Optimal. Leaf size=108 \[ -\frac {(2 A-2 B+C) \sin (c+d x)}{a d}+\frac {(3 A-2 B+2 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {(A-B+C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}+\frac {x (3 A-2 B+2 C)}{2 a} \]
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Rubi [A] time = 0.18, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {4084, 3787, 2635, 8, 2637} \[ -\frac {(2 A-2 B+C) \sin (c+d x)}{a d}+\frac {(3 A-2 B+2 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {(A-B+C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}+\frac {x (3 A-2 B+2 C)}{2 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2637
Rule 3787
Rule 4084
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+a \sec (c+d x)} \, dx &=-\frac {(A-B+C) \cos (c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac {\int \cos ^2(c+d x) (a (3 A-2 B+2 C)-a (2 A-2 B+C) \sec (c+d x)) \, dx}{a^2}\\ &=-\frac {(A-B+C) \cos (c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac {(2 A-2 B+C) \int \cos (c+d x) \, dx}{a}+\frac {(3 A-2 B+2 C) \int \cos ^2(c+d x) \, dx}{a}\\ &=-\frac {(2 A-2 B+C) \sin (c+d x)}{a d}+\frac {(3 A-2 B+2 C) \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {(A-B+C) \cos (c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}+\frac {(3 A-2 B+2 C) \int 1 \, dx}{2 a}\\ &=\frac {(3 A-2 B+2 C) x}{2 a}-\frac {(2 A-2 B+C) \sin (c+d x)}{a d}+\frac {(3 A-2 B+2 C) \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {(A-B+C) \cos (c+d x) \sin (c+d x)}{d (a+a \sec (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.58, size = 213, normalized size = 1.97 \[ \frac {\sec \left (\frac {c}{2}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \left (4 d x (3 A-2 B+2 C) \cos \left (c+\frac {d x}{2}\right )+4 d x (3 A-2 B+2 C) \cos \left (\frac {d x}{2}\right )-4 A \sin \left (c+\frac {d x}{2}\right )-3 A \sin \left (c+\frac {3 d x}{2}\right )-3 A \sin \left (2 c+\frac {3 d x}{2}\right )+A \sin \left (2 c+\frac {5 d x}{2}\right )+A \sin \left (3 c+\frac {5 d x}{2}\right )-20 A \sin \left (\frac {d x}{2}\right )+4 B \sin \left (c+\frac {d x}{2}\right )+4 B \sin \left (c+\frac {3 d x}{2}\right )+4 B \sin \left (2 c+\frac {3 d x}{2}\right )+20 B \sin \left (\frac {d x}{2}\right )-16 C \sin \left (\frac {d x}{2}\right )\right )}{8 a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 90, normalized size = 0.83 \[ \frac {{\left (3 \, A - 2 \, B + 2 \, C\right )} d x \cos \left (d x + c\right ) + {\left (3 \, A - 2 \, B + 2 \, C\right )} d x + {\left (A \cos \left (d x + c\right )^{2} - {\left (A - 2 \, B\right )} \cos \left (d x + c\right ) - 4 \, A + 4 \, B - 2 \, C\right )} \sin \left (d x + c\right )}{2 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 137, normalized size = 1.27 \[ \frac {\frac {{\left (d x + c\right )} {\left (3 \, A - 2 \, B + 2 \, C\right )}}{a} - \frac {2 \, {\left (A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a} - \frac {2 \, {\left (3 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.35, size = 248, normalized size = 2.30 \[ -\frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {3 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {2 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {3 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{a d}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 273, normalized size = 2.53 \[ -\frac {A {\left (\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} + B {\left (\frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {2 \, \sin \left (d x + c\right )}{{\left (a + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - C {\left (\frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.57, size = 111, normalized size = 1.03 \[ \frac {x\,\left (3\,A-2\,B+2\,C\right )}{2\,a}-\frac {\left (3\,A-2\,B\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (A-2\,B\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A-B+C\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {A \cos ^{2}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \cos ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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