Integrand size = 24, antiderivative size = 22 \[ \int \cos (c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=-\frac {(b+a \cos (c+d x))^2}{2 a d} \]
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Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4462, 12, 2718, 2644, 30} \[ \int \cos (c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=\frac {a \sin ^2(c+d x)}{2 d}-\frac {b \cos (c+d x)}{d} \]
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Rule 12
Rule 30
Rule 2644
Rule 2718
Rule 4462
Rubi steps \begin{align*} \text {integral}& = a \int \cos (c+d x) \sin (c+d x) \, dx+\int b \sin (c+d x) \, dx \\ & = b \int \sin (c+d x) \, dx+\frac {a \text {Subst}(\int x \, dx,x,\sin (c+d x))}{d} \\ & = -\frac {b \cos (c+d x)}{d}+\frac {a \sin ^2(c+d x)}{2 d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82 \[ \int \cos (c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=-\frac {b \cos (c) \cos (d x)}{d}-\frac {a \cos ^2(c+d x)}{2 d}+\frac {b \sin (c) \sin (d x)}{d} \]
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Time = 0.43 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18
method | result | size |
derivativedivides | \(\frac {-\frac {a \cos \left (d x +c \right )^{2}}{2}-\cos \left (d x +c \right ) b}{d}\) | \(26\) |
default | \(\frac {-\frac {a \cos \left (d x +c \right )^{2}}{2}-\cos \left (d x +c \right ) b}{d}\) | \(26\) |
risch | \(-\frac {b \cos \left (d x +c \right )}{d}-\frac {a \cos \left (2 d x +2 c \right )}{4 d}\) | \(29\) |
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Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \cos (c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=-\frac {a \cos \left (d x + c\right )^{2} + 2 \, b \cos \left (d x + c\right )}{2 \, d} \]
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\[ \int \cos (c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=\int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \cos (c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=-\frac {a \cos \left (d x + c\right )^{2} + 2 \, b \cos \left (d x + c\right )}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (20) = 40\).
Time = 0.34 (sec) , antiderivative size = 102, normalized size of antiderivative = 4.64 \[ \int \cos (c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=-\frac {a \cos \left (2 \, d x + 2 \, c\right )}{4 \, d} - \frac {b \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x\right )^{2} - 4 \, b \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, c\right )^{2} + b}{d \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d \tan \left (\frac {1}{2} \, d x\right )^{2} + d \tan \left (\frac {1}{2} \, c\right )^{2} + d} \]
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Time = 22.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \cos (c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx=-\frac {\left (\cos \left (c+d\,x\right )+1\right )\,\left (2\,b-a+a\,\cos \left (c+d\,x\right )\right )}{2\,d} \]
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