Optimal. Leaf size=34 \[ \frac {\sin (c+d x)}{b d}-\frac {a \log (a+b \sin (c+d x))}{b^2 d} \]
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Rubi [A] time = 0.05, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2833, 12, 43} \[ \frac {\sin (c+d x)}{b d}-\frac {a \log (a+b \sin (c+d x))}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{b (a+x)} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (1-\frac {a}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=-\frac {a \log (a+b \sin (c+d x))}{b^2 d}+\frac {\sin (c+d x)}{b d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.97 \[ -\frac {\frac {a \log (a+b \sin (c+d x))}{b^2}-\frac {\sin (c+d x)}{b}}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 31, normalized size = 0.91 \[ -\frac {a \log \left (b \sin \left (d x + c\right ) + a\right ) - b \sin \left (d x + c\right )}{b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 34, normalized size = 1.00 \[ -\frac {\frac {a \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{2}} - \frac {\sin \left (d x + c\right )}{b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 35, normalized size = 1.03 \[ -\frac {a \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{2}}+\frac {\sin \left (d x +c \right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 33, normalized size = 0.97 \[ -\frac {\frac {a \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{2}} - \frac {\sin \left (d x + c\right )}{b}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 31, normalized size = 0.91 \[ -\frac {a\,\ln \left (a+b\,\sin \left (c+d\,x\right )\right )-b\,\sin \left (c+d\,x\right )}{b^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.67, size = 66, normalized size = 1.94 \[ \begin {cases} \frac {x \sin {\relax (c )} \cos {\relax (c )}}{a} & \text {for}\: b = 0 \wedge d = 0 \\- \frac {\cos ^{2}{\left (c + d x \right )}}{2 a d} & \text {for}\: b = 0 \\\frac {x \sin {\relax (c )} \cos {\relax (c )}}{a + b \sin {\relax (c )}} & \text {for}\: d = 0 \\- \frac {a \log {\left (\frac {a}{b} + \sin {\left (c + d x \right )} \right )}}{b^{2} d} + \frac {\sin {\left (c + d x \right )}}{b d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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