Optimal. Leaf size=103 \[ \frac {(A-B+4 C) \sin (c+d x)}{3 a^2 d}-\frac {(B-2 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac {x (B-2 C)}{a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.26, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3041, 2968, 3023, 12, 2735, 2648} \[ \frac {(A-B+4 C) \sin (c+d x)}{3 a^2 d}-\frac {(B-2 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac {x (B-2 C)}{a^2}-\frac {(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2648
Rule 2735
Rule 2968
Rule 3023
Rule 3041
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx &=-\frac {(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {\int \frac {\cos (c+d x) (a (A+2 B-2 C)+a (A-B+4 C) \cos (c+d x))}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=-\frac {(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {\int \frac {a (A+2 B-2 C) \cos (c+d x)+a (A-B+4 C) \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{3 a^2}\\ &=\frac {(A-B+4 C) \sin (c+d x)}{3 a^2 d}-\frac {(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {\int \frac {3 a^2 (B-2 C) \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{3 a^3}\\ &=\frac {(A-B+4 C) \sin (c+d x)}{3 a^2 d}-\frac {(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {(B-2 C) \int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx}{a}\\ &=\frac {(B-2 C) x}{a^2}+\frac {(A-B+4 C) \sin (c+d x)}{3 a^2 d}-\frac {(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac {(B-2 C) \int \frac {1}{a+a \cos (c+d x)} \, dx}{a}\\ &=\frac {(B-2 C) x}{a^2}+\frac {(A-B+4 C) \sin (c+d x)}{3 a^2 d}-\frac {(A-B+C) \cos ^2(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac {(B-2 C) \sin (c+d x)}{d \left (a^2+a^2 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 0.69, size = 275, normalized size = 2.67 \[ \frac {\sec \left (\frac {c}{2}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \left (-12 A \sin \left (c+\frac {d x}{2}\right )+8 A \sin \left (c+\frac {3 d x}{2}\right )+12 A \sin \left (\frac {d x}{2}\right )+18 d x (B-2 C) \cos \left (c+\frac {d x}{2}\right )+24 B \sin \left (c+\frac {d x}{2}\right )-20 B \sin \left (c+\frac {3 d x}{2}\right )+6 B d x \cos \left (c+\frac {3 d x}{2}\right )+6 B d x \cos \left (2 c+\frac {3 d x}{2}\right )+18 d x (B-2 C) \cos \left (\frac {d x}{2}\right )-36 B \sin \left (\frac {d x}{2}\right )-30 C \sin \left (c+\frac {d x}{2}\right )+41 C \sin \left (c+\frac {3 d x}{2}\right )+9 C \sin \left (2 c+\frac {3 d x}{2}\right )+3 C \sin \left (2 c+\frac {5 d x}{2}\right )+3 C \sin \left (3 c+\frac {5 d x}{2}\right )-12 C d x \cos \left (c+\frac {3 d x}{2}\right )-12 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+66 C \sin \left (\frac {d x}{2}\right )\right )}{12 a^2 d (\cos (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 120, normalized size = 1.17 \[ \frac {3 \, {\left (B - 2 \, C\right )} d x \cos \left (d x + c\right )^{2} + 6 \, {\left (B - 2 \, C\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (B - 2 \, C\right )} d x + {\left (3 \, C \cos \left (d x + c\right )^{2} + {\left (2 \, A - 5 \, B + 14 \, C\right )} \cos \left (d x + c\right ) + A - 4 \, B + 10 \, C\right )} \sin \left (d x + c\right )}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.73, size = 151, normalized size = 1.47 \[ \frac {\frac {6 \, {\left (d x + c\right )} {\left (B - 2 \, C\right )}}{a^{2}} + \frac {12 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{2}} - \frac {A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 187, normalized size = 1.82 \[ -\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{6 d \,a^{2}}+\frac {B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d \,a^{2}}-\frac {C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d \,a^{2}}+\frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}-\frac {3 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}+\frac {5 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}+\frac {2 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{d \,a^{2}}-\frac {4 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 235, normalized size = 2.28 \[ \frac {C {\left (\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {24 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} + \frac {12 \, \sin \left (d x + c\right )}{{\left (a^{2} + \frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - B {\left (\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2}} - \frac {12 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )} + \frac {A {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 107, normalized size = 1.04 \[ \frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {A-B+C}{a^2}-\frac {A+B-3\,C}{2\,a^2}\right )}{d}+\frac {x\,\left (B-2\,C\right )}{a^2}+\frac {2\,C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (A-B+C\right )}{6\,a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.97, size = 536, normalized size = 5.20 \[ \begin {cases} - \frac {A \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} + \frac {2 A \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} + \frac {3 A \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} + \frac {6 B d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} + \frac {6 B d x}{6 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} + \frac {B \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} - \frac {8 B \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} - \frac {9 B \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} - \frac {12 C d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} - \frac {12 C d x}{6 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} - \frac {C \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} + \frac {14 C \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} + \frac {27 C \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 6 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \left (A + B \cos {\relax (c )} + C \cos ^{2}{\relax (c )}\right ) \cos {\relax (c )}}{\left (a \cos {\relax (c )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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