Integrand size = 17, antiderivative size = 24 \[ \int \cos (c+d x) (a+b \tan (c+d x)) \, dx=-\frac {b \cos (c+d x)}{d}+\frac {a \sin (c+d x)}{d} \]
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Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3567, 2717} \[ \int \cos (c+d x) (a+b \tan (c+d x)) \, dx=\frac {a \sin (c+d x)}{d}-\frac {b \cos (c+d x)}{d} \]
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Rule 2717
Rule 3567
Rubi steps \begin{align*} \text {integral}& = -\frac {b \cos (c+d x)}{d}+a \int \cos (c+d x) \, dx \\ & = -\frac {b \cos (c+d x)}{d}+\frac {a \sin (c+d x)}{d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92 \[ \int \cos (c+d x) (a+b \tan (c+d x)) \, dx=-\frac {b \cos (c) \cos (d x)}{d}+\frac {a \cos (d x) \sin (c)}{d}+\frac {a \cos (c) \sin (d x)}{d}+\frac {b \sin (c) \sin (d x)}{d} \]
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Time = 0.72 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {-b \cos \left (d x +c \right )+a \sin \left (d x +c \right )}{d}\) | \(23\) |
default | \(\frac {-b \cos \left (d x +c \right )+a \sin \left (d x +c \right )}{d}\) | \(23\) |
risch | \(-\frac {b \cos \left (d x +c \right )}{d}+\frac {a \sin \left (d x +c \right )}{d}\) | \(25\) |
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Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \cos (c+d x) (a+b \tan (c+d x)) \, dx=-\frac {b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )}{d} \]
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\[ \int \cos (c+d x) (a+b \tan (c+d x)) \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right ) \cos {\left (c + d x \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \cos (c+d x) (a+b \tan (c+d x)) \, dx=-\frac {b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (24) = 48\).
Time = 0.36 (sec) , antiderivative size = 129, normalized size of antiderivative = 5.38 \[ \int \cos (c+d x) (a+b \tan (c+d x)) \, dx=-\frac {b \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 2 \, a \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x\right )^{2} - 4 \, b \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, a \tan \left (\frac {1}{2} \, d x\right ) - 2 \, a \tan \left (\frac {1}{2} \, c\right ) + b}{d \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d \tan \left (\frac {1}{2} \, d x\right )^{2} + d \tan \left (\frac {1}{2} \, c\right )^{2} + d} \]
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Time = 4.21 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \cos (c+d x) (a+b \tan (c+d x)) \, dx=-\frac {2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-a\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
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