Optimal. Leaf size=25 \[ \frac {1}{2} \cos (a) \text {Ci}\left (b x^2\right )-\frac {1}{2} \sin (a) \text {Si}\left (b x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3378, 3376, 3375} \[ \frac {1}{2} \cos (a) \text {CosIntegral}\left (b x^2\right )-\frac {1}{2} \sin (a) \text {Si}\left (b x^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3375
Rule 3376
Rule 3378
Rubi steps
\begin {align*} \int \frac {\cos \left (a+b x^2\right )}{x} \, dx &=\cos (a) \int \frac {\cos \left (b x^2\right )}{x} \, dx-\sin (a) \int \frac {\sin \left (b x^2\right )}{x} \, dx\\ &=\frac {1}{2} \cos (a) \text {Ci}\left (b x^2\right )-\frac {1}{2} \sin (a) \text {Si}\left (b x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 24, normalized size = 0.96 \[ \frac {1}{2} \left (\cos (a) \text {Ci}\left (b x^2\right )-\sin (a) \text {Si}\left (b x^2\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.88, size = 29, normalized size = 1.16 \[ \frac {1}{4} \, {\left (\operatorname {Ci}\left (b x^{2}\right ) + \operatorname {Ci}\left (-b x^{2}\right )\right )} \cos \relax (a) - \frac {1}{2} \, \sin \relax (a) \operatorname {Si}\left (b x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.42, size = 21, normalized size = 0.84 \[ \frac {1}{2} \, \cos \relax (a) \operatorname {Ci}\left (b x^{2}\right ) - \frac {1}{2} \, \sin \relax (a) \operatorname {Si}\left (b x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 22, normalized size = 0.88 \[ \frac {\Ci \left (b \,x^{2}\right ) \cos \relax (a )}{2}-\frac {\Si \left (b \,x^{2}\right ) \sin \relax (a )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 1.14, size = 43, normalized size = 1.72 \[ \frac {1}{4} \, {\left ({\rm Ei}\left (i \, b x^{2}\right ) + {\rm Ei}\left (-i \, b x^{2}\right )\right )} \cos \relax (a) + \frac {1}{4} \, {\left (i \, {\rm Ei}\left (i \, b x^{2}\right ) - i \, {\rm Ei}\left (-i \, b x^{2}\right )\right )} \sin \relax (a) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \frac {\cos \relax (a)\,\mathrm {cosint}\left (b\,x^2\right )}{2}-\frac {\sin \relax (a)\,\mathrm {sinint}\left (b\,x^2\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (a + b x^{2} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________