Optimal. Leaf size=154 \[ -\frac{\text{CosIntegral}(2 a+2 b x)}{2 b^2}-\frac{a (a+b x) \text{Si}(a+b x)^2}{2 b^2}+\frac{a \text{Si}(2 a+2 b x)}{b^2}-\frac{\text{Si}(a+b x) \sin (a+b x)}{b^2}-\frac{a \text{Si}(a+b x) \cos (a+b x)}{b^2}+\frac{\log (a+b x)}{2 b^2}+\frac{\cos (2 a+2 b x)}{4 b^2}+\frac{x (a+b x) \text{Si}(a+b x)^2}{2 b}+\frac{x \text{Si}(a+b x) \cos (a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.333002, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 14, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.4, Rules used = {6509, 6513, 4573, 6741, 6742, 2638, 3299, 6517, 3312, 3302, 6505, 6511, 4406, 12} \[ -\frac{\text{CosIntegral}(2 a+2 b x)}{2 b^2}-\frac{a (a+b x) \text{Si}(a+b x)^2}{2 b^2}+\frac{a \text{Si}(2 a+2 b x)}{b^2}-\frac{\text{Si}(a+b x) \sin (a+b x)}{b^2}-\frac{a \text{Si}(a+b x) \cos (a+b x)}{b^2}+\frac{\log (a+b x)}{2 b^2}+\frac{\cos (2 a+2 b x)}{4 b^2}+\frac{x (a+b x) \text{Si}(a+b x)^2}{2 b}+\frac{x \text{Si}(a+b x) \cos (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6509
Rule 6513
Rule 4573
Rule 6741
Rule 6742
Rule 2638
Rule 3299
Rule 6517
Rule 3312
Rule 3302
Rule 6505
Rule 6511
Rule 4406
Rule 12
Rubi steps
\begin{align*} \int x \text{Si}(a+b x)^2 \, dx &=\frac{x (a+b x) \text{Si}(a+b x)^2}{2 b}-\frac{a \int \text{Si}(a+b x)^2 \, dx}{2 b}-\int x \sin (a+b x) \text{Si}(a+b x) \, dx\\ &=\frac{x \cos (a+b x) \text{Si}(a+b x)}{b}-\frac{a (a+b x) \text{Si}(a+b x)^2}{2 b^2}+\frac{x (a+b x) \text{Si}(a+b x)^2}{2 b}-\frac{\int \cos (a+b x) \text{Si}(a+b x) \, dx}{b}+\frac{a \int \sin (a+b x) \text{Si}(a+b x) \, dx}{b}-\int \frac{x \cos (a+b x) \sin (a+b x)}{a+b x} \, dx\\ &=-\frac{a \cos (a+b x) \text{Si}(a+b x)}{b^2}+\frac{x \cos (a+b x) \text{Si}(a+b x)}{b}-\frac{\sin (a+b x) \text{Si}(a+b x)}{b^2}-\frac{a (a+b x) \text{Si}(a+b x)^2}{2 b^2}+\frac{x (a+b x) \text{Si}(a+b x)^2}{2 b}-\frac{1}{2} \int \frac{x \sin (2 (a+b x))}{a+b x} \, dx+\frac{\int \frac{\sin ^2(a+b x)}{a+b x} \, dx}{b}+\frac{a \int \frac{\cos (a+b x) \sin (a+b x)}{a+b x} \, dx}{b}\\ &=-\frac{a \cos (a+b x) \text{Si}(a+b x)}{b^2}+\frac{x \cos (a+b x) \text{Si}(a+b x)}{b}-\frac{\sin (a+b x) \text{Si}(a+b x)}{b^2}-\frac{a (a+b x) \text{Si}(a+b x)^2}{2 b^2}+\frac{x (a+b x) \text{Si}(a+b x)^2}{2 b}-\frac{1}{2} \int \frac{x \sin (2 a+2 b x)}{a+b x} \, dx+\frac{\int \left (\frac{1}{2 (a+b x)}-\frac{\cos (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{b}+\frac{a \int \frac{\sin (2 a+2 b x)}{2 (a+b x)} \, dx}{b}\\ &=\frac{\log (a+b x)}{2 b^2}-\frac{a \cos (a+b x) \text{Si}(a+b x)}{b^2}+\frac{x \cos (a+b x) \text{Si}(a+b x)}{b}-\frac{\sin (a+b x) \text{Si}(a+b x)}{b^2}-\frac{a (a+b x) \text{Si}(a+b x)^2}{2 b^2}+\frac{x (a+b x) \text{Si}(a+b x)^2}{2 b}-\frac{1}{2} \int \left (\frac{\sin (2 a+2 b x)}{b}+\frac{a \sin (2 a+2 b x)}{b (-a-b x)}\right ) \, dx-\frac{\int \frac{\cos (2 a+2 b x)}{a+b x} \, dx}{2 b}+\frac{a \int \frac{\sin (2 a+2 b x)}{a+b x} \, dx}{2 b}\\ &=-\frac{\text{Ci}(2 a+2 b x)}{2 b^2}+\frac{\log (a+b x)}{2 b^2}-\frac{a \cos (a+b x) \text{Si}(a+b x)}{b^2}+\frac{x \cos (a+b x) \text{Si}(a+b x)}{b}-\frac{\sin (a+b x) \text{Si}(a+b x)}{b^2}-\frac{a (a+b x) \text{Si}(a+b x)^2}{2 b^2}+\frac{x (a+b x) \text{Si}(a+b x)^2}{2 b}+\frac{a \text{Si}(2 a+2 b x)}{2 b^2}-\frac{\int \sin (2 a+2 b x) \, dx}{2 b}-\frac{a \int \frac{\sin (2 a+2 b x)}{-a-b x} \, dx}{2 b}\\ &=\frac{\cos (2 a+2 b x)}{4 b^2}-\frac{\text{Ci}(2 a+2 b x)}{2 b^2}+\frac{\log (a+b x)}{2 b^2}-\frac{a \cos (a+b x) \text{Si}(a+b x)}{b^2}+\frac{x \cos (a+b x) \text{Si}(a+b x)}{b}-\frac{\sin (a+b x) \text{Si}(a+b x)}{b^2}-\frac{a (a+b x) \text{Si}(a+b x)^2}{2 b^2}+\frac{x (a+b x) \text{Si}(a+b x)^2}{2 b}+\frac{a \text{Si}(2 a+2 b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.280375, size = 95, normalized size = 0.62 \[ \frac{-2 \left (a^2-b^2 x^2\right ) \text{Si}(a+b x)^2-2 \text{CosIntegral}(2 (a+b x))+4 a \text{Si}(2 (a+b x))-4 \text{Si}(a+b x) (\sin (a+b x)+(a-b x) \cos (a+b x))+2 \log (a+b x)+\cos (2 (a+b x))}{4 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.066, size = 135, normalized size = 0.9 \begin{align*}{\frac{{x}^{2} \left ({\it Si} \left ( bx+a \right ) \right ) ^{2}}{2}}-{\frac{ \left ({\it Si} \left ( bx+a \right ) \right ) ^{2}{a}^{2}}{2\,{b}^{2}}}+{\frac{x\cos \left ( bx+a \right ){\it Si} \left ( bx+a \right ) }{b}}-{\frac{a\cos \left ( bx+a \right ){\it Si} \left ( bx+a \right ) }{{b}^{2}}}-{\frac{{\it Si} \left ( bx+a \right ) \sin \left ( bx+a \right ) }{{b}^{2}}}+{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{2}}{2\,{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) }{2\,{b}^{2}}}-{\frac{{\it Ci} \left ( 2\,bx+2\,a \right ) }{2\,{b}^{2}}}+{\frac{a{\it Si} \left ( 2\,bx+2\,a \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x{\rm Si}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x \operatorname{Si}\left (b x + a\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{Si}^{2}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x{\rm Si}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]