Integrand size = 18, antiderivative size = 184 \[ \int \frac {\cos \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=-\frac {\cos \left (a+b \sqrt {c+d x}\right )}{x}+\frac {b d \operatorname {CosIntegral}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right ) \sin \left (a-b \sqrt {c}\right )}{2 \sqrt {c}}-\frac {b d \operatorname {CosIntegral}\left (b \sqrt {c}-b \sqrt {c+d x}\right ) \sin \left (a+b \sqrt {c}\right )}{2 \sqrt {c}}+\frac {b d \cos \left (a-b \sqrt {c}\right ) \text {Si}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}+\frac {b d \cos \left (a+b \sqrt {c}\right ) \text {Si}\left (b \sqrt {c}-b \sqrt {c+d x}\right )}{2 \sqrt {c}} \]
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Time = 0.54 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3513, 3423, 3414, 3384, 3380, 3383} \[ \int \frac {\cos \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=\frac {b d \sin \left (a-b \sqrt {c}\right ) \operatorname {CosIntegral}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}-\frac {b d \sin \left (a+b \sqrt {c}\right ) \operatorname {CosIntegral}\left (b \sqrt {c}-b \sqrt {c+d x}\right )}{2 \sqrt {c}}+\frac {b d \cos \left (a-b \sqrt {c}\right ) \text {Si}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}+\frac {b d \cos \left (a+b \sqrt {c}\right ) \text {Si}\left (b \sqrt {c}-b \sqrt {c+d x}\right )}{2 \sqrt {c}}-\frac {\cos \left (a+b \sqrt {c+d x}\right )}{x} \]
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Rule 3380
Rule 3383
Rule 3384
Rule 3414
Rule 3423
Rule 3513
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x \cos (a+b x)}{\left (-\frac {c}{d}+\frac {x^2}{d}\right )^2} \, dx,x,\sqrt {c+d x}\right )}{d} \\ & = -\frac {\cos \left (a+b \sqrt {c+d x}\right )}{x}-b \text {Subst}\left (\int \frac {\sin (a+b x)}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right ) \\ & = -\frac {\cos \left (a+b \sqrt {c+d x}\right )}{x}-b \text {Subst}\left (\int \left (-\frac {d \sin (a+b x)}{2 \sqrt {c} \left (\sqrt {c}-x\right )}-\frac {d \sin (a+b x)}{2 \sqrt {c} \left (\sqrt {c}+x\right )}\right ) \, dx,x,\sqrt {c+d x}\right ) \\ & = -\frac {\cos \left (a+b \sqrt {c+d x}\right )}{x}+\frac {(b d) \text {Subst}\left (\int \frac {\sin (a+b x)}{\sqrt {c}-x} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt {c}}+\frac {(b d) \text {Subst}\left (\int \frac {\sin (a+b x)}{\sqrt {c}+x} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt {c}} \\ & = -\frac {\cos \left (a+b \sqrt {c+d x}\right )}{x}+\frac {\left (b d \cos \left (a-b \sqrt {c}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (b \sqrt {c}+b x\right )}{\sqrt {c}+x} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt {c}}-\frac {\left (b d \cos \left (a+b \sqrt {c}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (b \sqrt {c}-b x\right )}{\sqrt {c}-x} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt {c}}+\frac {\left (b d \sin \left (a-b \sqrt {c}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (b \sqrt {c}+b x\right )}{\sqrt {c}+x} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt {c}}+\frac {\left (b d \sin \left (a+b \sqrt {c}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (b \sqrt {c}-b x\right )}{\sqrt {c}-x} \, dx,x,\sqrt {c+d x}\right )}{2 \sqrt {c}} \\ & = -\frac {\cos \left (a+b \sqrt {c+d x}\right )}{x}+\frac {b d \operatorname {CosIntegral}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right ) \sin \left (a-b \sqrt {c}\right )}{2 \sqrt {c}}-\frac {b d \operatorname {CosIntegral}\left (b \sqrt {c}-b \sqrt {c+d x}\right ) \sin \left (a+b \sqrt {c}\right )}{2 \sqrt {c}}+\frac {b d \cos \left (a-b \sqrt {c}\right ) \text {Si}\left (b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )}{2 \sqrt {c}}+\frac {b d \cos \left (a+b \sqrt {c}\right ) \text {Si}\left (b \sqrt {c}-b \sqrt {c+d x}\right )}{2 \sqrt {c}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.38 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.30 \[ \int \frac {\cos \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=\frac {i \left (e^{-i a} \left (2 i \sqrt {c} e^{-i b \sqrt {c+d x}}-b d e^{-i b \sqrt {c}} x \operatorname {ExpIntegralEi}\left (-i b \left (-\sqrt {c}+\sqrt {c+d x}\right )\right )+b d e^{i b \sqrt {c}} x \operatorname {ExpIntegralEi}\left (-i b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )\right )+e^{i \left (a-b \sqrt {c}\right )} \left (2 i \sqrt {c} e^{i b \left (\sqrt {c}+\sqrt {c+d x}\right )}+b d e^{2 i b \sqrt {c}} x \operatorname {ExpIntegralEi}\left (i b \left (-\sqrt {c}+\sqrt {c+d x}\right )\right )-b d x \operatorname {ExpIntegralEi}\left (i b \left (\sqrt {c}+\sqrt {c+d x}\right )\right )\right )\right )}{4 \sqrt {c} x} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(713\) vs. \(2(142)=284\).
