Integrand size = 16, antiderivative size = 213 \[ \int \frac {\sin (a+b x)}{c+d x^2} \, dx=-\frac {\operatorname {CosIntegral}\left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right ) \sin \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {CosIntegral}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right ) \sin \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right )}{2 \sqrt {-c} \sqrt {d}} \]
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Time = 0.61 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3414, 3384, 3380, 3383} \[ \int \frac {\sin (a+b x)}{c+d x^2} \, dx=-\frac {\sin \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \operatorname {CosIntegral}\left (x b+\frac {\sqrt {-c} b}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\sin \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \operatorname {CosIntegral}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (x b+\frac {\sqrt {-c} b}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]
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Rule 3380
Rule 3383
Rule 3384
Rule 3414
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {-c} \sin (a+b x)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \sin (a+b x)}{2 c \left (\sqrt {-c}+\sqrt {d} x\right )}\right ) \, dx \\ & = -\frac {\int \frac {\sin (a+b x)}{\sqrt {-c}-\sqrt {d} x} \, dx}{2 \sqrt {-c}}-\frac {\int \frac {\sin (a+b x)}{\sqrt {-c}+\sqrt {d} x} \, dx}{2 \sqrt {-c}} \\ & = -\frac {\cos \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \int \frac {\sin \left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right )}{\sqrt {-c}+\sqrt {d} x} \, dx}{2 \sqrt {-c}}+\frac {\cos \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \int \frac {\sin \left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{\sqrt {-c}-\sqrt {d} x} \, dx}{2 \sqrt {-c}}-\frac {\sin \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \int \frac {\cos \left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right )}{\sqrt {-c}+\sqrt {d} x} \, dx}{2 \sqrt {-c}}-\frac {\sin \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \int \frac {\cos \left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{\sqrt {-c}-\sqrt {d} x} \, dx}{2 \sqrt {-c}} \\ & = -\frac {\operatorname {CosIntegral}\left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right ) \sin \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\operatorname {CosIntegral}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right ) \sin \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Si}\left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right )}{2 \sqrt {-c} \sqrt {d}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.77 \[ \int \frac {\sin (a+b x)}{c+d x^2} \, dx=\frac {e^{-i a-\frac {b \sqrt {c}}{\sqrt {d}}} \left (e^{\frac {2 b \sqrt {c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (-\frac {b \sqrt {c}}{\sqrt {d}}-i b x\right )-\operatorname {ExpIntegralEi}\left (\frac {b \sqrt {c}}{\sqrt {d}}-i b x\right )+e^{2 i a} \left (e^{\frac {2 b \sqrt {c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (-\frac {b \sqrt {c}}{\sqrt {d}}+i b x\right )-\operatorname {ExpIntegralEi}\left (\frac {b \sqrt {c}}{\sqrt {d}}+i b x\right )\right )\right )}{4 \sqrt {c} \sqrt {d}} \]
