Optimal. Leaf size=26 \[ \frac {\cos (a+b x)}{b}+\frac {(a+b x) \text {Si}(a+b x)}{b} \]
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Rubi [A]
time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6634}
\begin {gather*} \frac {(a+b x) \text {Si}(a+b x)}{b}+\frac {\cos (a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 6634
Rubi steps
\begin {align*} \int \text {Si}(a+b x) \, dx &=\frac {\cos (a+b x)}{b}+\frac {(a+b x) \text {Si}(a+b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 41, normalized size = 1.58 \begin {gather*} \frac {\cos (a) \cos (b x)}{b}-\frac {\sin (a) \sin (b x)}{b}+\frac {a \text {Si}(a+b x)}{b}+x \text {Si}(a+b x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 24, normalized size = 0.92
method | result | size |
derivativedivides | \(\frac {\sinIntegral \left (b x +a \right ) \left (b x +a \right )+\cos \left (b x +a \right )}{b}\) | \(24\) |
default | \(\frac {\sinIntegral \left (b x +a \right ) \left (b x +a \right )+\cos \left (b x +a \right )}{b}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 23, normalized size = 0.88 \begin {gather*} \frac {{\left (b x + a\right )} \operatorname {Si}\left (b x + a\right ) + \cos \left (b x + a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 23, normalized size = 0.88 \begin {gather*} \frac {{\left (b x + a\right )} \operatorname {Si}\left (b x + a\right ) + \cos \left (b x + a\right )}{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 \operatorname {Si}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.41, size = 303, normalized size = 11.65 \begin {gather*} x \operatorname {Si}\left (b x + a\right ) + \frac {{\left (a \Im \left ( \operatorname {Ci}\left (b x + a\right ) \right ) \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} - a \Im \left ( \operatorname {Ci}\left (-b x - a\right ) \right ) \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} + 2 \, a \operatorname {Si}\left (b x + a\right ) \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} + a \Im \left ( \operatorname {Ci}\left (b x + a\right ) \right ) \tan \left (\frac {1}{2} \, b x\right )^{2} - a \Im \left ( \operatorname {Ci}\left (-b x - a\right ) \right ) \tan \left (\frac {1}{2} \, b x\right )^{2} + 2 \, a \operatorname {Si}\left (b x + a\right ) \tan \left (\frac {1}{2} \, b x\right )^{2} + a \Im \left ( \operatorname {Ci}\left (b x + a\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} - a \Im \left ( \operatorname {Ci}\left (-b x - a\right ) \right ) \tan \left (\frac {1}{2} \, a\right )^{2} + 2 \, a \operatorname {Si}\left (b x + a\right ) \tan \left (\frac {1}{2} \, a\right )^{2} + 2 \, \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} + a \Im \left ( \operatorname {Ci}\left (b x + a\right ) \right ) - a \Im \left ( \operatorname {Ci}\left (-b x - a\right ) \right ) + 2 \, a \operatorname {Si}\left (b x + a\right ) - 2 \, \tan \left (\frac {1}{2} \, b x\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, b x\right ) \tan \left (\frac {1}{2} \, a\right ) - 2 \, \tan \left (\frac {1}{2} \, a\right )^{2} + 2\right )} b}{2 \, {\left (b^{2} \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} + b^{2} \tan \left (\frac {1}{2} \, b x\right )^{2} + b^{2} \tan \left (\frac {1}{2} \, a\right )^{2} + b^{2}\right )}} \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 = -0.04 \begin {gather*} x\,\mathrm {sinint}\left (a+b\,x\right )+\frac {\cos \left (a+b\,x\right )+a\,\mathrm {sinint}\left (a+b\,x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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