Time = 1.34 (sec) , antiderivative size = 714, normalized size of antiderivative = 3.88
method | result | size |
derivativedivides | \(\frac {2 d \left (\frac {\cos \left (a +b \sqrt {d x +c}\right ) \left (-\frac {a \,b^{2} \left (a +b \sqrt {d x +c}\right )}{2 c}+\frac {b^{2} \left (-b^{2} c +a^{2}\right )}{2 c}\right )}{-b^{2} c +a^{2}-2 \left (a +b \sqrt {d x +c}\right ) a +\left (a +b \sqrt {d x +c}\right )^{2}}-\frac {a b \left (\operatorname {Si}\left (b \sqrt {c}-b \sqrt {d x +c}\right ) \sin \left (a +b \sqrt {c}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}-b \sqrt {c}\right ) \cos \left (a +b \sqrt {c}\right )\right )}{4 c^{\frac {3}{2}}}+\frac {a b \left (-\operatorname {Si}\left (b \sqrt {d x +c}+b \sqrt {c}\right ) \sin \left (a -b \sqrt {c}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}+b \sqrt {c}\right ) \cos \left (a -b \sqrt {c}\right )\right )}{4 c^{\frac {3}{2}}}+\frac {b \left (-b^{2} c +a^{2}-\left (a +b \sqrt {c}\right ) a \right ) \left (-\operatorname {Si}\left (b \sqrt {c}-b \sqrt {d x +c}\right ) \cos \left (a +b \sqrt {c}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}-b \sqrt {c}\right ) \sin \left (a +b \sqrt {c}\right )\right )}{4 c^{\frac {3}{2}}}-\frac {b \left (-b^{2} c +a^{2}-\left (a -b \sqrt {c}\right ) a \right ) \left (\operatorname {Si}\left (b \sqrt {d x +c}+b \sqrt {c}\right ) \cos \left (a -b \sqrt {c}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}+b \sqrt {c}\right ) \sin \left (a -b \sqrt {c}\right )\right )}{4 c^{\frac {3}{2}}}-a \,b^{4} \left (\frac {\cos \left (a +b \sqrt {d x +c}\right ) \left (-\frac {a +b \sqrt {d x +c}}{2 c \,b^{2}}+\frac {a}{2 c \,b^{2}}\right )}{-b^{2} c +a^{2}-2 \left (a +b \sqrt {d x +c}\right ) a +\left (a +b \sqrt {d x +c}\right )^{2}}-\frac {\operatorname {Si}\left (b \sqrt {c}-b \sqrt {d x +c}\right ) \sin \left (a +b \sqrt {c}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}-b \sqrt {c}\right ) \cos \left (a +b \sqrt {c}\right )}{4 c^{\frac {3}{2}} b^{3}}+\frac {-\operatorname {Si}\left (b \sqrt {d x +c}+b \sqrt {c}\right ) \sin \left (a -b \sqrt {c}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}+b \sqrt {c}\right ) \cos \left (a -b \sqrt {c}\right )}{4 c^{\frac {3}{2}} b^{3}}-\frac {-\operatorname {Si}\left (b \sqrt {c}-b \sqrt {d x +c}\right ) \cos \left (a +b \sqrt {c}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}-b \sqrt {c}\right ) \sin \left (a +b \sqrt {c}\right )}{4 c \,b^{2}}-\frac {\operatorname {Si}\left (b \sqrt {d x +c}+b \sqrt {c}\right ) \cos \left (a -b \sqrt {c}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}+b \sqrt {c}\right ) \sin \left (a -b \sqrt {c}\right )}{4 c \,b^{2}}\right )\right )}{b^{2}}\) | \(714\) |
default | \(\frac {2 d \left (\frac {\cos \left (a +b \sqrt {d x +c}\right ) \left (-\frac {a \,b^{2} \left (a +b \sqrt {d x +c}\right )}{2 c}+\frac {b^{2} \left (-b^{2} c +a^{2}\right )}{2 c}\right )}{-b^{2} c +a^{2}-2 \left (a +b \sqrt {d x +c}\right ) a +\left (a +b \sqrt {d x +c}\right )^{2}}-\frac {a b \left (\operatorname {Si}\left (b \sqrt {c}-b \sqrt {d x +c}\right ) \sin \left (a +b \sqrt {c}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}-b \sqrt {c}\right ) \cos \left (a +b \sqrt {c}\right )\right )}{4 c^{\frac {3}{2}}}+\frac {a b \left (-\operatorname {Si}\left (b \sqrt {d x +c}+b \sqrt {c}\right ) \sin \left (a -b \sqrt {c}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}+b \sqrt {c}\right ) \cos \left (a -b \sqrt {c}\right )\right )}{4 c^{\frac {3}{2}}}+\frac {b \left (-b^{2} c +a^{2}-\left (a +b \sqrt {c}\right ) a \right ) \left (-\operatorname {Si}\left (b \sqrt {c}-b \sqrt {d x +c}\right ) \cos \left (a +b \sqrt {c}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}-b \sqrt {c}\right ) \sin \left (a +b \sqrt {c}\right )\right )}{4 c^{\frac {3}{2}}}-\frac {b \left (-b^{2} c +a^{2}-\left (a -b \sqrt {c}\right ) a \right ) \left (\operatorname {Si}\left (b \sqrt {d x +c}+b \sqrt {c}\right ) \cos \left (a -b \sqrt {c}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}+b \sqrt {c}\right ) \sin \left (a -b \sqrt {c}\right )\right )}{4 c^{\frac {3}{2}}}-a \,b^{4} \left (\frac {\cos \left (a +b \sqrt {d x +c}\right ) \left (-\frac {a +b \sqrt {d x +c}}{2 c \,b^{2}}+\frac {a}{2 c \,b^{2}}\right )}{-b^{2} c +a^{2}-2 \left (a +b \sqrt {d x +c}\right ) a +\left (a +b \sqrt {d x +c}\right )^{2}}-\frac {\operatorname {Si}\left (b \sqrt {c}-b \sqrt {d x +c}\right ) \sin \left (a +b \sqrt {c}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}-b \sqrt {c}\right ) \cos \left (a +b \sqrt {c}\right )}{4 c^{\frac {3}{2}} b^{3}}+\frac {-\operatorname {Si}\left (b \sqrt {d x +c}+b \sqrt {c}\right ) \sin \left (a -b \sqrt {c}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}+b \sqrt {c}\right ) \cos \left (a -b \sqrt {c}\right )}{4 c^{\frac {3}{2}} b^{3}}-\frac {-\operatorname {Si}\left (b \sqrt {c}-b \sqrt {d x +c}\right ) \cos \left (a +b \sqrt {c}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}-b \sqrt {c}\right ) \sin \left (a +b \sqrt {c}\right )}{4 c \,b^{2}}-\frac {\operatorname {Si}\left (b \sqrt {d x +c}+b \sqrt {c}\right ) \cos \left (a -b \sqrt {c}\right )+\operatorname {Ci}\left (b \sqrt {d x +c}+b \sqrt {c}\right ) \sin \left (a -b \sqrt {c}\right )}{4 c \,b^{2}}\right )\right )}{b^{2}}\) | \(714\) |
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Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.14 \[ \int \frac {\cos \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=\frac {\sqrt {-b^{2} c} d x {\rm Ei}\left (i \, \sqrt {d x + c} b - \sqrt {-b^{2} c}\right ) e^{\left (i \, a + \sqrt {-b^{2} c}\right )} - \sqrt {-b^{2} c} d x {\rm Ei}\left (i \, \sqrt {d x + c} b + \sqrt {-b^{2} c}\right ) e^{\left (i \, a - \sqrt {-b^{2} c}\right )} + \sqrt {-b^{2} c} d x {\rm Ei}\left (-i \, \sqrt {d x + c} b - \sqrt {-b^{2} c}\right ) e^{\left (-i \, a + \sqrt {-b^{2} c}\right )} - \sqrt {-b^{2} c} d x {\rm Ei}\left (-i \, \sqrt {d x + c} b + \sqrt {-b^{2} c}\right ) e^{\left (-i \, a - \sqrt {-b^{2} c}\right )} - 4 \, c \cos \left (\sqrt {d x + c} b + a\right )}{4 \, c x} \]
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\[ \int \frac {\cos \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=\int \frac {\cos {\left (a + b \sqrt {c + d x} \right )}}{x^{2}}\, dx \]
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\[ \int \frac {\cos \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=\int { \frac {\cos \left (\sqrt {d x + c} b + a\right )}{x^{2}} \,d x } \]
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\[ \int \frac {\cos \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=\int { \frac {\cos \left (\sqrt {d x + c} b + a\right )}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\cos \left (a+b \sqrt {c+d x}\right )}{x^2} \, dx=\int \frac {\cos \left (a+b\,\sqrt {c+d\,x}\right )}{x^2} \,d x \]
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