Optimal. Leaf size=31 \[ x \text {Ci}(b x)^2-\frac {2 \text {Ci}(b x) \sin (b x)}{b}+\frac {\text {Si}(2 b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6506, 6512, 12, 4406, 3299} \[ x \text {CosIntegral}(b x)^2-\frac {2 \text {CosIntegral}(b x) \sin (b x)}{b}+\frac {\text {Si}(2 b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 3299
Rule 4406
Rule 6506
Rule 6512
Rubi steps
\begin {align*} \int \text {Ci}(b x)^2 \, dx &=x \text {Ci}(b x)^2-2 \int \cos (b x) \text {Ci}(b x) \, dx\\ &=x \text {Ci}(b x)^2-\frac {2 \text {Ci}(b x) \sin (b x)}{b}+2 \int \frac {\cos (b x) \sin (b x)}{b x} \, dx\\ &=x \text {Ci}(b x)^2-\frac {2 \text {Ci}(b x) \sin (b x)}{b}+\frac {2 \int \frac {\cos (b x) \sin (b x)}{x} \, dx}{b}\\ &=x \text {Ci}(b x)^2-\frac {2 \text {Ci}(b x) \sin (b x)}{b}+\frac {2 \int \frac {\sin (2 b x)}{2 x} \, dx}{b}\\ &=x \text {Ci}(b x)^2-\frac {2 \text {Ci}(b x) \sin (b x)}{b}+\frac {\int \frac {\sin (2 b x)}{x} \, dx}{b}\\ &=x \text {Ci}(b x)^2-\frac {2 \text {Ci}(b x) \sin (b x)}{b}+\frac {\text {Si}(2 b x)}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 31, normalized size = 1.00 \[ x \text {Ci}(b x)^2-\frac {2 \text {Ci}(b x) \sin (b x)}{b}+\frac {\text {Si}(2 b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {Ci}\left (b x\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [C] time = 0.21, size = 49, normalized size = 1.58 \[ x \operatorname {Ci}\left (b x\right )^{2} - \frac {2 \, \operatorname {Ci}\left (b x\right ) \sin \left (b x\right )}{b} + \frac {\Im \left (\operatorname {Ci}\left (2 \, b x\right ) \right ) - \Im \left (\operatorname {Ci}\left (-2 \, b x\right ) \right ) + 2 \, \operatorname {Si}\left (2 \, b x\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 30, normalized size = 0.97 \[ \frac {b x \Ci \left (b x \right )^{2}-2 \Ci \left (b x \right ) \sin \left (b x \right )+\Si \left (2 b x \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm Ci}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\mathrm {cosint}\left (b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {Ci}^{2}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________