Optimal. Leaf size=52 \[ \frac {(2 A+C) \sin (c+d x)}{a d}-\frac {(A+C) \sin (c+d x)}{d (a \sec (c+d x)+a)}-\frac {A x}{a} \]
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Rubi [A] time = 0.11, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {4085, 3787, 2637, 8} \[ \frac {(2 A+C) \sin (c+d x)}{a d}-\frac {(A+C) \sin (c+d x)}{d (a \sec (c+d x)+a)}-\frac {A x}{a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2637
Rule 3787
Rule 4085
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \left (A+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx &=-\frac {(A+C) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac {\int \cos (c+d x) (-a (2 A+C)+a A \sec (c+d x)) \, dx}{a^2}\\ &=-\frac {(A+C) \sin (c+d x)}{d (a+a \sec (c+d x))}-\frac {A \int 1 \, dx}{a}+\frac {(2 A+C) \int \cos (c+d x) \, dx}{a}\\ &=-\frac {A x}{a}+\frac {(2 A+C) \sin (c+d x)}{a d}-\frac {(A+C) \sin (c+d x)}{d (a+a \sec (c+d x))}\\ \end {align*}
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Mathematica [B] time = 0.30, size = 108, normalized size = 2.08 \[ \frac {\sec \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \left (A \sin \left (c+\frac {d x}{2}\right )+A \sin \left (c+\frac {3 d x}{2}\right )+A \sin \left (2 c+\frac {3 d x}{2}\right )-2 A d x \cos \left (c+\frac {d x}{2}\right )+5 A \sin \left (\frac {d x}{2}\right )-2 A d x \cos \left (\frac {d x}{2}\right )+4 C \sin \left (\frac {d x}{2}\right )\right )}{4 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 53, normalized size = 1.02 \[ -\frac {A d x \cos \left (d x + c\right ) + A d x - {\left (A \cos \left (d x + c\right ) + 2 \, A + C\right )} \sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.57, size = 74, normalized size = 1.42 \[ -\frac {\frac {{\left (d x + c\right )} A}{a} - \frac {A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} - \frac {2 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.12, size = 88, normalized size = 1.69 \[ \frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {2 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {2 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 117, normalized size = 2.25 \[ -\frac {A {\left (\frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {2 \, \sin \left (d x + c\right )}{{\left (a + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - \frac {C \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.60, size = 59, normalized size = 1.13 \[ \frac {2\,A\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )}-\frac {A\,x}{a}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A+C\right )}{a\,d} \]
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 \[ \frac {\int \frac {A \cos {\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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