Optimal. Leaf size=48 \[ \frac {B \log (a+b \sin (c+d x))}{b^2 d}-\frac {A b-a B}{b^2 d (a+b \sin (c+d x))} \]
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Rubi [A] time = 0.08, antiderivative size = 48, 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} \[ \frac {B \log (a+b \sin (c+d x))}{b^2 d}-\frac {A b-a B}{b^2 d (a+b \sin (c+d x))} \]
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))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {A+\frac {B x}{b}}{(a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {A b-a B}{b (a+x)^2}+\frac {B}{b (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac {B \log (a+b \sin (c+d x))}{b^2 d}-\frac {A b-a B}{b^2 d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 42, normalized size = 0.88 \[ \frac {\frac {a B-A b}{a+b \sin (c+d x)}+B \log (a+b \sin (c+d x))}{b^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 54, normalized size = 1.12 \[ \frac {B a - A b + {\left (B b \sin \left (d x + c\right ) + B a\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{3} d \sin \left (d x + c\right ) + a b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 80, normalized size = 1.67 \[ -\frac {\frac {B {\left (\frac {\log \left (\frac {{\left | b \sin \left (d x + c\right ) + a \right |}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2} {\left | b \right |}}\right )}{b} - \frac {a}{{\left (b \sin \left (d x + c\right ) + a\right )} b}\right )}}{b} + \frac {A}{{\left (b \sin \left (d x + c\right ) + a\right )} b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 63, normalized size = 1.31 \[ \frac {B \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{2} d}-\frac {A}{d b \left (a +b \sin \left (d x +c \right )\right )}+\frac {a B}{d \,b^{2} \left (a +b \sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 48, normalized size = 1.00 \[ \frac {\frac {B a - A b}{b^{3} \sin \left (d x + c\right ) + a b^{2}} + \frac {B \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 48, normalized size = 1.00 \[ \frac {B\,\ln \left (a+b\,\sin \left (c+d\,x\right )\right )}{b^2\,d}-\frac {A\,b-B\,a}{b^2\,d\,\left (a+b\,\sin \left (c+d\,x\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.36, size = 178, normalized size = 3.71 \[ \begin {cases} \frac {x \left (A + B \sin {\relax (c )}\right ) \cos {\relax (c )}}{a^{2}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\frac {A \sin {\left (c + d x \right )}}{d} - \frac {B \cos ^{2}{\left (c + d x \right )}}{2 d}}{a^{2}} & \text {for}\: b = 0 \\\frac {x \left (A + B \sin {\relax (c )}\right ) \cos {\relax (c )}}{\left (a + b \sin {\relax (c )}\right )^{2}} & \text {for}\: d = 0 \\- \frac {A b}{a b^{2} d + b^{3} d \sin {\left (c + d x \right )}} + \frac {B a \log {\left (\frac {a}{b} + \sin {\left (c + d x \right )} \right )}}{a b^{2} d + b^{3} d \sin {\left (c + d x \right )}} + \frac {B a}{a b^{2} d + b^{3} d \sin {\left (c + d x \right )}} + \frac {B b \log {\left (\frac {a}{b} + \sin {\left (c + d x \right )} \right )} \sin {\left (c + d x \right )}}{a b^{2} d + b^{3} d \sin {\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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