {(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) (3 a (B-C)-a (3 B-4 C) \cos (c+d x)) \, dx}{a^2}\\ &=\frac {(B-C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac {(3 B-4 C) \int \cos ^3(c+d x) \, dx}{a}+\frac {(3 (B-C)) \int \cos ^2(c+d x) \, dx}{a}\\ &=\frac {3 (B-C) \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {(B-C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {(3 (B-C)) \int 1 \, dx}{2 a}+\frac {(3 B-4 C) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a d}\\ &=\frac {3 (B-C) x}{2 a}-\frac {(3 B-4 C) \sin (c+d x)}{a d}+\frac {3 (B-C) \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {(B-C) \cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {(3 B-4 C) \sin ^3(c+d x)}{3 a d}\\ \end {align*}
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Mathematica [B] time = 0.57, size = 249, normalized size = 2.04 \[ \frac {\sec \left (\frac {c}{2}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \left (36 d x (B-C) \cos \left (c+\frac {d x}{2}\right )-12 B \sin \left (c+\frac {d x}{2}\right )-9 B \sin \left (c+\frac {3 d x}{2}\right )-9 B \sin \left (2 c+\frac {3 d x}{2}\right )+3 B \sin \left (2 c+\frac {5 d x}{2}\right )+3 B \sin \left (3 c+\frac {5 d x}{2}\right )+36 d x (B-C) \cos \left (\frac {d x}{2}\right )-60 B \sin \left (\frac {d x}{2}\right )+21 C \sin \left (c+\frac {d x}{2}\right )+18 C \sin \left (c+\frac {3 d x}{2}\right )+18 C \sin \left (2 c+\frac {3 d x}{2}\right )-2 C \sin \left (2 c+\frac {5 d x}{2}\right )-2 C \sin \left (3 c+\frac {5 d x}{2}\right )+C \sin \left (3 c+\frac {7 d x}{2}\right )+C \sin \left (4 c+\frac {7 d x}{2}\right )+69 C \sin \left (\frac {d x}{2}\right )\right )}{24 a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 98, normalized size = 0.80 \[ \frac {9 \, {\left (B - C\right )} d x \cos \left (d x + c\right ) + 9 \, {\left (B - C\right )} d x + {\left (2 \, C \cos \left (d x + c\right )^{3} + {\left (3 \, B - C\right )} \cos \left (d x + c\right )^{2} - {\left (3 \, B - 7 \, C\right )} \cos \left (d x + c\right ) - 12 \, B + 16 \, C\right )} \sin \left (d x + c\right )}{6 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 151, normalized size = 1.24 \[ \frac {\frac {9 \, {\left (d x + c\right )} {\left (B - C\right )}}{a} - \frac {6 \, {\left (B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a} - \frac {2 \, {\left (9 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} a}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 281, normalized size = 2.30 \[ -\frac {B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {5 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {4 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {16 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{3 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {3 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{a d}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.08, size = 310, normalized size = 2.54 \[ \frac {C {\left (\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a + \frac {3 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {9 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {3 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - 3 \, B {\left (\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.22, size = 138, normalized size = 1.13 \[ \frac {3\,x\,\left (B-C\right )}{2\,a}-\frac {\left (3\,B-5\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (4\,B-\frac {16\,C}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (B-3\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (B-C\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.48, size = 1166, normalized size = 9.56 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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