Optimal. Leaf size=185 \[ -\frac{a^2 \text{Si}(2 a+2 b x)}{2 b^3}+\frac{a \text{CosIntegral}(2 a+2 b x)}{b^3}-\frac{2 \text{CosIntegral}(a+b x) \sin (a+b x)}{b^3}+\frac{2 x \text{CosIntegral}(a+b x) \cos (a+b x)}{b^2}+\frac{\text{Si}(2 a+2 b x)}{b^3}+\frac{a \log (a+b x)}{b^3}-\frac{\sin (2 a+2 b x)}{8 b^3}-\frac{a \cos (2 a+2 b x)}{4 b^3}+\frac{x \cos (2 a+2 b x)}{4 b^2}-\frac{\sin (a+b x) \cos (a+b x)}{b^3}+\frac{x^2 \text{CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac{x}{b^2} \]
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Rubi [A] time = 0.585556, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 16, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6514, 4573, 6741, 6742, 2638, 3296, 2637, 3299, 6520, 2635, 8, 3312, 3302, 6512, 4406, 12} \[ -\frac{a^2 \text{Si}(2 a+2 b x)}{2 b^3}+\frac{a \text{CosIntegral}(2 a+2 b x)}{b^3}-\frac{2 \text{CosIntegral}(a+b x) \sin (a+b x)}{b^3}+\frac{2 x \text{CosIntegral}(a+b x) \cos (a+b x)}{b^2}+\frac{\text{Si}(2 a+2 b x)}{b^3}+\frac{a \log (a+b x)}{b^3}-\frac{\sin (2 a+2 b x)}{8 b^3}-\frac{a \cos (2 a+2 b x)}{4 b^3}+\frac{x \cos (2 a+2 b x)}{4 b^2}-\frac{\sin (a+b x) \cos (a+b x)}{b^3}+\frac{x^2 \text{CosIntegral}(a+b x) \sin (a+b x)}{b}-\frac{x}{b^2} \]
Antiderivative was successfully verified.
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Rule 6514
Rule 4573
Rule 6741
Rule 6742
Rule 2638
Rule 3296
Rule 2637
Rule 3299
Rule 6520
Rule 2635
Rule 8
Rule 3312
Rule 3302
Rule 6512
Rule 4406
Rule 12
Rubi steps
\begin{align*} \int x^2 \cos (a+b x) \text{Ci}(a+b x) \, dx &=\frac{x^2 \text{Ci}(a+b x) \sin (a+b x)}{b}-\frac{2 \int x \text{Ci}(a+b x) \sin (a+b x) \, dx}{b}-\int \frac{x^2 \cos (a+b x) \sin (a+b x)}{a+b x} \, dx\\ &=\frac{2 x \cos (a+b x) \text{Ci}(a+b x)}{b^2}+\frac{x^2 \text{Ci}(a+b x) \sin (a+b x)}{b}-\frac{1}{2} \int \frac{x^2 \sin (2 (a+b x))}{a+b x} \, dx-\frac{2 \int \cos (a+b x) \text{Ci}(a+b x) \, dx}{b^2}-\frac{2 \int \frac{x \cos ^2(a+b x)}{a+b x} \, dx}{b}\\ &=\frac{2 x \cos (a+b x) \text{Ci}(a+b x)}{b^2}-\frac{2 \text{Ci}(a+b x) \sin (a+b x)}{b^3}+\frac{x^2 \text{Ci}(a+b x) \sin (a+b x)}{b}-\frac{1}{2} \int \frac{x^2 \sin (2 a+2 b x)}{a+b x} \, dx+\frac{2 \int \frac{\cos (a+b x) \sin (a+b x)}{a+b x} \, dx}{b^2}-\frac{2 \int \left (\frac{\cos ^2(a+b x)}{b}-\frac{a \cos ^2(a+b x)}{b (a+b x)}\right ) \, dx}{b}\\ &=\frac{2 x \cos (a+b x) \text{Ci}(a+b x)}{b^2}-\frac{2 \text{Ci}(a+b x) \sin (a+b x)}{b^3}+\frac{x^2 \text{Ci}(a+b x) \sin (a+b x)}{b}-\frac{1}{2} \int \left (-\frac{a \sin (2 a+2 b x)}{b^2}+\frac{x \sin (2 a+2 b x)}{b}+\frac{a^2 \sin (2 a+2 b x)}{b^2 (a+b x)}\right ) \, dx-\frac{2 \int \cos ^2(a+b x) \, dx}{b^2}+\frac{2 \int \frac{\sin (2 a+2 b x)}{2 (a+b x)} \, dx}{b^2}+\frac{(2 a) \int \frac{\cos ^2(a+b x)}{a+b x} \, dx}{b^2}\\ &=\frac{2 x \cos (a+b x) \text{Ci}(a+b x)}{b^2}-\frac{\cos (a+b x) \sin (a+b x)}{b^3}-\frac{2 \text{Ci}(a+b x) \sin (a+b x)}{b^3}+\frac{x^2 \text{Ci}(a+b x) \sin (a+b x)}{b}-\frac{\int 1 \, dx}{b^2}+\frac{\int \frac{\sin (2 a+2 b x)}{a+b x} \, dx}{b^2}+\frac{a \int \sin (2 a+2 b x) \, dx}{2 b^2}+\frac{(2 a) \int \left (\frac{1}{2 (a+b x)}+\frac{\cos (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{b^2}-\frac{a^2 \int \frac{\sin (2 a+2 b x)}{a+b x} \, dx}{2 b^2}-\frac{\int x \sin (2 a+2 b x) \, dx}{2 b}\\ &=-\frac{x}{b^2}-\frac{a \cos (2 a+2 b x)}{4 b^3}+\frac{x \cos (2 a+2 b x)}{4 