Optimal. Leaf size=370 \[ \frac {\sin \left (a-\frac {b c}{d}\right ) \text {Ci}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b^2}-\frac {\sin \left (a-\frac {b c}{d}\right ) \text {Ci}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b^2}+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b^2}+\frac {\cos (a+b x) \text {Si}(c+d x)}{b^2}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b^2}+\frac {c \cos \left (a-\frac {b c}{d}\right ) \text {Ci}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b d}-\frac {c \cos \left (a-\frac {b c}{d}\right ) \text {Ci}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b d}-\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b d}+\frac {x \sin (a+b x) \text {Si}(c+d x)}{b}+\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b d}-\frac {\sin (a+x (b-d)-c)}{2 b (b-d)}+\frac {\sin (a+x (b+d)+c)}{2 b (b+d)} \]
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Rubi [A] time = 1.24, antiderivative size = 370, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6519, 4608, 6742, 2637, 3303, 3299, 3302, 6511, 4430} \[ \frac {\sin \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {c (b-d)}{d}+x (b-d)\right )}{2 b^2}-\frac {\sin \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {c (b+d)}{d}+x (b+d)\right )}{2 b^2}+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b^2}+\frac {\cos (a+b x) \text {Si}(c+d x)}{b^2}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b^2}+\frac {c \cos \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {c (b-d)}{d}+x (b-d)\right )}{2 b d}-\frac {c \cos \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {c (b+d)}{d}+x (b+d)\right )}{2 b d}-\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b-d)+\frac {c (b-d)}{d}\right )}{2 b d}+\frac {x \sin (a+b x) \text {Si}(c+d x)}{b}+\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (x (b+d)+\frac {c (b+d)}{d}\right )}{2 b d}-\frac {\sin (a+x (b-d)-c)}{2 b (b-d)}+\frac {\sin (a+x (b+d)+c)}{2 b (b+d)} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3299
Rule 3302
Rule 3303
Rule 4430
Rule 4608
Rule 6511
Rule 6519
Rule 6742
Rubi steps
\begin {align*} \int x \cos (a+b x) \text {Si}(c+d x) \, dx &=\frac {x \sin (a+b x) \text {Si}(c+d x)}{b}-\frac {\int \sin (a+b x) \text {Si}(c+d x) \, dx}{b}-\frac {d \int \frac {x \sin (a+b x) \sin (c+d x)}{c+d x} \, dx}{b}\\ &=\frac {\cos (a+b x) \text {Si}(c+d x)}{b^2}+\frac {x \sin (a+b x) \text {Si}(c+d x)}{b}-\frac {d \int \frac {\cos (a+b x) \sin (c+d x)}{c+d x} \, dx}{b^2}-\frac {d \int \left (\frac {x \cos (a-c+(b-d) x)}{2 (c+d x)}-\frac {x \cos (a+c+(b+d) x)}{2 (c+d x)}\right ) \, dx}{b}\\ &=\frac {\cos (a+b x) \text {Si}(c+d x)}{b^2}+\frac {x \sin (a+b x) \text {Si}(c+d x)}{b}-\frac {d \int \left (-\frac {\sin (a-c+(b-d) x)}{2 (c+d x)}+\frac {\sin (a+c+(b+d) x)}{2 (c+d x)}\right ) \, dx}{b^2}-\frac {d \int \frac {x \cos (a-c+(b-d) x)}{c+d x} \, dx}{2 b}+\frac {d \int \frac {x \cos (a+c+(b+d) x)}{c+d x} \, dx}{2 b}\\ &=\frac {\cos (a+b x) \text {Si}(c+d x)}{b^2}+\frac {x \sin (a+b x) \text {Si}(c+d x)}{b}+\frac {d \int \frac {\sin (a-c+(b-d) x)}{c+d x} \, dx}{2 b^2}-\frac {d \int \frac {\sin (a+c+(b+d) x)}{c+d x} \, dx}{2 b^2}-\frac {d \int \left (\frac {\cos (a-c+(b-d) x)}{d}-\frac {c \cos (a-c+(b-d) x)}{d (c+d x)}\right ) \, dx}{2 b}+\frac {d \int \left (\frac {\cos (a+c+(b+d) x)}{d}-\frac {c \cos (a+c+(b+d) x)}{d (c+d x)}\right ) \, dx}{2 b}\\ &=\frac {\cos (a+b x) \text {Si}(c+d x)}{b^2}+\frac {x \sin (a+b x) \text {Si}(c+d x)}{b}-\frac {\int \cos (a-c+(b-d) x) \, dx}{2 b}+\frac {\int \cos (a+c+(b+d) x) \, dx}{2 b}+\frac {c \int \frac {\cos (a-c+(b-d) x)}{c+d x} \, dx}{2 b}-\frac {c \int \frac {\cos (a+c+(b+d) x)}{c+d x} \, dx}{2 b}+\frac {\left (d \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {c (b-d)}{d}+(b-d) x\right )}{c+d x} \, dx}{2 b^2}-\frac {\left (d \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b^2}+\frac {\left (d \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {c (b-d)}{d}+(b-d) x\right )}{c+d x} \, dx}{2 b^2}-\frac {\left (d \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b^2}\\ &=\frac {\text {Ci}\left (\frac {c (b-d)}{d}+(b-d) x\right ) \sin \left (a-\frac {b c}{d}\right )}{2 b^2}-\frac {\text {Ci}\left (\frac {c (b+d)}{d}+(b+d) x\right ) \sin \left (a-\frac {b c}{d}\right )}{2 b^2}-\frac {\sin (a-c+(b-d) x)}{2 b (b-d)}+\frac {\sin (a+c+(b+d) x)}{2 b (b+d)}+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}+\frac {\cos (a+b x) \text {Si}(c+d x)}{b^2}+\frac {x \sin (a+b x) \text {Si}(c+d x)}{b}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b^2}+\frac {\left (c \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {c (b-d)}{d}+(b-d) x\right )}{c+d x} \, dx}{2 b}-\frac {\left (c \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b}-\frac {\left (c \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {c (b-d)}{d}+(b-d) x\right )}{c+d x} \, dx}{2 b}+\frac {\left (c \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {c (b+d)}{d}+(b+d) x\right )}{c+d x} \, dx}{2 b}\\ &=\frac {c \cos \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b d}-\frac {c \cos \left (a-\frac {b c}{d}\right ) \text {Ci}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b d}+\frac {\text {Ci}\left (\frac {c (b-d)}{d}+(b-d) x\right ) \sin \left (a-\frac {b c}{d}\right )}{2 b^2}-\frac {\text {Ci}\left (\frac {c (b+d)}{d}+(b+d) x\right ) \sin \left (a-\frac {b c}{d}\right )}{2 b^2}-\frac {\sin (a-c+(b-d) x)}{2 b (b-d)}+\frac {\sin (a+c+(b+d) x)}{2 b (b+d)}+\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b^2}-\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b-d)}{d}+(b-d) x\right )}{2 b d}+\frac {\cos (a+b x) \text {Si}(c+d x)}{b^2}+\frac {x \sin (a+b x) \text {Si}(c+d x)}{b}-\frac {\cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b^2}+\frac {c \sin \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {c (b+d)}{d}+(b+d) x\right )}{2 b d}\\ \end {align*}
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Mathematica [C] time = 5.73, size = 343, normalized size = 0.93 \[ -\frac {e^{-i (a+c)} \left ((-b c+i d) e^{\frac {i (d (2 a+c)-b c)}{d}} \text {Ei}\left (\frac {i (b-d) (c+d x)}{d}\right )-i b d \left (\frac {e^{i (2 a+x (b-d))}}{b-d}+\frac {e^{-i x (b+d)}}{b+d}\right )+(b c+i d) e^{\frac {i c (b+d)}{d}} \text {Ei}\left (-\frac {i (b+d) (c+d x)}{d}\right )\right )}{4 b^2 d}+\frac {e^{-i (a-c)} \left ((-b c+i d) e^{2 i a-\frac {i c (b+d)}{d}} \text {Ei}\left (\frac {i (b+d) (c+d x)}{d}\right )-i b d \left (\frac {e^{i (2 a+x (b+d))}}{b+d}+\frac {e^{-i x (b-d)}}{b-d}\right )+(b c+i d) e^{\frac {i c (b-d)}{d}} \text {Ei}\left (-\frac {i (b-d) (c+d x)}{d}\right )\right )}{4 b^2 d}+\frac {\text {Si}(c+d x) (b x \sin (a+b x)+\cos (a+b x))}{b^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \cos \left (b x + a\right ) \operatorname {Si}\left (d x + c\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 1208, normalized size = 3.26 \[ \text {result too large to display} \]
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 \[ \int x {\rm Si}\left (d x + c\right ) \cos \left (b x + a\right )\,{d x} \]
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 \[ \int x\,\mathrm {sinint}\left (c+d\,x\right )\,\cos \left (a+b\,x\right ) \,d x \]
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 \[ \int x \cos {\left (a + b x \right )} \operatorname {Si}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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