Optimal. Leaf size=89 \[ -\frac {1}{2} a d^2 \sin (c) \text {Ci}(d x)-\frac {1}{2} a d^2 \cos (c) \text {Si}(d x)-\frac {a \sin (c+d x)}{2 x^2}-\frac {a d \cos (c+d x)}{2 x}+b d \cos (c) \text {Ci}(d x)-b d \sin (c) \text {Si}(d x)-\frac {b \sin (c+d x)}{x} \]
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Rubi [A] time = 0.27, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 11, 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} \[ -\frac {1}{2} a d^2 \sin (c) \text {CosIntegral}(d x)-\frac {1}{2} a d^2 \cos (c) \text {Si}(d x)-\frac {a \sin (c+d x)}{2 x^2}-\frac {a d \cos (c+d x)}{2 x}+b d \cos (c) \text {CosIntegral}(d x)-b d \sin (c) \text {Si}(d x)-\frac {b \sin (c+d x)}{x} \]
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^3} \, dx &=\int \left (\frac {a \sin (c+d x)}{x^3}+\frac {b \sin (c+d x)}{x^2}\right ) \, dx\\ &=a \int \frac {\sin (c+d x)}{x^3} \, dx+b \int \frac {\sin (c+d x)}{x^2} \, dx\\ &=-\frac {a \sin (c+d x)}{2 x^2}-\frac {b \sin (c+d x)}{x}+\frac {1}{2} (a d) \int \frac {\cos (c+d x)}{x^2} \, dx+(b d) \int \frac {\cos (c+d x)}{x} \, dx\\ &=-\frac {a d \cos (c+d x)}{2 x}-\frac {a \sin (c+d x)}{2 x^2}-\frac {b \sin (c+d x)}{x}-\frac {1}{2} \left (a d^2\right ) \int \frac {\sin (c+d x)}{x} \, dx+(b d \cos (c)) \int \frac {\cos (d x)}{x} \, dx-(b d \sin (c)) \int \frac {\sin (d x)}{x} \, dx\\ &=-\frac {a d \cos (c+d x)}{2 x}+b d \cos (c) \text {Ci}(d x)-\frac {a \sin (c+d x)}{2 x^2}-\frac {b \sin (c+d x)}{x}-b d \sin (c) \text {Si}(d x)-\frac {1}{2} \left (a d^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx-\frac {1}{2} \left (a d^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx\\ &=-\frac {a d \cos (c+d x)}{2 x}+b d \cos (c) \text {Ci}(d x)-\frac {1}{2} a d^2 \text {Ci}(d x) \sin (c)-\frac {a \sin (c+d x)}{2 x^2}-\frac {b \sin (c+d x)}{x}-\frac {1}{2} a d^2 \cos (c) \text {Si}(d x)-b d \sin (c) \text {Si}(d x)\\ \end {align*}
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Mathematica [A] time = 0.27, size = 76, normalized size = 0.85 \[ -\frac {d x^2 \text {Ci}(d x) (a d \sin (c)-2 b \cos (c))+d x^2 \text {Si}(d x) (a d \cos (c)+2 b \sin (c))+a \sin (c+d x)+a d x \cos (c+d x)+2 b x \sin (c+d x)}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 111, normalized size = 1.25 \[ -\frac {2 \, a d x \cos \left (d x + c\right ) + 2 \, {\left (a d^{2} x^{2} \operatorname {Si}\left (d x\right ) - b d x^{2} \operatorname {Ci}\left (d x\right ) - b d x^{2} \operatorname {Ci}\left (-d x\right )\right )} \cos \relax (c) + 2 \, {\left (2 \, b x + a\right )} \sin \left (d x + c\right ) + {\left (a d^{2} x^{2} \operatorname {Ci}\left (d x\right ) + a d^{2} x^{2} \operatorname {Ci}\left (-d x\right ) + 4 \, b d x^{2} \operatorname {Si}\left (d x\right )\right )} \sin \relax (c)}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.54, size = 796, normalized size = 8.94 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 88, normalized size = 0.99 \[ d^{2} \left (\frac {b \left (-\frac {\sin \left (d x +c \right )}{x d}-\Si \left (d x \right ) \sin \relax (c )+\Ci \left (d x \right ) \cos \relax (c )\right )}{d}+a \left (-\frac {\sin \left (d x +c \right )}{2 x^{2} d^{2}}-\frac {\cos \left (d x +c \right )}{2 x d}-\frac {\Si \left (d x \right ) \cos \relax (c )}{2}-\frac {\Ci \left (d x \right ) \sin \relax (c )}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.60, size = 111, normalized size = 1.25 \[ -\frac {{\left ({\left (a {\left (-i \, \Gamma \left (-2, i \, d x\right ) + i \, \Gamma \left (-2, -i \, d x\right )\right )} \cos \relax (c) - a {\left (\Gamma \left (-2, i \, d x\right ) + \Gamma \left (-2, -i \, d x\right )\right )} \sin \relax (c)\right )} d^{3} + {\left (2 \, b {\left (\Gamma \left (-2, i \, d x\right ) + \Gamma \left (-2, -i \, d x\right )\right )} \cos \relax (c) + b {\left (-2 i \, \Gamma \left (-2, i \, d x\right ) + 2 i \, \Gamma \left (-2, -i \, d x\right )\right )} \sin \relax (c)\right )} d^{2}\right )} x^{2} + 2 \, b \cos \left (d x + c\right )}{2 \, d x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sin \left (c+d\,x\right )\,\left (a+b\,x\right )}{x^3} \,d x \]
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 \[ \int \frac {\left (a + b x\right ) \sin {\left (c + d x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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