Optimal. Leaf size=91 \[ \frac{3}{8} b \cos (a) \text{CosIntegral}\left (b x^2\right )-\frac{3}{8} b \cos (3 a) \text{CosIntegral}\left (3 b x^2\right )-\frac{3}{8} b \sin (a) \text{Si}\left (b x^2\right )+\frac{3}{8} b \sin (3 a) \text{Si}\left (3 b x^2\right )-\frac{3 \sin \left (a+b x^2\right )}{8 x^2}+\frac{\sin \left (3 \left (a+b x^2\right )\right )}{8 x^2} \]
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Rubi [A] time = 0.22005, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3403, 3379, 3297, 3303, 3299, 3302} \[ \frac{3}{8} b \cos (a) \text{CosIntegral}\left (b x^2\right )-\frac{3}{8} b \cos (3 a) \text{CosIntegral}\left (3 b x^2\right )-\frac{3}{8} b \sin (a) \text{Si}\left (b x^2\right )+\frac{3}{8} b \sin (3 a) \text{Si}\left (3 b x^2\right )-\frac{3 \sin \left (a+b x^2\right )}{8 x^2}+\frac{\sin \left (3 \left (a+b x^2\right )\right )}{8 x^2} \]
Antiderivative was successfully verified.
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Rule 3403
Rule 3379
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{\sin ^3\left (a+b x^2\right )}{x^3} \, dx &=\int \left (\frac{3 \sin \left (a+b x^2\right )}{4 x^3}-\frac{\sin \left (3 a+3 b x^2\right )}{4 x^3}\right ) \, dx\\ &=-\left (\frac{1}{4} \int \frac{\sin \left (3 a+3 b x^2\right )}{x^3} \, dx\right )+\frac{3}{4} \int \frac{\sin \left (a+b x^2\right )}{x^3} \, dx\\ &=-\left (\frac{1}{8} \operatorname{Subst}\left (\int \frac{\sin (3 a+3 b x)}{x^2} \, dx,x,x^2\right )\right )+\frac{3}{8} \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^2} \, dx,x,x^2\right )\\ &=-\frac{3 \sin \left (a+b x^2\right )}{8 x^2}+\frac{\sin \left (3 \left (a+b x^2\right )\right )}{8 x^2}+\frac{1}{8} (3 b) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x} \, dx,x,x^2\right )-\frac{1}{8} (3 b) \operatorname{Subst}\left (\int \frac{\cos (3 a+3 b x)}{x} \, dx,x,x^2\right )\\ &=-\frac{3 \sin \left (a+b x^2\right )}{8 x^2}+\frac{\sin \left (3 \left (a+b x^2\right )\right )}{8 x^2}+\frac{1}{8} (3 b \cos (a)) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,x^2\right )-\frac{1}{8} (3 b \cos (3 a)) \operatorname{Subst}\left (\int \frac{\cos (3 b x)}{x} \, dx,x,x^2\right )-\frac{1}{8} (3 b \sin (a)) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,x^2\right )+\frac{1}{8} (3 b \sin (3 a)) \operatorname{Subst}\left (\int \frac{\sin (3 b x)}{x} \, dx,x,x^2\right )\\ &=\frac{3}{8} b \cos (a) \text{Ci}\left (b x^2\right )-\frac{3}{8} b \cos (3 a) \text{Ci}\left (3 b x^2\right )-\frac{3 \sin \left (a+b x^2\right )}{8 x^2}+\frac{\sin \left (3 \left (a+b x^2\right )\right )}{8 x^2}-\frac{3}{8} b \sin (a) \text{Si}\left (b x^2\right )+\frac{3}{8} b \sin (3 a) \text{Si}\left (3 b x^2\right )\\ \end{align*}
Mathematica [A] time = 0.1296, size = 90, normalized size = 0.99 \[ \frac{3 b x^2 \cos (a) \text{CosIntegral}\left (b x^2\right )-3 b x^2 \cos (3 a) \text{CosIntegral}\left (3 b x^2\right )-3 b x^2 \sin (a) \text{Si}\left (b x^2\right )+3 b x^2 \sin (3 a) \text{Si}\left (3 b x^2\right )-3 \sin \left (a+b x^2\right )+\sin \left (3 \left (a+b x^2\right )\right )}{8 x^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.194, size = 162, normalized size = 1.8 \begin{align*} -{\frac{3\,i}{16}}{{\rm