Optimal. Leaf size=48 \[ a d \cos (c) \text {Ci}(d x)-a d \sin (c) \text {Si}(d x)-\frac {a \sin (c+d x)}{x}+b \sin (c) \text {Ci}(d x)+b \cos (c) \text {Si}(d x) \]
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Rubi [A] time = 0.22, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6742, 3297, 3303, 3299, 3302} \[ a d \cos (c) \text {CosIntegral}(d x)-a d \sin (c) \text {Si}(d x)-\frac {a \sin (c+d x)}{x}+b \sin (c) \text {CosIntegral}(d x)+b \cos (c) \text {Si}(d x) \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 6742
Rubi steps
\begin {align*} \int \frac {(a+b x) \sin (c+d x)}{x^2} \, dx &=\int \left (\frac {a \sin (c+d x)}{x^2}+\frac {b \sin (c+d x)}{x}\right ) \, dx\\ &=a \int \frac {\sin (c+d x)}{x^2} \, dx+b \int \frac {\sin (c+d x)}{x} \, dx\\ &=-\frac {a \sin (c+d x)}{x}+(a d) \int \frac {\cos (c+d x)}{x} \, dx+(b \cos (c)) \int \frac {\sin (d x)}{x} \, dx+(b \sin (c)) \int \frac {\cos (d x)}{x} \, dx\\ &=b \text {Ci}(d x) \sin (c)-\frac {a \sin (c+d x)}{x}+b \cos (c) \text {Si}(d x)+(a d \cos (c)) \int \frac {\cos (d x)}{x} \, dx-(a d \sin (c)) \int \frac {\sin (d x)}{x} \, dx\\ &=a d \cos (c) \text {Ci}(d x)+b \text {Ci}(d x) \sin (c)-\frac {a \sin (c+d x)}{x}+b \cos (c) \text {Si}(d x)-a d \sin (c) \text {Si}(d x)\\ \end {align*}
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Mathematica [A] time = 0.14, size = 60, normalized size = 1.25 \[ a d (\cos (c) \text {Ci}(d x)-\sin (c) \text {Si}(d x))-\frac {a \sin (c) \cos (d x)}{x}-\frac {a \cos (c) \sin (d x)}{x}+b \sin (c) \text {Ci}(d x)+b \cos (c) \text {Si}(d x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 75, normalized size = 1.56 \[ \frac {{\left (a d x \operatorname {Ci}\left (d x\right ) + a d x \operatorname {Ci}\left (-d x\right ) + 2 \, b x \operatorname {Si}\left (d x\right )\right )} \cos \relax (c) - 2 \, a \sin \left (d x + c\right ) - {\left (2 \, a d x \operatorname {Si}\left (d x\right ) - b x \operatorname {Ci}\left (d x\right ) - b x \operatorname {Ci}\left (-d x\right )\right )} \sin \relax (c)}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.49, size = 569, normalized size = 11.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 56, normalized size = 1.17 \[ d \left (\frac {b \left (\Si \left (d x \right ) \cos \relax (c )+\Ci \left (d x \right ) \sin \relax (c )\right )}{d}+a \left (-\frac {\sin \left (d x +c \right )}{x d}-\Si \left (d x \right ) \sin \relax (c )+\Ci \left (d x \right ) \cos \relax (c )\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.25, size = 108, normalized size = 2.25 \[ \frac {{\left ({\left (a {\left (\Gamma \left (-1, i \, d x\right ) + \Gamma \left (-1, -i \, d x\right )\right )} \cos \relax (c) - a {\left (i \, \Gamma \left (-1, i \, d x\right ) - i \, \Gamma \left (-1, -i \, d x\right )\right )} \sin \relax (c)\right )} d^{2} - {\left (b {\left (-i \, \Gamma \left (-1, i \, d x\right ) + i \, \Gamma \left (-1, -i \, d x\right )\right )} \cos \relax (c) - b {\left (\Gamma \left (-1, i \, d x\right ) + \Gamma \left (-1, -i \, d x\right )\right )} \sin \relax (c)\right )} d\right )} x - 2 \, b \cos \left (d x + c\right )}{2 \, d x} \]
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 \[ \int \frac {\sin \left (c+d\,x\right )\,\left (a+b\,x\right )}{x^2} \,d x \]
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 \[ \int \frac {\left (a + b x\right ) \sin {\left (c + d x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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