Optimal. Leaf size=34 \[ \frac {\text {CosIntegral}(2 b x)}{2 b}-\frac {\log (x)}{2 b}+\frac {\sin (b x) \text {Si}(b x)}{b} \]
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Rubi [A]
time = 0.04, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6652, 12, 3393,
3383} \begin {gather*} \frac {\text {CosIntegral}(2 b x)}{2 b}+\frac {\text {Si}(b x) \sin (b x)}{b}-\frac {\log (x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3383
Rule 3393
Rule 6652
Rubi steps
\begin {align*} \int \cos (b x) \text {Si}(b x) \, dx &=\frac {\sin (b x) \text {Si}(b x)}{b}-\int \frac {\sin ^2(b x)}{b x} \, dx\\ &=\frac {\sin (b x) \text {Si}(b x)}{b}-\frac {\int \frac {\sin ^2(b x)}{x} \, dx}{b}\\ &=\frac {\sin (b x) \text {Si}(b x)}{b}-\frac {\int \left (\frac {1}{2 x}-\frac {\cos (2 b x)}{2 x}\right ) \, dx}{b}\\ &=-\frac {\log (x)}{2 b}+\frac {\sin (b x) \text {Si}(b x)}{b}+\frac {\int \frac {\cos (2 b x)}{x} \, dx}{2 b}\\ &=\frac {\text {Ci}(2 b x)}{2 b}-\frac {\log (x)}{2 b}+\frac {\sin (b x) \text {Si}(b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 36, normalized size = 1.06 \begin {gather*} \frac {\text {CosIntegral}(2 b x)}{2 b}-\frac {\log (b x)}{2 b}+\frac {\sin (b x) \text {Si}(b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 28, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {\sinIntegral \left (b x \right ) \sin \left (b x \right )-\frac {\ln \left (b x \right )}{2}+\frac {\cosineIntegral \left (2 b x \right )}{2}}{b}\) | \(28\) |
default | \(\frac {\sinIntegral \left (b x \right ) \sin \left (b x \right )-\frac {\ln \left (b x \right )}{2}+\frac {\cosineIntegral \left (2 b x \right )}{2}}{b}\) | \(28\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 30, normalized size = 0.88 \begin {gather*} \frac {4 \, \sin \left (b x\right ) \operatorname {Si}\left (b x\right ) + \operatorname {Ci}\left (2 \, b x\right ) + \operatorname {Ci}\left (-2 \, b x\right ) - 2 \, \log \left (x\right )}{4 \, b} \end {gather*}
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 \begin {gather*} \int \cos {\left (b x \right )} \operatorname {Si}{\left (b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 33, normalized size = 0.97 \begin {gather*} \frac {\sin \left (b x\right ) \operatorname {Si}\left (b x\right )}{b} + \frac {\operatorname {Ci}\left (2 \, b x\right ) + \operatorname {Ci}\left (-2 \, b x\right ) - 2 \, \log \left (x\right )}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \frac {\mathrm {cosint}\left (2\,b\,x\right )-\ln \left (x\right )+2\,\mathrm {sinint}\left (b\,x\right )\,\sin \left (b\,x\right )}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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