Optimal. Leaf size=47 \[ \frac {\text {Ci}(2 a+2 b x)}{2 b}-\frac {\text {Ci}(a+b x) \cos (a+b x)}{b}+\frac {\log (a+b x)}{2 b} \]
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Rubi [A] time = 0.08, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6518, 3312, 3302} \[ \frac {\text {CosIntegral}(2 a+2 b x)}{2 b}-\frac {\text {CosIntegral}(a+b x) \cos (a+b x)}{b}+\frac {\log (a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 3302
Rule 3312
Rule 6518
Rubi steps
\begin {align*} \int \text {Ci}(a+b x) \sin (a+b x) \, dx &=-\frac {\cos (a+b x) \text {Ci}(a+b x)}{b}+\int \frac {\cos ^2(a+b x)}{a+b x} \, dx\\ &=-\frac {\cos (a+b x) \text {Ci}(a+b x)}{b}+\int \left (\frac {1}{2 (a+b x)}+\frac {\cos (2 a+2 b x)}{2 (a+b x)}\right ) \, dx\\ &=-\frac {\cos (a+b x) \text {Ci}(a+b x)}{b}+\frac {\log (a+b x)}{2 b}+\frac {1}{2} \int \frac {\cos (2 a+2 b x)}{a+b x} \, dx\\ &=-\frac {\cos (a+b x) \text {Ci}(a+b x)}{b}+\frac {\text {Ci}(2 a+2 b x)}{2 b}+\frac {\log (a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 46, normalized size = 0.98 \[ \frac {\text {Ci}(2 (a+b x))}{2 b}-\frac {\text {Ci}(a+b x) \cos (a+b x)}{b}+\frac {\log (a+b x)}{2 b} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {Ci}\left (b x + a\right ) \sin \left (b x + a\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 96, normalized size = 2.04 \[ -\frac {\cos \left (b x + a\right ) \operatorname {Ci}\left (b x + a\right )}{b} + \frac {\cos \left (2 \, a\right )^{2} \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) + \cos \left (2 \, a\right )^{2} \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) + \operatorname {Ci}\left (2 \, b x + 2 \, a\right ) \sin \left (2 \, a\right )^{2} + \operatorname {Ci}\left (-2 \, b x - 2 \, a\right ) \sin \left (2 \, a\right )^{2} + 2 \, \log \left (b x + a\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 44, normalized size = 0.94 \[ \frac {\Ci \left (2 b x +2 a \right )}{2 b}-\frac {\Ci \left (b x +a \right ) \cos \left (b x +a \right )}{b}+\frac {\ln \left (b x +a \right )}{2 b} \]
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 \[ \int {\rm Ci}\left (b x + a\right ) \sin \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \frac {\ln \left (a+b\,x\right )}{2\,b}+\frac {\mathrm {cosint}\left (2\,a+2\,b\,x\right )}{2\,b}-\frac {\mathrm {cosint}\left (a+b\,x\right )\,\cos \left (a+b\,x\right )}{b} \]
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 \[ \int \sin {\left (a + b x \right )} \operatorname {Ci}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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