Optimal. Leaf size=121 \[ -\frac{1}{2} a^2 d^2 \sin (c) \text{CosIntegral}(d x)-\frac{1}{2} a^2 d^2 \cos (c) \text{Si}(d x)-\frac{a^2 \sin (c+d x)}{2 x^2}-\frac{a^2 d \cos (c+d x)}{2 x}+2 a b d \cos (c) \text{CosIntegral}(d x)-2 a b d \sin (c) \text{Si}(d x)-\frac{2 a b \sin (c+d x)}{x}+b^2 \sin (c) \text{CosIntegral}(d x)+b^2 \cos (c) \text{Si}(d x) \]
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Rubi [A] time = 0.339819, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {6742, 3297, 3303, 3299, 3302} \[ -\frac{1}{2} a^2 d^2 \sin (c) \text{CosIntegral}(d x)-\frac{1}{2} a^2 d^2 \cos (c) \text{Si}(d x)-\frac{a^2 \sin (c+d x)}{2 x^2}-\frac{a^2 d \cos (c+d x)}{2 x}+2 a b d \cos (c) \text{CosIntegral}(d x)-2 a b d \sin (c) \text{Si}(d x)-\frac{2 a b \sin (c+d x)}{x}+b^2 \sin (c) \text{CosIntegral}(d x)+b^2 \cos (c) \text{Si}(d x) \]
Antiderivative was successfully verified.
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Rule 6742
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{(a+b x)^2 \sin (c+d x)}{x^3} \, dx &=\int \left (\frac{a^2 \sin (c+d x)}{x^3}+\frac{2 a b \sin (c+d x)}{x^2}+\frac{b^2 \sin (c+d x)}{x}\right ) \, dx\\ &=a^2 \int \frac{\sin (c+d x)}{x^3} \, dx+(2 a b) \int \frac{\sin (c+d x)}{x^2} \, dx+b^2 \int \frac{\sin (c+d x)}{x} \, dx\\ &=-\frac{a^2 \sin (c+d x)}{2 x^2}-\frac{2 a b \sin (c+d x)}{x}+\frac{1}{2} \left (a^2 d\right ) \int \frac{\cos (c+d x)}{x^2} \, dx+(2 a b d) \int \frac{\cos (c+d x)}{x} \, dx+\left (b^2 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx+\left (b^2 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx\\ &=-\frac{a^2 d \cos (c+d x)}{2 x}+b^2 \text{Ci}(d x) \sin (c)-\frac{a^2 \sin (c+d x)}{2 x^2}-\frac{2 a b \sin (c+d x)}{x}+b^2 \cos (c) \text{Si}(d x)-\frac{1}{2} \left (a^2 d^2\right ) \int \frac{\sin (c+d x)}{x} \, dx+(2 a b d \cos (c)) \int \frac{\cos (d x)}{x} \, dx-(2 a b d \sin (c)) \int \frac{\sin (d x)}{x} \, dx\\ &=-\frac{a^2 d \cos (c+d x)}{2 x}+2 a b d \cos (c) \text{Ci}(d x)+b^2 \text{Ci}(d x) \sin (c)-\frac{a^2 \sin (c+d x)}{2 x^2}-\frac{2 a b \sin (c+d x)}{x}+b^2 \cos (c) \text{Si}(d x)-2 a b d \sin (c) \text{Si}(d x)-\frac{1}{2} \left (a^2 d^2 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx-\frac{1}{2} \left (a^2 d^2 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx\\ &=-\frac{a^2 d \cos (c+d x)}{2 x}+2 a b d \cos (c) \text{Ci}(d x)+b^2 \text{Ci}(d x) \sin (c)-\frac{1}{2} a^2 d^2 \text{Ci}(d x) \sin (c)-\frac{a^2 \sin (c+d x)}{2 x^2}-\frac{2 a b \sin (c+d x)}{x}+b^2 \cos (c) \text{Si}(d x)-\frac{1}{2} a^2 d^2 \cos (c) \text{Si}(d x)-2 a b d \sin (c) \text{Si}(d x)\\ \end{align*}
Mathematica [A] time = 0.403643, size = 95, normalized size = 0.79 \[ \frac{1}{2} \left (\text{CosIntegral}(d x) \left (\sin (c) \left (2 b^2-a^2 d^2\right )+4 a b d \cos (c)\right )+\text{Si}(d x) \left (\cos (c) \left (2 b^2-a^2 d^2\right )-4 a b d \sin (c)\right )-\frac{a ((a+4 b x) \sin (c+d x)+a d x \cos (c+d x))}{x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 114, normalized size = 0.9 \begin{align*}{d}^{2} \left ({\frac{{b}^{2} \left ({\it Si} \left ( dx \right ) \cos \left ( c \right ) +{\it Ci} \left ( dx \right ) \sin \left ( c \right ) \right ) }{{d}^{2}}}+2\,{\frac{ab}{d} \left ( -{\frac{\sin \left ( dx+c \right ) }{dx}}-{\it Si} \left ( dx \right ) \sin \left ( c \right ) +{\it Ci} \left ( dx \right ) \cos \left ( c \right ) \right ) }+{a}^{2} \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 = 5.30843, size = 252, normalized size = 2.08 \begin{align*} -\frac{{\left ({\left (a^{2}{\left (-i \, \Gamma \left (-2, i \, d x\right ) + i \, \Gamma \left (-2, -i \, d x\right )\right )} \cos \left (c\right ) - a^{2}{\left (\Gamma \left (-2, i \, d x\right ) + \Gamma \left (-2, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{4} +{\left (4 \, a b{\left (\Gamma \left (-2, i \, d x\right ) + \Gamma \left (-2, -i \, d x\right )\right )} \cos \left (c\right ) + a b{\left (-4 i \, \Gamma \left (-2, i \, d x\right ) + 4 i \, \Gamma \left (-2, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{3} +{\left (b^{2}{\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^{2}{\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^{2} \sin \left (d x + c\right ) + 2 \,{\left (b^{2} d x + 2 \, a b d\right )} \cos \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 = 1.73733, size = 421, normalized size = 3.48 \begin{align*} -\frac{2 \, a^{2} d x \cos \left (d x + c\right ) - 2 \,{\left (2 \, a b d x^{2} \operatorname{Ci}\left (d x\right ) + 2 \, a b d x^{2} \operatorname{Ci}\left (-d x\right ) -{\left (a^{2} d^{2} - 2 \, b^{2}\right )} x^{2} \operatorname{Si}\left (d x\right )\right )} \cos \left (c\right ) + 2 \,{\left (4 \, a b x + a^{2}\right )} \sin \left (d x + c\right ) +{\left (8 \, a b d x^{2} \operatorname{Si}\left (d x\right ) +{\left (a^{2} d^{2} - 2 \, b^{2}\right )} x^{2} \operatorname{Ci}\left (d x\right ) +{\left (a^{2} d^{2} - 2 \, b^{2}\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\right )^{2} \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 = 1.14839, size = 1596, normalized size = 13.19 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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