Optimal. Leaf size=29 \[ \frac {(A+B) \sec (c+d x) (a \sin (c+d x)+a)}{d}-a B x \]
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Rubi [A] time = 0.05, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2855, 8} \[ \frac {(A+B) \sec (c+d x) (a \sin (c+d x)+a)}{d}-a B x \]
Antiderivative was successfully verified.
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Rule 8
Rule 2855
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=\frac {(A+B) \sec (c+d x) (a+a \sin (c+d x))}{d}-(a B) \int 1 \, dx\\ &=-a B x+\frac {(A+B) \sec (c+d x) (a+a \sin (c+d x))}{d}\\ \end {align*}
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Mathematica [B] time = 0.35, size = 85, normalized size = 2.93 \[ \frac {a \left (2 (A+B) \sin \left (\frac {d x}{2}\right )+B d x \sin \left (c+\frac {d x}{2}\right )-B d x \cos \left (\frac {d x}{2}\right )\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 73, normalized size = 2.52 \[ -\frac {B a d x - {\left (A + B\right )} a + {\left (B a d x - {\left (A + B\right )} a\right )} \cos \left (d x + c\right ) - {\left (B a d x + {\left (A + B\right )} a\right )} \sin \left (d x + c\right )}{d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 36, normalized size = 1.24 \[ -\frac {{\left (d x + c\right )} B a + \frac {2 \, {\left (A a + B a\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 54, normalized size = 1.86 \[ \frac {\frac {a A}{\cos \left (d x +c \right )}+a B \left (\tan \left (d x +c \right )-d x -c \right )+a A \tan \left (d x +c \right )+\frac {a B}{\cos \left (d x +c \right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 56, normalized size = 1.93 \[ -\frac {{\left (d x + c - \tan \left (d x + c\right )\right )} B a - A a \tan \left (d x + c\right ) - \frac {A a}{\cos \left (d x + c\right )} - \frac {B a}{\cos \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.17, size = 33, normalized size = 1.14 \[ -\frac {2\,A\,a+2\,B\,a}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}-B\,a\,x \]
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 \[ a \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int A \sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \sin {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int B \sin ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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