Optimal. Leaf size=53 \[ -\frac{a}{2 x^2}+\frac{1}{2} b d \cos (c) \text{CosIntegral}\left (d x^2\right )-\frac{1}{2} b d \sin (c) \text{Si}\left (d x^2\right )-\frac{b \sin \left (c+d x^2\right )}{2 x^2} \]
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Rubi [A] time = 0.0911441, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {14, 3379, 3297, 3303, 3299, 3302} \[ -\frac{a}{2 x^2}+\frac{1}{2} b d \cos (c) \text{CosIntegral}\left (d x^2\right )-\frac{1}{2} b d \sin (c) \text{Si}\left (d x^2\right )-\frac{b \sin \left (c+d x^2\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 14
Rule 3379
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{a+b \sin \left (c+d x^2\right )}{x^3} \, dx &=\int \left (\frac{a}{x^3}+\frac{b \sin \left (c+d x^2\right )}{x^3}\right ) \, dx\\ &=-\frac{a}{2 x^2}+b \int \frac{\sin \left (c+d x^2\right )}{x^3} \, dx\\ &=-\frac{a}{2 x^2}+\frac{1}{2} b \operatorname{Subst}\left (\int \frac{\sin (c+d x)}{x^2} \, dx,x,x^2\right )\\ &=-\frac{a}{2 x^2}-\frac{b \sin \left (c+d x^2\right )}{2 x^2}+\frac{1}{2} (b d) \operatorname{Subst}\left (\int \frac{\cos (c+d x)}{x} \, dx,x,x^2\right )\\ &=-\frac{a}{2 x^2}-\frac{b \sin \left (c+d x^2\right )}{2 x^2}+\frac{1}{2} (b d \cos (c)) \operatorname{Subst}\left (\int \frac{\cos (d x)}{x} \, dx,x,x^2\right )-\frac{1}{2} (b d \sin (c)) \operatorname{Subst}\left (\int \frac{\sin (d x)}{x} \, dx,x,x^2\right )\\ &=-\frac{a}{2 x^2}+\frac{1}{2} b d \cos (c) \text{Ci}\left (d x^2\right )-\frac{b \sin \left (c+d x^2\right )}{2 x^2}-\frac{1}{2} b d \sin (c) \text{Si}\left (d x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0812176, size = 48, normalized size = 0.91 \[ -\frac{a-b d x^2 \cos (c) \text{CosIntegral}\left (d x^2\right )+b d x^2 \sin (c) \text{Si}\left (d x^2\right )+b \sin \left (c+d x^2\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 47, normalized size = 0.9 \begin{align*} -{\frac{a}{2\,{x}^{2}}}+b \left ( -{\frac{\sin \left ( d{x}^{2}+c \right ) }{2\,{x}^{2}}}+d \left ({\frac{\cos \left ( c \right ){\it Ci} \left ( d{x}^{2} \right ) }{2}}-{\frac{\sin \left ( c \right ){\it Si} \left ( d{x}^{2} \right ) }{2}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.14813, size = 77, normalized size = 1.45 \begin{align*} \frac{1}{4} \,{\left ({\left (\Gamma \left (-1, i \, d x^{2}\right ) + \Gamma \left (-1, -i \, d x^{2}\right )\right )} \cos \left (c\right ) -{\left (i \, \Gamma \left (-1, i \, d x^{2}\right ) - i \, \Gamma \left (-1, -i \, d x^{2}\right )\right )} \sin \left (c\right )\right )} b d - \frac{a}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9283, size = 197, normalized size = 3.72 \begin{align*} -\frac{2 \, b d x^{2} \sin \left (c\right ) \operatorname{Si}\left (d x^{2}\right ) -{\left (b d x^{2} \operatorname{Ci}\left (d x^{2}\right ) + b d x^{2} \operatorname{Ci}\left (-d x^{2}\right )\right )} \cos \left (c\right ) + 2 \, b \sin \left (d x^{2} + c\right ) + 2 \, a}{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{a + b \sin{\left (c + d x^{2} \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14906, size = 134, normalized size = 2.53 \begin{align*} \frac{{\left (d x^{2} + c\right )} b d^{2} \cos \left (c\right ) \operatorname{Ci}\left (d x^{2}\right ) - b c d^{2} \cos \left (c\right ) \operatorname{Ci}\left (d x^{2}\right ) -{\left (d x^{2} + c\right )} b d^{2} \sin \left (c\right ) \operatorname{Si}\left (d x^{2}\right ) + b c d^{2} \sin \left (c\right ) \operatorname{Si}\left (d x^{2}\right ) - b d^{2} \sin \left (d x^{2} + c\right ) - a d^{2}}{2 \, d^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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