Optimal. Leaf size=35 \[ \frac {A x}{a}-\frac {(A-B) \tan (c+d x)}{d (a \sec (c+d x)+a)} \]
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Rubi [A] time = 0.06, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3919, 3794} \[ \frac {A x}{a}-\frac {(A-B) \tan (c+d x)}{d (a \sec (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 3794
Rule 3919
Rubi steps
\begin {align*} \int \frac {A+B \sec (c+d x)}{a+a \sec (c+d x)} \, dx &=\frac {A x}{a}-(A-B) \int \frac {\sec (c+d x)}{a+a \sec (c+d x)} \, dx\\ &=\frac {A x}{a}-\frac {(A-B) \tan (c+d x)}{d (a+a \sec (c+d x))}\\ \end {align*}
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Mathematica [B] time = 0.16, size = 72, normalized size = 2.06 \[ \frac {\sec \left (\frac {c}{2}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \left (2 (B-A) \sin \left (\frac {d x}{2}\right )+A d x \cos \left (c+\frac {d x}{2}\right )+A d x \cos \left (\frac {d x}{2}\right )\right )}{a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 44, normalized size = 1.26 \[ \frac {A d x \cos \left (d x + c\right ) + A d x - {\left (A - B\right )} \sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 44, normalized size = 1.26 \[ \frac {\frac {{\left (d x + c\right )} A}{a} - \frac {A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.73, size = 56, normalized size = 1.60 \[ -\frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {2 A \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 73, normalized size = 2.09 \[ \frac {A {\left (\frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} + \frac {B \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.90, size = 32, normalized size = 0.91 \[ -\frac {\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A-B\right )}{a}-\frac {A\,d\,x}{a}}{d} \]
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 \[ \frac {\int \frac {A}{\sec {\left (c + d x \right )} + 1}\, dx + \int \frac {B \sec {\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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