Optimal. Leaf size=151 \[ -\frac{1}{6} a^2 d^3 \cos (c) \text{CosIntegral}(d x)+\frac{1}{6} a^2 d^3 \sin (c) \text{Si}(d x)+\frac{a^2 d^2 \sin (c+d x)}{6 x}-\frac{a^2 \sin (c+d x)}{3 x^3}-\frac{a^2 d \cos (c+d x)}{6 x^2}+2 a b \sin (c) \text{CosIntegral}(d x)+2 a b \cos (c) \text{Si}(d x)+\frac{2 b^2 x \sin (c+d x)}{d^2}+\frac{2 b^2 \cos (c+d x)}{d^3}-\frac{b^2 x^2 \cos (c+d x)}{d} \]
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Rubi [A] time = 0.251689, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {3339, 3297, 3303, 3299, 3302, 3296, 2638} \[ -\frac{1}{6} a^2 d^3 \cos (c) \text{CosIntegral}(d x)+\frac{1}{6} a^2 d^3 \sin (c) \text{Si}(d x)+\frac{a^2 d^2 \sin (c+d x)}{6 x}-\frac{a^2 \sin (c+d x)}{3 x^3}-\frac{a^2 d \cos (c+d x)}{6 x^2}+2 a b \sin (c) \text{CosIntegral}(d x)+2 a b \cos (c) \text{Si}(d x)+\frac{2 b^2 x \sin (c+d x)}{d^2}+\frac{2 b^2 \cos (c+d x)}{d^3}-\frac{b^2 x^2 \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3339
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^2 \sin (c+d x)}{x^4} \, dx &=\int \left (\frac{a^2 \sin (c+d x)}{x^4}+\frac{2 a b \sin (c+d x)}{x}+b^2 x^2 \sin (c+d x)\right ) \, dx\\ &=a^2 \int \frac{\sin (c+d x)}{x^4} \, dx+(2 a b) \int \frac{\sin (c+d x)}{x} \, dx+b^2 \int x^2 \sin (c+d x) \, dx\\ &=-\frac{b^2 x^2 \cos (c+d x)}{d}-\frac{a^2 \sin (c+d x)}{3 x^3}+\frac{\left (2 b^2\right ) \int x \cos (c+d x) \, dx}{d}+\frac{1}{3} \left (a^2 d\right ) \int \frac{\cos (c+d x)}{x^3} \, dx+(2 a b \cos (c)) \int \frac{\sin (d x)}{x} \, dx+(2 a b \sin (c)) \int \frac{\cos (d x)}{x} \, dx\\ &=-\frac{a^2 d \cos (c+d x)}{6 x^2}-\frac{b^2 x^2 \cos (c+d x)}{d}+2 a b \text{Ci}(d x) \sin (c)-\frac{a^2 \sin (c+d x)}{3 x^3}+\frac{2 b^2 x \sin (c+d x)}{d^2}+2 a b \cos (c) \text{Si}(d x)-\frac{\left (2 b^2\right ) \int \sin (c+d x) \, dx}{d^2}-\frac{1}{6} \left (a^2 d^2\right ) \int \frac{\sin (c+d x)}{x^2} \, dx\\ &=\frac{2 b^2 \cos (c+d x)}{d^3}-\frac{a^2 d \cos (c+d x)}{6 x^2}-\frac{b^2 x^2 \cos (c+d x)}{d}+2 a b \text{Ci}(d x) \sin (c)-\frac{a^2 \sin (c+d x)}{3 x^3}+\frac{a^2 d^2 \sin (c+d x)}{6 x}+\frac{2 b^2 x \sin (c+d x)}{d^2}+2 a b \cos (c) \text{Si}(d x)-\frac{1}{6} \left (a^2 d^3\right ) \int \frac{\cos (c+d x)}{x} \, dx\\ &=\frac{2 b^2 \cos (c+d x)}{d^3}-\frac{a^2 d \cos (c+d x)}{6 x^2}-\frac{b^2 x^2 \cos (c+d x)}{d}+2 a b \text{Ci}(d x) \sin (c)-\frac{a^2 \sin (c+d x)}{3 x^3}+\frac{a^2 d^2 \sin (c+d x)}{6 x}+\frac{2 b^2 x \sin (c+d x)}{d^2}+2 a b \cos (c) \text{Si}(d x)-\frac{1}{6} \left (a^2 d^3 \cos (c)\right ) \int \frac{\cos (d x)}{x} \, dx+\frac{1}{6} \left (a^2 d^3 \sin (c)\right ) \int \frac{\sin (d x)}{x} \, dx\\ &=\frac{2 b^2 \cos (c+d x)}{d^3}-\frac{a^2 d \cos (c+d x)}{6 x^2}-\frac{b^2 x^2 \cos (c+d x)}{d}-\frac{1}{6} a^2 d^3 \cos (c) \text{Ci}(d x)+2 a b \text{Ci}(d x) \sin (c)-\frac{a^2 \sin (c+d x)}{3 x^3}+\frac{a^2 d^2 \sin (c+d x)}{6 x}+\frac{2 b^2 x \sin (c+d x)}{d^2}+2 a b \cos (c) \text{Si}(d x)+\frac{1}{6} a^2 d^3 \sin (c) \text{Si}(d x)\\ \end{align*}