e}^{-3\,ia}}{\it csgn} \left ( b{x}^{2} \right ) \pi \,b+{\frac{3\,i}{8}}{{\rm e}^{-3\,ia}}{\it Si} \left ( 3\,b{x}^{2} \right ) b+{\frac{3\,{{\rm e}^{-3\,ia}}{\it Ei} \left ( 1,-3\,ib{x}^{2} \right ) b}{16}}+{\frac{3\,{{\rm e}^{3\,ia}}b{\it Ei} \left ( 1,-3\,ib{x}^{2} \right ) }{16}}-{\frac{3\,{{\rm e}^{ia}}b{\it Ei} \left ( 1,-ib{x}^{2} \right ) }{16}}+{\frac{3\,i}{16}}{\it csgn} \left ( b{x}^{2} \right ){{\rm e}^{-ia}}\pi \,b-{\frac{3\,i}{8}}{{\rm e}^{-ia}}{\it Si} \left ( b{x}^{2} \right ) b-{\frac{3\,{\it Ei} \left ( 1,-ib{x}^{2} \right ){{\rm e}^{-ia}}b}{16}}-{\frac{3\,\sin \left ( b{x}^{2}+a \right ) }{8\,{x}^{2}}}+{\frac{\sin \left ( 3\,b{x}^{2}+3\,a \right ) }{8\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.2149, size = 135, normalized size = 1.48 \begin{align*} -\frac{1}{16} \,{\left (3 \,{\left (\Gamma \left (-1, 3 i \, b x^{2}\right ) + \Gamma \left (-1, -3 i \, b x^{2}\right )\right )} \cos \left (3 \, a\right ) - 3 \,{\left (\Gamma \left (-1, i \, b x^{2}\right ) + \Gamma \left (-1, -i \, b x^{2}\right )\right )} \cos \left (a\right ) -{\left (3 i \, \Gamma \left (-1, 3 i \, b x^{2}\right ) - 3 i \, \Gamma \left (-1, -3 i \, b x^{2}\right )\right )} \sin \left (3 \, a\right ) -{\left (-3 i \, \Gamma \left (-1, i \, b x^{2}\right ) + 3 i \, \Gamma \left (-1, -i \, b x^{2}\right )\right )} \sin \left (a\right )\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.29476, size = 367, normalized size = 4.03 \begin{align*} \frac{6 \, b x^{2} \sin \left (3 \, a\right ) \operatorname{Si}\left (3 \, b x^{2}\right ) - 6 \, b x^{2} \sin \left (a\right ) \operatorname{Si}\left (b x^{2}\right ) - 3 \,{\left (b x^{2} \operatorname{Ci}\left (3 \, b x^{2}\right ) + b x^{2} \operatorname{Ci}\left (-3 \, b x^{2}\right )\right )} \cos \left (3 \, a\right ) + 3 \,{\left (b x^{2} \operatorname{Ci}\left (b x^{2}\right ) + b x^{2} \operatorname{Ci}\left (-b x^{2}\right )\right )} \cos \left (a\right ) + 8 \,{\left (\cos \left (b x^{2} + a\right )^{2} - 1\right )} \sin \left (b x^{2} + a\right )}{16 \, 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{\sin ^{3}{\left (a + b 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 = 1.10732, size = 251, normalized size = 2.76 \begin{align*} -\frac{3 \,{\left (b x^{2} + a\right )} b^{2} \cos \left (3 \, a\right ) \operatorname{Ci}\left (3 \, b x^{2}\right ) - 3 \, a b^{2} \cos \left (3 \, a\right ) \operatorname{Ci}\left (3 \, b x^{2}\right ) - 3 \,{\left (b x^{2} + a\right )} b^{2} \cos \left (a\right ) \operatorname{Ci}\left (b x^{2}\right ) + 3 \, a b^{2} \cos \left (a\right ) \operatorname{Ci}\left (b x^{2}\right ) + 3 \,{\left (b x^{2} + a\right )} b^{2} \sin \left (a\right ) \operatorname{Si}\left (b x^{2}\right ) - 3 \, a b^{2} \sin \left (a\right ) \operatorname{Si}\left (b x^{2}\right ) + 3 \,{\left (b x^{2} + a\right )} b^{2} \sin \left (3 \, a\right ) \operatorname{Si}\left (-3 \, b x^{2}\right ) - 3 \, a b^{2} \sin \left (3 \, a\right ) \operatorname{Si}\left (-3 \, b x^{2}\right ) - b^{2} \sin \left (3 \, b x^{2} + 3 \, a\right ) + 3 \, b^{2} \sin \left (b x^{2} + a\right )}{8 \, b^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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