b^2}+\frac{2 x \cos (a+b x) \text{Ci}(a+b x)}{b^2}+\frac{a \log (a+b x)}{b^3}-\frac{\cos (a+b x) \sin (a+b x)}{b^3}-\frac{2 \text{Ci}(a+b x) \sin (a+b x)}{b^3}+\frac{x^2 \text{Ci}(a+b x) \sin (a+b x)}{b}+\frac{\text{Si}(2 a+2 b x)}{b^3}-\frac{a^2 \text{Si}(2 a+2 b x)}{2 b^3}-\frac{\int \cos (2 a+2 b x) \, dx}{4 b^2}+\frac{a \int \frac{\cos (2 a+2 b x)}{a+b x} \, dx}{b^2}\\ &=-\frac{x}{b^2}-\frac{a \cos (2 a+2 b x)}{4 b^3}+\frac{x \cos (2 a+2 b x)}{4 b^2}+\frac{2 x \cos (a+b x) \text{Ci}(a+b x)}{b^2}+\frac{a \text{Ci}(2 a+2 b x)}{b^3}+\frac{a \log (a+b x)}{b^3}-\frac{\cos (a+b x) \sin (a+b x)}{b^3}-\frac{2 \text{Ci}(a+b x) \sin (a+b x)}{b^3}+\frac{x^2 \text{Ci}(a+b x) \sin (a+b x)}{b}-\frac{\sin (2 a+2 b x)}{8 b^3}+\frac{\text{Si}(2 a+2 b x)}{b^3}-\frac{a^2 \text{Si}(2 a+2 b x)}{2 b^3}\\ \end{align*}
Mathematica [A] time = 0.300744, size = 123, normalized size = 0.66 \[ \frac{-4 a^2 \text{Si}(2 (a+b x))+8 \text{CosIntegral}(a+b x) \left (\left (b^2 x^2-2\right ) \sin (a+b x)+2 b x \cos (a+b x)\right )+8 a \text{CosIntegral}(2 (a+b x))+8 \text{Si}(2 (a+b x))+8 a \log (a+b x)-5 \sin (2 (a+b x))-2 a \cos (2 (a+b x))+2 b x \cos (2 (a+b x))-8 b x}{8 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.083, size = 168, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}{\it Ci} \left ( bx+a \right ) \sin \left ( bx+a \right ) }{b}}+2\,{\frac{x{\it Ci} \left ( bx+a \right ) \cos \left ( bx+a \right ) }{{b}^{2}}}-2\,{\frac{{\it Ci} \left ( bx+a \right ) \sin \left ( bx+a \right ) }{{b}^{3}}}+{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{2}x}{2\,{b}^{2}}}-{\frac{a \left ( \cos \left ( bx+a \right ) \right ) ^{2}}{2\,{b}^{3}}}-{\frac{5\,\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{4\,{b}^{3}}}-{\frac{5\,x}{4\,{b}^{2}}}-{\frac{5\,a}{4\,{b}^{3}}}-{\frac{{a}^{2}{\it Si} \left ( 2\,bx+2\,a \right ) }{2\,{b}^{3}}}+{\frac{a\ln \left ( bx+a \right ) }{{b}^{3}}}+{\frac{a{\it Ci} \left ( 2\,bx+2\,a \right ) }{{b}^{3}}}+{\frac{{\it Si} \left ( 2\,bx+2\,a \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2}{\rm Ci}\left (b x + a\right ) \cos \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \cos \left (b x + a\right ) \operatorname{Ci}\left (b x + a\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \cos{\left (a + b x \right )} \operatorname{Ci}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.20779, size = 221, normalized size = 1.19 \begin{align*}{\left (\frac{2 \, x \cos \left (b x + a\right )}{b^{2}} + \frac{{\left (b^{2} x^{2} - 2\right )} \sin \left (b x + a\right )}{b^{3}}\right )} \operatorname{Ci}\left (b x + a\right ) - \frac{a^{2} \Im \left ( \operatorname{Ci}\left (2 \, b x + 2 \, a\right ) \right ) - a^{2} \Im \left ( \operatorname{Ci}\left (-2 \, b x - 2 \, a\right ) \right ) + 2 \, a^{2} \operatorname{Si}\left (2 \, b x + 2 \, a\right ) + 4 \, b x - 4 \, a \log \left ({\left | b x + a \right |}\right ) - 2 \, a \Re \left ( \operatorname{Ci}\left (2 \, b x + 2 \, a\right ) \right ) - 2 \, a \Re \left ( \operatorname{Ci}\left (-2 \, b x - 2 \, a\right ) \right ) - 2 \, \Im \left ( \operatorname{Ci}\left (2 \, b x + 2 \, a\right ) \right ) + 2 \, \Im \left ( \operatorname{Ci}\left (-2 \, b x - 2 \, a\right ) \right ) - 4 \, \operatorname{Si}\left (2 \, b x + 2 \, a\right )}{4 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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