Optimal. Leaf size=29 \[ a \sin (c) \text {Ci}(d x)+a \cos (c) \text {Si}(d x)-\frac {b \cos (c+d x)}{d} \]
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Rubi [A] time = 0.15, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6742, 2638, 3303, 3299, 3302} \[ a \sin (c) \text {CosIntegral}(d x)+a \cos (c) \text {Si}(d x)-\frac {b \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3299
Rule 3302
Rule 3303
Rule 6742
Rubi steps
\begin {align*} \int \frac {(a+b x) \sin (c+d x)}{x} \, dx &=\int \left (b \sin (c+d x)+\frac {a \sin (c+d x)}{x}\right ) \, dx\\ &=a \int \frac {\sin (c+d x)}{x} \, dx+b \int \sin (c+d x) \, dx\\ &=-\frac {b \cos (c+d x)}{d}+(a \cos (c)) \int \frac {\sin (d x)}{x} \, dx+(a \sin (c)) \int \frac {\cos (d x)}{x} \, dx\\ &=-\frac {b \cos (c+d x)}{d}+a \text {Ci}(d x) \sin (c)+a \cos (c) \text {Si}(d x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 40, normalized size = 1.38 \[ a \sin (c) \text {Ci}(d x)+a \cos (c) \text {Si}(d x)+\frac {b \sin (c) \sin (d x)}{d}-\frac {b \cos (c) \cos (d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 44, normalized size = 1.52 \[ \frac {2 \, a d \cos \relax (c) \operatorname {Si}\left (d x\right ) - 2 \, b \cos \left (d x + c\right ) + {\left (a d \operatorname {Ci}\left (d x\right ) + a d \operatorname {Ci}\left (-d x\right )\right )} \sin \relax (c)}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.48, size = 339, normalized size = 11.69 \[ -\frac {a d \Im \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - a d \Im \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a d \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, a d \Re \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 2 \, a d \Re \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - a d \Im \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} + a d \Im \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} - 2 \, a d \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, d x\right )^{2} + a d \Im \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} - a d \Im \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a d \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, a d \Re \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, a d \Re \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) - a d \Im \left (\operatorname {Ci}\left (d x\right ) \right ) + a d \Im \left (\operatorname {Ci}\left (-d x\right ) \right ) - 2 \, a d \operatorname {Si}\left (d x\right ) - 2 \, b \tan \left (\frac {1}{2} \, d x\right )^{2} - 8 \, b \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, b \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, b}{2 \, {\left (d \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d \tan \left (\frac {1}{2} \, d x\right )^{2} + d \tan \left (\frac {1}{2} \, c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 31, normalized size = 1.07 \[ -\frac {b \cos \left (d x +c \right )}{d}+a \left (\Si \left (d x \right ) \cos \relax (c )+\Ci \left (d x \right ) \sin \relax (c )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.64, size = 522, normalized size = 18.00 \[ -\frac {1}{2} \, {\left ({\left (i \, E_{1}\left (i \, d x\right ) - i \, E_{1}\left (-i \, d x\right )\right )} \cos \relax (c) + {\left (E_{1}\left (i \, d x\right ) + E_{1}\left (-i \, d x\right )\right )} \sin \relax (c)\right )} a + \frac {{\left ({\left (i \, E_{1}\left (i \, d x\right ) - i \, E_{1}\left (-i \, d x\right )\right )} \cos \relax (c) + {\left (E_{1}\left (i \, d x\right ) + E_{1}\left (-i \, d x\right )\right )} \sin \relax (c)\right )} b c}{2 \, d} - \frac {{\left (2 \, {\left (d x + c\right )} {\left (\cos \relax (c)^{2} + \sin \relax (c)^{2}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (d x + c\right )} {\left (\cos \relax (c)^{2} + \sin \relax (c)^{2}\right )} \cos \left (d x + c\right ) - {\left (c {\left (E_{2}\left (i \, d x\right ) + E_{2}\left (-i \, d x\right )\right )} \cos \relax (c)^{3} + c {\left (E_{2}\left (i \, d x\right ) + E_{2}\left (-i \, d x\right )\right )} \cos \relax (c) \sin \relax (c)^{2} - c {\left (i \, E_{2}\left (i \, d x\right ) - i \, E_{2}\left (-i \, d x\right )\right )} \sin \relax (c)^{3} + c {\left (E_{2}\left (i \, d x\right ) + E_{2}\left (-i \, d x\right )\right )} \cos \relax (c) - {\left (c {\left (i \, E_{2}\left (i \, d x\right ) - i \, E_{2}\left (-i \, d x\right )\right )} \cos \relax (c)^{2} + c {\left (i \, E_{2}\left (i \, d x\right ) - i \, E_{2}\left (-i \, d x\right )\right )}\right )} \sin \relax (c)\right )} \cos \left (d x + c\right )^{2} - {\left (c {\left (E_{2}\left (i \, d x\right ) + E_{2}\left (-i \, d x\right )\right )} \cos \relax (c)^{3} + c {\left (E_{2}\left (i \, d x\right ) + E_{2}\left (-i \, d x\right )\right )} \cos \relax (c) \sin \relax (c)^{2} - c {\left (i \, E_{2}\left (i \, d x\right ) - i \, E_{2}\left (-i \, d x\right )\right )} \sin \relax (c)^{3} - 2 \, {\left (d x + c\right )} {\left (\cos \relax (c)^{2} + \sin \relax (c)^{2}\right )} \cos \left (d x + c\right ) + c {\left (E_{2}\left (i \, d x\right ) + E_{2}\left (-i \, d x\right )\right )} \cos \relax (c) - {\left (c {\left (i \, E_{2}\left (i \, d x\right ) - i \, E_{2}\left (-i \, d x\right )\right )} \cos \relax (c)^{2} + c {\left (i \, E_{2}\left (i \, d x\right ) - i \, E_{2}\left (-i \, d x\right )\right )}\right )} \sin \relax (c)\right )} \sin \left (d x + c\right )^{2}\right )} b}{4 \, {\left ({\left ({\left (d x + c\right )} {\left (\cos \relax (c)^{2} + \sin \relax (c)^{2}\right )} d - {\left (c \cos \relax (c)^{2} + c \sin \relax (c)^{2}\right )} d\right )} \cos \left (d x + c\right )^{2} + {\left ({\left (d x + c\right )} {\left (\cos \relax (c)^{2} + \sin \relax (c)^{2}\right )} d - {\left (c \cos \relax (c)^{2} + c \sin \relax (c)^{2}\right )} d\right )} \sin \left (d x + c\right )^{2}\right )}} \]
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 \[ a\,\mathrm {cosint}\left (d\,x\right )\,\sin \relax (c)+a\,\mathrm {sinint}\left (d\,x\right )\,\cos \relax (c)-\frac {b\,\cos \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.40, size = 37, normalized size = 1.28 \[ - a \left (- \sin {\relax (c )} \operatorname {Ci}{\left (d x \right )} - \cos {\relax (c )} \operatorname {Si}{\left (d x \right )}\right ) - b \left (\begin {cases} - x \sin {\relax (c )} & \text {for}\: d = 0 \\\frac {\cos {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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