Optimal. Leaf size=44 \[ b \text {Ci}(2 b x)-\frac {1}{2} b \text {Si}(b x)^2-\frac {\text {Si}(b x) \cos (b x)}{x}-\frac {\sin (2 b x)}{2 x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6521, 6686, 12, 4406, 3297, 3302} \[ b \text {CosIntegral}(2 b x)-\frac {1}{2} b \text {Si}(b x)^2-\frac {\text {Si}(b x) \cos (b x)}{x}-\frac {\sin (2 b x)}{2 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 3297
Rule 3302
Rule 4406
Rule 6521
Rule 6686
Rubi steps
\begin {align*} \int \frac {\cos (b x) \text {Si}(b x)}{x^2} \, dx &=-\frac {\cos (b x) \text {Si}(b x)}{x}+b \int \frac {\cos (b x) \sin (b x)}{b x^2} \, dx-b \int \frac {\sin (b x) \text {Si}(b x)}{x} \, dx\\ &=-\frac {\cos (b x) \text {Si}(b x)}{x}-\frac {1}{2} b \text {Si}(b x)^2+\int \frac {\cos (b x) \sin (b x)}{x^2} \, dx\\ &=-\frac {\cos (b x) \text {Si}(b x)}{x}-\frac {1}{2} b \text {Si}(b x)^2+\int \frac {\sin (2 b x)}{2 x^2} \, dx\\ &=-\frac {\cos (b x) \text {Si}(b x)}{x}-\frac {1}{2} b \text {Si}(b x)^2+\frac {1}{2} \int \frac {\sin (2 b x)}{x^2} \, dx\\ &=-\frac {\sin (2 b x)}{2 x}-\frac {\cos (b x) \text {Si}(b x)}{x}-\frac {1}{2} b \text {Si}(b x)^2+b \int \frac {\cos (2 b x)}{x} \, dx\\ &=b \text {Ci}(2 b x)-\frac {\sin (2 b x)}{2 x}-\frac {\cos (b x) \text {Si}(b x)}{x}-\frac {1}{2} b \text {Si}(b x)^2\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.69, size = 0, normalized size = 0.00 \[ \int \frac {\cos (b x) \text {Si}(b x)}{x^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.31, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (b x\right ) \operatorname {Si}\left (b x\right )}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Si}\left (b x\right ) \cos \left (b x\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (b x \right ) \Si \left (b x \right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm Si}\left (b x\right ) \cos \left (b x\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {sinint}\left (b\,x\right )\,\cos \left (b\,x\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos {\left (b x \right )} \operatorname {Si}{\left (b x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________