Integrand size = 24, antiderivative size = 78 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{c+d x} \, dx=-\frac {\cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right )}{8 d}+\frac {\log (c+d x)}{8 d}+\frac {\sin \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{8 d} \]
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Time = 0.16 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4491, 3384, 3380, 3383} \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{c+d x} \, dx=-\frac {\cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right )}{8 d}+\frac {\sin \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{8 d}+\frac {\log (c+d x)}{8 d} \]
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Rule 3380
Rule 3383
Rule 3384
Rule 4491
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{8 (c+d x)}-\frac {\cos (4 a+4 b x)}{8 (c+d x)}\right ) \, dx \\ & = \frac {\log (c+d x)}{8 d}-\frac {1}{8} \int \frac {\cos (4 a+4 b x)}{c+d x} \, dx \\ & = \frac {\log (c+d x)}{8 d}-\frac {1}{8} \cos \left (4 a-\frac {4 b c}{d}\right ) \int \frac {\cos \left (\frac {4 b c}{d}+4 b x\right )}{c+d x} \, dx+\frac {1}{8} \sin \left (4 a-\frac {4 b c}{d}\right ) \int \frac {\sin \left (\frac {4 b c}{d}+4 b x\right )}{c+d x} \, dx \\ & = -\frac {\cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 b c}{d}+4 b x\right )}{8 d}+\frac {\log (c+d x)}{8 d}+\frac {\sin \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b c}{d}+4 b x\right )}{8 d} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{c+d x} \, dx=\frac {-\cos \left (4 a-\frac {4 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 b (c+d x)}{d}\right )+\log (c+d x)+\sin \left (4 a-\frac {4 b c}{d}\right ) \text {Si}\left (\frac {4 b (c+d x)}{d}\right )}{8 d} \]
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Result contains complex when optimal does not.
Time = 0.92 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.37
method | result | size |
risch | \(\frac {\ln \left (d x +c \right )}{8 d}+\frac {{\mathrm e}^{-\frac {4 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (4 i b x +4 i a -\frac {4 i \left (a d -c b \right )}{d}\right )}{16 d}+\frac {{\mathrm e}^{\frac {4 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-4 i b x -4 i a -\frac {4 \left (-i a d +i c b \right )}{d}\right )}{16 d}\) | \(107\) |
derivativedivides | \(\frac {-\frac {b \left (-\frac {4 \,\operatorname {Si}\left (-4 x b -4 a -\frac {4 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-4 a d +4 c b}{d}\right )}{d}+\frac {4 \,\operatorname {Ci}\left (4 x b +4 a +\frac {-4 a d +4 c b}{d}\right ) \cos \left (\frac {-4 a d +4 c b}{d}\right )}{d}\right )}{32}+\frac {b \ln \left (-a d +c b +d \left (x b +a \right )\right )}{8 d}}{b}\) | \(114\) |
default | \(\frac {-\frac {b \left (-\frac {4 \,\operatorname {Si}\left (-4 x b -4 a -\frac {4 \left (-a d +c b \right )}{d}\right ) \sin \left (\frac {-4 a d +4 c b}{d}\right )}{d}+\frac {4 \,\operatorname {Ci}\left (4 x b +4 a +\frac {-4 a d +4 c b}{d}\right ) \cos \left (\frac {-4 a d +4 c b}{d}\right )}{d}\right )}{32}+\frac {b \ln \left (-a d +c b +d \left (x b +a \right )\right )}{8 d}}{b}\) | \(114\) |
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none
Time = 0.25 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{c+d x} \, dx=-\frac {\cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {4 \, {\left (b d x + b c\right )}}{d}\right ) - \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {4 \, {\left (b d x + b c\right )}}{d}\right ) - \log \left (d x + c\right )}{8 \, d} \]
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\[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{c+d x} \, dx=\int \frac {\sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{c + d x}\, dx \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.08 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{c+d x} \, dx=\frac {b {\left (E_{1}\left (\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{1}\left (-\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + b {\left (i \, E_{1}\left (\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) - i \, E_{1}\left (-\frac {4 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {4 \, {\left (b c - a d\right )}}{d}\right ) + 2 \, b \log \left (b c + {\left (b x + a\right )} d - a d\right )}{16 \, b d} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.33 (sec) , antiderivative size = 669, normalized size of antiderivative = 8.58 \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{c+d x} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\cos ^2(a+b x) \sin ^2(a+b x)}{c+d x} \, dx=\int \frac {{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^2}{c+d\,x} \,d x \]
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