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Time = 1.14 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(b \left (-\frac {-\operatorname {Si}\left (-x b -a +\frac {b \sqrt {-c d}+a d}{d}\right ) \cos \left (\frac {b \sqrt {-c d}+a d}{d}\right )+\operatorname {Ci}\left (x b +a -\frac {b \sqrt {-c d}+a d}{d}\right ) \sin \left (\frac {b \sqrt {-c d}+a d}{d}\right )}{2 d \left (-\frac {b \sqrt {-c d}+a d}{d}+a \right )}-\frac {-\operatorname {Si}\left (-x b -a -\frac {b \sqrt {-c d}-a d}{d}\right ) \cos \left (\frac {b \sqrt {-c d}-a d}{d}\right )-\operatorname {Ci}\left (x b +a +\frac {b \sqrt {-c d}-a d}{d}\right ) \sin \left (\frac {b \sqrt {-c d}-a d}{d}\right )}{2 d \left (\frac {b \sqrt {-c d}-a d}{d}+a \right )}\right )\) | \(233\) |
default | \(b \left (-\frac {-\operatorname {Si}\left (-x b -a +\frac {b \sqrt {-c d}+a d}{d}\right ) \cos \left (\frac {b \sqrt {-c d}+a d}{d}\right )+\operatorname {Ci}\left (x b +a -\frac {b \sqrt {-c d}+a d}{d}\right ) \sin \left (\frac {b \sqrt {-c d}+a d}{d}\right )}{2 d \left (-\frac {b \sqrt {-c d}+a d}{d}+a \right )}-\frac {-\operatorname {Si}\left (-x b -a -\frac {b \sqrt {-c d}-a d}{d}\right ) \cos \left (\frac {b \sqrt {-c d}-a d}{d}\right )-\operatorname {Ci}\left (x b +a +\frac {b \sqrt {-c d}-a d}{d}\right ) \sin \left (\frac {b \sqrt {-c d}-a d}{d}\right )}{2 d \left (\frac {b \sqrt {-c d}-a d}{d}+a \right )}\right )\) | \(233\) |
risch | \(-\frac {\sqrt {c d}\, {\mathrm e}^{\frac {i a d +b \sqrt {c d}}{d}} \operatorname {Ei}_{1}\left (\frac {i a d +b \sqrt {c d}-\left (i b x +i a \right ) d}{d}\right )}{4 c d}+\frac {\sqrt {c d}\, {\mathrm e}^{\frac {i a d -b \sqrt {c d}}{d}} \operatorname {Ei}_{1}\left (\frac {i a d -b \sqrt {c d}-\left (i b x +i a \right ) d}{d}\right )}{4 c d}+\frac {\sqrt {c d}\, \operatorname {Ei}_{1}\left (-\frac {i a d +b \sqrt {c d}-\left (i b x +i a \right ) d}{d}\right ) {\mathrm e}^{-\frac {i a d +b \sqrt {c d}}{d}}}{4 c d}-\frac {\sqrt {c d}\, \operatorname {Ei}_{1}\left (-\frac {i a d -b \sqrt {c d}-\left (i b x +i a \right ) d}{d}\right ) {\mathrm e}^{-\frac {i a d -b \sqrt {c d}}{d}}}{4 c d}\) | \(262\) |
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.88 \[ \int \frac {\sin (a+b x)}{c+d x^2} \, dx=\frac {\sqrt {\frac {b^{2} c}{d}} {\rm Ei}\left (i \, b x - \sqrt {\frac {b^{2} c}{d}}\right ) e^{\left (i \, a + \sqrt {\frac {b^{2} c}{d}}\right )} - \sqrt {\frac {b^{2} c}{d}} {\rm Ei}\left (i \, b x + \sqrt {\frac {b^{2} c}{d}}\right ) e^{\left (i \, a - \sqrt {\frac {b^{2} c}{d}}\right )} + \sqrt {\frac {b^{2} c}{d}} {\rm Ei}\left (-i \, b x - \sqrt {\frac {b^{2} c}{d}}\right ) e^{\left (-i \, a + \sqrt {\frac {b^{2} c}{d}}\right )} - \sqrt {\frac {b^{2} c}{d}} {\rm Ei}\left (-i \, b x + \sqrt {\frac {b^{2} c}{d}}\right ) e^{\left (-i \, a - \sqrt {\frac {b^{2} c}{d}}\right )}}{4 \, b c} \]
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\[ \int \frac {\sin (a+b x)}{c+d x^2} \, dx=\int \frac {\sin {\left (a + b x \right )}}{c + d x^{2}}\, dx \]
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\[ \int \frac {\sin (a+b x)}{c+d x^2} \, dx=\int { \frac {\sin \left (b x + a\right )}{d x^{2} + c} \,d x } \]
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\[ \int \frac {\sin (a+b x)}{c+d x^2} \, dx=\int { \frac {\sin \left (b x + a\right )}{d x^{2} + c} \,d x } \]
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Timed out. \[ \int \frac {\sin (a+b x)}{c+d x^2} \, dx=\int \frac {\sin \left (a+b\,x\right )}{d\,x^2+c} \,d x \]
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