Optimal. Leaf size=41 \[ \frac {(A b-a B) \log (a+b \sin (c+d x))}{b^2 d}+\frac {B \sin (c+d x)}{b d} \]
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Rubi [A] time = 0.07, antiderivative size = 41, 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} \[ \frac {(A b-a B) \log (a+b \sin (c+d x))}{b^2 d}+\frac {B \sin (c+d x)}{b d} \]
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+b \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {A+\frac {B x}{b}}{a+x} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {B}{b}+\frac {A b-a B}{b (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac {(A b-a B) \log (a+b \sin (c+d x))}{b^2 d}+\frac {B \sin (c+d x)}{b d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 39, normalized size = 0.95 \[ \frac {\frac {(A b-a B) \log (a+b \sin (c+d x))}{b}+B \sin (c+d x)}{b d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 38, normalized size = 0.93 \[ \frac {B b \sin \left (d x + c\right ) - {\left (B a - A b\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 41, normalized size = 1.00 \[ \frac {\frac {B \sin \left (d x + c\right )}{b} - \frac {{\left (B a - A b\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 56, normalized size = 1.37 \[ \frac {B \sin \left (d x +c \right )}{b d}+\frac {\ln \left (a +b \sin \left (d x +c \right )\right ) A}{d b}-\frac {\ln \left (a +b \sin \left (d x +c \right )\right ) B a}{d \,b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 40, normalized size = 0.98 \[ \frac {\frac {B \sin \left (d x + c\right )}{b} - \frac {{\left (B a - A b\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 41, normalized size = 1.00 \[ \frac {B\,\sin \left (c+d\,x\right )}{b\,d}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (A\,b-B\,a\right )}{b^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.71, size = 104, normalized size = 2.54 \[ \begin {cases} \frac {x \left (A + B \sin {\relax (c )}\right ) \cos {\relax (c )}}{a} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\frac {A \sin {\left (c + d x \right )}}{d} - \frac {B \cos ^{2}{\left (c + d x \right )}}{2 d}}{a} & \text {for}\: b = 0 \\\frac {x \left (A + B \sin {\relax (c )}\right ) \cos {\relax (c )}}{a + b \sin {\relax (c )}} & \text {for}\: d = 0 \\\frac {A \log {\left (\frac {a}{b} + \sin {\left (c + d x \right )} \right )}}{b d} - \frac {B a \log {\left (\frac {a}{b} + \sin {\left (c + d x \right )} \right )}}{b^{2} d} + \frac {B \sin {\left (c + d x \right )}}{b d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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