Optimal. Leaf size=220 \[ \frac {a^2 \text {Ci}(2 a+2 b x)}{2 b^3}+\frac {a^2 \log (a+b x)}{2 b^3}-\frac {\text {Ci}(2 a+2 b x)}{b^3}+\frac {2 \text {Ci}(a+b x) \cos (a+b x)}{b^3}+\frac {a \text {Si}(2 a+2 b x)}{b^3}-\frac {\log (a+b x)}{b^3}+\frac {\cos ^2(a+b x)}{4 b^3}+\frac {\cos (2 a+2 b x)}{2 b^3}-\frac {a \sin (a+b x) \cos (a+b x)}{2 b^3}+\frac {2 x \text {Ci}(a+b x) \sin (a+b x)}{b^2}-\frac {a x}{2 b^2}+\frac {x \sin (a+b x) \cos (a+b x)}{2 b^2}-\frac {x^2 \text {Ci}(a+b x) \cos (a+b x)}{b}+\frac {x^2}{4 b} \]
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Rubi [A] time = 0.72, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6520, 6742, 2635, 8, 3310, 30, 3312, 3302, 6514, 4573, 6741, 2638, 3299, 6518} \[ \frac {a^2 \text {CosIntegral}(2 a+2 b x)}{2 b^3}+\frac {a^2 \log (a+b x)}{2 b^3}-\frac {\text {CosIntegral}(2 a+2 b x)}{b^3}+\frac {2 x \text {CosIntegral}(a+b x) \sin (a+b x)}{b^2}+\frac {2 \text {CosIntegral}(a+b x) \cos (a+b x)}{b^3}+\frac {a \text {Si}(2 a+2 b x)}{b^3}-\frac {a x}{2 b^2}-\frac {\log (a+b x)}{b^3}+\frac {\cos ^2(a+b x)}{4 b^3}+\frac {\cos (2 a+2 b x)}{2 b^3}-\frac {a \sin (a+b x) \cos (a+b x)}{2 b^3}+\frac {x \sin (a+b x) \cos (a+b x)}{2 b^2}-\frac {x^2 \text {CosIntegral}(a+b x) \cos (a+b x)}{b}+\frac {x^2}{4 b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2635
Rule 2638
Rule 3299
Rule 3302
Rule 3310
Rule 3312
Rule 4573
Rule 6514
Rule 6518
Rule 6520
Rule 6741
Rule 6742
Rubi steps
\begin {align*} \int x^2 \text {Ci}(a+b x) \sin (a+b x) \, dx &=-\frac {x^2 \cos (a+b x) \text {Ci}(a+b x)}{b}+\frac {2 \int x \cos (a+b x) \text {Ci}(a+b x) \, dx}{b}+\int \frac {x^2 \cos ^2(a+b x)}{a+b x} \, dx\\ &=-\frac {x^2 \cos (a+b x) \text {Ci}(a+b x)}{b}+\frac {2 x \text {Ci}(a+b x) \sin (a+b x)}{b^2}-\frac {2 \int \text {Ci}(a+b x) \sin (a+b x) \, dx}{b^2}-\frac {2 \int \frac {x \cos (a+b x) \sin (a+b x)}{a+b x} \, dx}{b}+\int \left (-\frac {a \cos ^2(a+b x)}{b^2}+\frac {x \cos ^2(a+b x)}{b}+\frac {a^2 \cos ^2(a+b x)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac {2 \cos (a+b x) \text {Ci}(a+b x)}{b^3}-\frac {x^2 \cos (a+b x) \text {Ci}(a+b x)}{b}+\frac {2 x \text {Ci}(a+b x) \sin (a+b x)}{b^2}-\frac {2 \int \frac {\cos ^2(a+b x)}{a+b x} \, dx}{b^2}-\frac {a \int \cos ^2(a+b x) \, dx}{b^2}+\frac {a^2 \int \frac {\cos ^2(a+b x)}{a+b x} \, dx}{b^2}+\frac {\int x \cos ^2(a+b x) \, dx}{b}-\frac {\int \frac {x \sin (2 (a+b x))}{a+b x} \, dx}{b}\\ &=\frac {\cos ^2(a+b x)}{4 b^3}+\frac {2 \cos (a+b x) \text {Ci}(a+b x)}{b^3}-\frac {x^2 \cos (a+b x) \text {Ci}(a+b x)}{b}-\frac {a \cos (a+b x) \sin (a+b x)}{2 b^3}+\frac {x \cos (a+b x) \sin (a+b x)}{2 b^2}+\frac {2 x \text {Ci}(a+b x) \sin (a+b x)}{b^2}-\frac {2 \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 \int 1 \, dx}{2 b^2}+\frac {a^2 \int \left (\frac {1}{2 (a+b x)}+\frac {\cos (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{b^2}+\frac {\int x \, dx}{2 b}-\frac {\int \frac {x \sin (2 a+2 b x)}{a+b x} \, dx}{b}\\ &=-\frac {a x}{2 b^2}+\frac {x^2}{4 b}+\frac {\cos ^2(a+b x)}{4 b^3}+\frac {2 \cos (a+b x) \text {Ci}(a+b x)}{b^3}-\frac {x^2 \cos (a+b x) \text {Ci}(a+b x)}{b}-\frac {\log (a+b x)}{b^3}+\frac {a^2 \log (a+b x)}{2 b^3}-\frac {a \cos (a+b x) \sin (a+b x)}{2 b^3}+\frac {x \cos (a+b x) \sin (a+b x)}{2 b^2}+\frac {2 x \text {Ci}(a+b x) \sin (a+b x)}{b^2}-\frac {\int \frac {\cos (2 a+2 b x)}{a+b x} \, dx}{b^2}+\frac {a^2 \int \frac {\cos (2 a+2 b x)}{a+b x} \, dx}{2 b^2}-\frac {\int \left (\frac {\sin (2 a+2 b x)}{b}+\frac {a \sin (2 a+2 b x)}{b (-a-b x)}\right ) \, dx}{b}\\ &=-\frac {a x}{2 b^2}+\frac {x^2}{4 b}+\frac {\cos ^2(a+b x)}{4 b^3}+\frac {2 \cos (a+b x) \text {Ci}(a+b x)}{b^3}-\frac {x^2 \cos (a+b x) \text {Ci}(a+b x)}{b}-\frac {\text {Ci}(2 a+2 b x)}{b^3}+\frac {a^2 \text {Ci}(2 a+2 b x)}{2 b^3}-\frac {\log (a+b x)}{b^3}+\frac {a^2 \log (a+b x)}{2 b^3}-\frac {a \cos (a+b x) \sin (a+b x)}{2 b^3}+\frac {x \cos (a+b x) \sin (a+b x)}{2 b^2}+\frac {2 x \text {Ci}(a+b x) \sin (a+b x)}{b^2}-\frac {\int \sin (2 a+2 b x) \, dx}{b^2}-\frac {a \int \frac {\sin (2 a+2 b x)}{-a-b x} \, dx}{b^2}\\ &=-\frac {a x}{2 b^2}+\frac {x^2}{4 b}+\frac {\cos ^2(a+b x)}{4 b^3}+\frac {\cos (2 a+2 b x)}{2 b^3}+\frac {2 \cos (a+b x) \text {Ci}(a+b x)}{b^3}-\frac {x^2 \cos (a+b x) \text {Ci}(a+b x)}{b}-\frac {\text {Ci}(2 a+2 b x)}{b^3}+\frac {a^2 \text {Ci}(2 a+2 b x)}{2 b^3}-\frac {\log (a+b x)}{b^3}+\frac {a^2 \log (a+b x)}{2 b^3}-\frac {a \cos (a+b x) \sin (a+b x)}{2 b^3}+\frac {x \cos (a+b x) \sin (a+b x)}{2 b^2}+\frac {2 x \text {Ci}(a+b x) \sin (a+b x)}{b^2}+\frac {a \text {Si}(2 a+2 b x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 134, normalized size = 0.61 \[ \frac {4 \left (a^2-2\right ) \text {Ci}(2 (a+b x))+4 a^2 \log (a+b x)-8 \text {Ci}(a+b x) \left (\left (b^2 x^2-2\right ) \cos (a+b x)-2 b x \sin (a+b x)\right )+8 a \text {Si}(2 (a+b x))-4 a b x-8 \log (a+b x)-2 a \sin (2 (a+b x))+2 b x \sin (2 (a+b x))+5 \cos (2 (a+b x))+2 b^2 x^2}{8 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \operatorname {Ci}\left (b x + a\right ) \sin \left (b x + a\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.23, size = 432, normalized size = 1.96 \[ {\left (\frac {2 \, x \sin \left (b x + a\right )}{b^{2}} - \frac {{\left (b^{2} x^{2} - 2\right )} \cos \left (b x + a\right )}{b^{3}}\right )} \operatorname {Ci}\left (b x + a\right ) + \frac {2 \, b^{2} x^{2} \tan \left (b x + a\right )^{2} - 4 \, a b x \tan \left (b x + a\right )^{2} + 4 \, a^{2} \log \left ({\left | b x + a \right |}\right ) \tan \left (b x + a\right )^{2} + 2 \, a^{2} \Re \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} + 2 \, a^{2} \Re \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} + 2 \, b^{2} x^{2} + 4 \, a \Im \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} - 4 \, a \Im \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} + 8 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) \tan \left (b x + a\right )^{2} - 4 \, a b x + 4 \, a^{2} \log \left ({\left | b x + a \right |}\right ) + 2 \, a^{2} \Re \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) + 2 \, a^{2} \Re \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) + 4 \, b x \tan \left (b x + a\right ) - 8 \, \log \left ({\left | b x + a \right |}\right ) \tan \left (b x + a\right )^{2} - 4 \, \Re \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} - 4 \, \Re \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) \tan \left (b x + a\right )^{2} + 4 \, a \Im \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) - 4 \, a \Im \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) + 8 \, a \operatorname {Si}\left (2 \, b x + 2 \, a\right ) - 4 \, a \tan \left (b x + a\right ) - 5 \, \tan \left (b x + a\right )^{2} - 8 \, \log \left ({\left | b x + a \right |}\right ) - 4 \, \Re \left (\operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \right ) - 4 \, \Re \left (\operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \right ) + 5}{8 \, {\left (b^{3} \tan \left (b x + a\right )^{2} + b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 211, normalized size = 0.96 \[ -\frac {x^{2} \Ci \left (b x +a \right ) \cos \left (b x +a \right )}{b}+\frac {2 x \Ci \left (b x +a \right ) \sin \left (b x +a \right )}{b^{2}}+\frac {2 \Ci \left (b x +a \right ) \cos \left (b x +a \right )}{b^{3}}+\frac {x \cos \left (b x +a \right ) \sin \left (b x +a \right )}{2 b^{2}}-\frac {a \cos \left (b x +a \right ) \sin \left (b x +a \right )}{2 b^{3}}+\frac {x^{2}}{4 b}-\frac {a x}{2 b^{2}}-\frac {3 a^{2}}{4 b^{3}}-\frac {\sin ^{2}\left (b x +a \right )}{4 b^{3}}+\frac {a^{2} \ln \left (b x +a \right )}{2 b^{3}}+\frac {a^{2} \Ci \left (2 b x +2 a \right )}{2 b^{3}}+\frac {\cos ^{2}\left (b x +a \right )}{b^{3}}+\frac {a \Si \left (2 b x +2 a \right )}{b^{3}}-\frac {\ln \left (b x +a \right )}{b^{3}}-\frac {\Ci \left (2 b x +2 a \right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} {\rm Ci}\left (b x + a\right ) \sin \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,\mathrm {cosint}\left (a+b\,x\right )\,\sin \left (a+b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \sin {\left (a + b x \right )} \operatorname {Ci}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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