Mathematica [A] time = 0.580123, size = 135, normalized size = 0.89 \[ \frac{1}{6} \left (\frac{a^2 d^2 \sin (c+d x)}{x}-\frac{2 a^2 \sin (c+d x)}{x^3}-\frac{a^2 d \cos (c+d x)}{x^2}-a \text{CosIntegral}(d x) \left (a d^3 \cos (c)-12 b \sin (c)\right )+a \text{Si}(d x) \left (a d^3 \sin (c)+12 b \cos (c)\right )+\frac{12 b^2 x \sin (c+d x)}{d^2}+\frac{12 b^2 \cos (c+d x)}{d^3}-\frac{6 b^2 x^2 \cos (c+d x)}{d}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 196, normalized size = 1.3 \begin{align*}{d}^{3} \left ({\frac{ \left ( 10\,{c}^{2}+4\,c+1 \right ){b}^{2} \left ( - \left ( dx+c \right ) ^{2}\cos \left ( dx+c \right ) +2\,\cos \left ( dx+c \right ) +2\, \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) }{{d}^{6}}}-6\,{\frac{c{b}^{2} \left ( 1+4\,c \right ) \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{{d}^{6}}}-15\,{\frac{{c}^{2}{b}^{2}\cos \left ( dx+c \right ) }{{d}^{6}}}+2\,{\frac{ab \left ({\it Si} \left ( dx \right ) \cos \left ( c \right ) +{\it Ci} \left ( dx \right ) \sin \left ( c \right ) \right ) }{{d}^{3}}}+{a}^{2} \left ( -{\frac{\sin \left ( dx+c \right ) }{3\,{d}^{3}{x}^{3}}}-{\frac{\cos \left ( dx+c \right ) }{6\,{d}^{2}{x}^{2}}}+{\frac{\sin \left ( dx+c \right ) }{6\,dx}}+{\frac{{\it Si} \left ( dx \right ) \sin \left ( c \right ) }{6}}-{\frac{{\it Ci} \left ( dx \right ) \cos \left ( c \right ) }{6}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 45.2172, size = 234, normalized size = 1.55 \begin{align*} -\frac{{\left ({\left (a^{2}{\left (\Gamma \left (-3, i \, d x\right ) + \Gamma \left (-3, -i \, d x\right )\right )} \cos \left (c\right ) - a^{2}{\left (i \, \Gamma \left (-3, i \, d x\right ) - i \, \Gamma \left (-3, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{6} -{\left (a b{\left (12 i \, \Gamma \left (-3, i \, d x\right ) - 12 i \, \Gamma \left (-3, -i \, d x\right )\right )} \cos \left (c\right ) + 12 \, a b{\left (\Gamma \left (-3, i \, d x\right ) + \Gamma \left (-3, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{3}\right )} x^{3} + 2 \,{\left (b^{2} d^{2} x^{5} + 2 \, a b d^{2} x^{2} - 2 \, b^{2} x^{3} - 4 \, a b\right )} \cos \left (d x + c\right ) - 4 \,{\left (b^{2} d x^{4} - a b d x\right )} \sin \left (d x + c\right )}{2 \, d^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02818, size = 478, normalized size = 3.17 \begin{align*} -\frac{2 \,{\left (6 \, b^{2} d^{2} x^{5} + a^{2} d^{4} x - 12 \, b^{2} x^{3}\right )} \cos \left (d x + c\right ) +{\left (a^{2} d^{6} x^{3} \operatorname{Ci}\left (d x\right ) + a^{2} d^{6} x^{3} \operatorname{Ci}\left (-d x\right ) - 24 \, a b d^{3} x^{3} \operatorname{Si}\left (d x\right )\right )} \cos \left (c\right ) - 2 \,{\left (a^{2} d^{5} x^{2} + 12 \, b^{2} d x^{4} - 2 \, a^{2} d^{3}\right )} \sin \left (d x + c\right ) - 2 \,{\left (a^{2} d^{6} x^{3} \operatorname{Si}\left (d x\right ) + 6 \, a b d^{3} x^{3} \operatorname{Ci}\left (d x\right ) + 6 \, a b d^{3} x^{3} \operatorname{Ci}\left (-d x\right )\right )} \sin \left (c\right )}{12 \, d^{3} x^{3}} \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^{3}\right )^{2} \sin{\left (c + d x \right )}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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