Optimal. Leaf size=36 \[ \frac {(A-B) \log (\sin (c+d x)+1)}{a d}+\frac {B \sin (c+d x)}{a d} \]
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Rubi [A] time = 0.08, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2833, 43} \[ \frac {(A-B) \log (\sin (c+d x)+1)}{a d}+\frac {B \sin (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {A+\frac {B x}{a}}{a+x} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {B}{a}+\frac {A-B}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {(A-B) \log (1+\sin (c+d x))}{a d}+\frac {B \sin (c+d x)}{a d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 31, normalized size = 0.86 \[ \frac {(A-B) \log (\sin (c+d x)+1)+B \sin (c+d x)}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 31, normalized size = 0.86 \[ \frac {{\left (A - B\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + B \sin \left (d x + c\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 35, normalized size = 0.97 \[ \frac {\frac {{\left (A - B\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} + \frac {B \sin \left (d x + c\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 51, normalized size = 1.42 \[ \frac {\ln \left (1+\sin \left (d x +c \right )\right ) A}{d a}-\frac {\ln \left (1+\sin \left (d x +c \right )\right ) B}{d a}+\frac {B \sin \left (d x +c \right )}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 34, normalized size = 0.94 \[ \frac {\frac {{\left (A - B\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {B \sin \left (d x + c\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.23, size = 36, normalized size = 1.00 \[ \frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (A-B\right )}{a\,d}+\frac {B\,\sin \left (c+d\,x\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 60, normalized size = 1.67 \[ \begin {cases} \frac {A \log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{a d} - \frac {B \log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{a d} + \frac {B \sin {\left (c + d x \right )}}{a d} & \text {for}\: d \neq 0 \\\frac {x \left (A + B \sin {\relax (c )}\right ) \cos {\relax (c )}}{a \sin {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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