Optimal. Leaf size=74 \[ -\frac{1}{2} a d^2 \sin (c) \text{CosIntegral}(d x)-\frac{1}{2} a d^2 \cos (c) \text{Si}(d x)-\frac{a \sin (c+d x)}{2 x^2}-\frac{a d \cos (c+d x)}{2 x}+b \sin (c) \text{CosIntegral}(d x)+b \cos (c) \text{Si}(d x) \]
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Rubi [A] time = 0.16056, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {3339, 3297, 3303, 3299, 3302} \[ -\frac{1}{2} a d^2 \sin (c) \text{CosIntegral}(d x)-\frac{1}{2} a d^2 \cos (c) \text{Si}(d x)-\frac{a \sin (c+d x)}{2 x^2}-\frac{a d \cos (c+d x)}{2 x}+b \sin (c) \text{CosIntegral}(d x)+b \cos (c) \text{Si}(d x) \]
Antiderivative was successfully verified.
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Rule 3339
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right ) \sin (c+d x)}{x^3} \, dx &=\int \left (\frac{a \sin (c+d x)}{x^3}+\frac{b \sin (c+d x)}{x}\right ) \, dx\\ &=a \int \frac{\sin (c+d x)}{x^3} \, dx+b \int \frac{\sin (c+d x)}{x} \, dx\\ &=-\frac{a \sin (c+d x)}{2 x^2}+\frac{1}{2} (a d) \int \frac{\cos (c+d x)}{x^2} \, dx+(b \cos (c)) \int \frac{\sin (d x)}{x} \, dx+(b \sin (c)) \int \frac{\cos (d x)}{x} \, dx\\ &=-\frac{a d \cos (c+d x)}{2 x}+b \text{Ci}(d x) \sin (c)-\frac{a \sin (c+d x)}{2 x^2}+b \cos (c) \text{Si}(d x)-\frac{1}{2} \left (a d^2\right ) \int \frac{\sin (c+d x)}{x} \, dx\\ &=-\frac{a d \cos (c+d x)}{2 x}+b \text{Ci}(d x) \sin (c)-\frac{a \sin (c+d x)}{2 x^2}+b \cos (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 \text{Ci}(d x) \sin (c)-\frac{1}{2} a d^2 \text{Ci}(d x) \sin (c)-\frac{a \sin (c+d x)}{2 x^2}+b \cos (c) \text{Si}(d x)-\frac{1}{2} a d^2 \cos (c) \text{Si}(d x)\\ \end{align*}
Mathematica [A] time = 0.186828, size = 82, normalized size = 1.11 \[ -\frac{1}{2} a d^2 (\sin (c) \text{CosIntegral}(d x)+\cos (c) \text{Si}(d x))-\frac{a \cos (d x) (d x \cos (c)+\sin (c))}{2 x^2}+\frac{a \sin (d x) (d x \sin (c)-\cos (c))}{2 x^2}+b \sin (c) \text{CosIntegral}(d x)+b \cos (c) \text{Si}(d x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 73, normalized size = 1. \begin{align*}{d}^{2} \left ({\frac{b \left ({\it Si} \left ( dx \right ) \cos \left ( c \right ) +{\it Ci} \left ( dx \right ) \sin \left ( c \right ) \right ) }{{d}^{2}}}+a \left ( -{\frac{\sin \left ( dx+c \right ) }{2\,{d}^{2}{x}^{2}}}-{\frac{\cos \left ( dx+c \right ) }{2\,dx}}-{\frac{{\it Si} \left ( dx \right ) \cos \left ( c \right ) }{2}}-{\frac{{\it Ci} \left ( dx \right ) \sin \left ( c \right ) }{2}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 3.10349, size = 165, normalized size = 2.23 \begin{align*} -\frac{2 \, b d x \cos \left (d x + c\right ) +{\left ({\left (a{\left (-i \, \Gamma \left (-2, i \, d x\right ) + i \, \Gamma \left (-2, -i \, d x\right )\right )} \cos \left (c\right ) - a{\left (\Gamma \left (-2, i \, d x\right ) + \Gamma \left (-2, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{4} +{\left (b{\left (2 i \, \Gamma \left (-2, i \, d x\right ) - 2 i \, \Gamma \left (-2, -i \, d x\right )\right )} \cos \left (c\right ) + 2 \, b{\left (\Gamma \left (-2, i \, d x\right ) + \Gamma \left (-2, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{2}\right )} x^{2} + 2 \, b \sin \left (d x + c\right )}{2 \, d^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64548, size = 250, normalized size = 3.38 \begin{align*} -\frac{2 \,{\left (a d^{2} - 2 \, b\right )} x^{2} \cos \left (c\right ) \operatorname{Si}\left (d x\right ) + 2 \, a d x \cos \left (d x + c\right ) + 2 \, a \sin \left (d x + c\right ) +{\left ({\left (a d^{2} - 2 \, b\right )} x^{2} \operatorname{Ci}\left (d x\right ) +{\left (a d^{2} - 2 \, b\right )} x^{2} \operatorname{Ci}\left (-d x\right )\right )} \sin \left (c\right )}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{2}\right ) \sin{\left (c + d x \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.17029, size = 1034, normalized size = 13.97 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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