Integrand size = 14, antiderivative size = 73 \[ \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx=-\frac {\cos (a+b x)}{d (c+d x)}-\frac {b \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{d^2}-\frac {b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^2} \]
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Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3378, 3384, 3380, 3383} \[ \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx=-\frac {b \sin \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right )}{d^2}-\frac {b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^2}-\frac {\cos (a+b x)}{d (c+d x)} \]
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Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (a+b x)}{d (c+d x)}-\frac {b \int \frac {\sin (a+b x)}{c+d x} \, dx}{d} \\ & = -\frac {\cos (a+b x)}{d (c+d x)}-\frac {\left (b \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{d}-\frac {\left (b \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{d} \\ & = -\frac {\cos (a+b x)}{d (c+d x)}-\frac {b \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{d^2}-\frac {b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^2} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.89 \[ \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx=-\frac {\frac {d \cos (a+b x)}{c+d x}+b \operatorname {CosIntegral}\left (b \left (\frac {c}{d}+x\right )\right ) \sin \left (a-\frac {b c}{d}\right )+b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )}{d^2} \]
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Time = 0.77 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.56
method | result | size |
derivativedivides | \(b \left (-\frac {\cos \left (b x +a \right )}{\left (-a d +b c +d \left (b x +a \right )\right ) d}-\frac {-\frac {\operatorname {Si}\left (-b x -a -\frac {-a d +b c}{d}\right ) \cos \left (\frac {-a d +b c}{d}\right )}{d}-\frac {\operatorname {Ci}\left (b x +a +\frac {-a d +b c}{d}\right ) \sin \left (\frac {-a d +b c}{d}\right )}{d}}{d}\right )\) | \(114\) |
default | \(b \left (-\frac {\cos \left (b x +a \right )}{\left (-a d +b c +d \left (b x +a \right )\right ) d}-\frac {-\frac {\operatorname {Si}\left (-b x -a -\frac {-a d +b c}{d}\right ) \cos \left (\frac {-a d +b c}{d}\right )}{d}-\frac {\operatorname {Ci}\left (b x +a +\frac {-a d +b c}{d}\right ) \sin \left (\frac {-a d +b c}{d}\right )}{d}}{d}\right )\) | \(114\) |
risch | \(\frac {i b \,{\mathrm e}^{-\frac {i \left (a d -b c \right )}{d}} \operatorname {Ei}_{1}\left (i x b +i a -\frac {i \left (a d -b c \right )}{d}\right )}{2 d^{2}}-\frac {i b \,{\mathrm e}^{\frac {i \left (a d -b c \right )}{d}} \operatorname {Ei}_{1}\left (-i x b -i a -\frac {-i a d +i b c}{d}\right )}{2 d^{2}}-\frac {\left (-2 b x d -2 b c \right ) \cos \left (b x +a \right )}{2 d \left (d x +c \right ) \left (-b x d -b c \right )}\) | \(140\) |
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none
Time = 0.28 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.32 \[ \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx=-\frac {{\left (b d x + b c\right )} \operatorname {Ci}\left (\frac {b d x + b c}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + {\left (b d x + b c\right )} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (\frac {b d x + b c}{d}\right ) + d \cos \left (b x + a\right )}{d^{3} x + c d^{2}} \]
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\[ \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx=\int \frac {\cos {\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.25 \[ \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx=-\frac {b^{2} {\left (E_{2}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{2}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + b^{2} {\left (-i \, E_{2}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + i \, E_{2}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )}{2 \, {\left (b c d + {\left (b x + a\right )} d^{2} - a d^{2}\right )} b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 523 vs. \(2 (73) = 146\).
Time = 0.31 (sec) , antiderivative size = 523, normalized size of antiderivative = 7.16 \[ \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx=-\frac {{\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} \operatorname {Ci}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + b^{3} c \operatorname {Ci}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - a b^{2} d \operatorname {Ci}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) - b^{3} c \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) + a b^{2} d \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) + b^{2} d \cos \left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )\right )} d^{2}}{{\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} d^{4} + b c d^{4} - a d^{5}\right )} b} \]
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Timed out. \[ \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx=\int \frac {\cos \left (a+b\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \]
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