Optimal. Leaf size=167 \[ \frac{1}{24} a^2 d^4 \sin (c) \text{CosIntegral}(d x)+\frac{1}{24} a^2 d^4 \cos (c) \text{Si}(d x)+\frac{a^2 d^2 \sin (c+d x)}{24 x^2}+\frac{a^2 d^3 \cos (c+d x)}{24 x}-\frac{a^2 \sin (c+d x)}{4 x^4}-\frac{a^2 d \cos (c+d x)}{12 x^3}+2 a b d \cos (c) \text{CosIntegral}(d x)-2 a b d \sin (c) \text{Si}(d x)-\frac{2 a b \sin (c+d x)}{x}+\frac{b^2 \sin (c+d x)}{d^2}-\frac{b^2 x \cos (c+d x)}{d} \]
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Rubi [A] time = 0.282767, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 15, 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, 2637} \[ \frac{1}{24} a^2 d^4 \sin (c) \text{CosIntegral}(d x)+\frac{1}{24} a^2 d^4 \cos (c) \text{Si}(d x)+\frac{a^2 d^2 \sin (c+d x)}{24 x^2}+\frac{a^2 d^3 \cos (c+d x)}{24 x}-\frac{a^2 \sin (c+d x)}{4 x^4}-\frac{a^2 d \cos (c+d x)}{12 x^3}+2 a b d \cos (c) \text{CosIntegral}(d x)-2 a b d \sin (c) \text{Si}(d x)-\frac{2 a b \sin (c+d x)}{x}+\frac{b^2 \sin (c+d x)}{d^2}-\frac{b^2 x \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3339
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^2 \sin (c+d x)}{x^5} \, dx &=\int \left (\frac{a^2 \sin (c+d x)}{x^5}+\frac{2 a b \sin (c+d x)}{x^2}+b^2 x \sin (c+d x)\right ) \, dx\\ &=a^2 \int \frac{\sin (c+d x)}{x^5} \, dx+(2 a b) \int \frac{\sin (c+d x)}{x^2} \, dx+b^2 \int x \sin (c+d x) \, dx\\ &=-\frac{b^2 x \cos (c+d x)}{d}-\frac{a^2 \sin (c+d x)}{4 x^4}-\frac{2 a b \sin (c+d x)}{x}+\frac{b^2 \int \cos (c+d x) \, dx}{d}+\frac{1}{4} \left (a^2 d\right ) \int \frac{\cos (c+d x)}{x^4} \, dx+(2 a b d) \int \frac{\cos (c+d x)}{x} \, dx\\ &=-\frac{a^2 d \cos (c+d x)}{12 x^3}-\frac{b^2 x \cos (c+d x)}{d}+\frac{b^2 \sin (c+d x)}{d^2}-\frac{a^2 \sin (c+d x)}{4 x^4}-\frac{2 a b \sin (c+d x)}{x}-\frac{1}{12} \left (a^2 d^2\right ) \int \frac{\sin (c+d x)}{x^3} \, 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)}{12 x^3}-\frac{b^2 x \cos (c+d x)}{d}+2 a b d \cos (c) \text{Ci}(d x)+\frac{b^2 \sin (c+d x)}{d^2}-\frac{a^2 \sin (c+d x)}{4 x^4}+\frac{a^2 d^2 \sin (c+d x)}{24 x^2}-\frac{2 a b \sin (c+d x)}{x}-2 a b d \sin (c) \text{Si}(d x)-\frac{1}{24} \left (a^2 d^3\right ) \int \frac{\cos (c+d x)}{x^2} \, dx\\ &=-\frac{a^2 d \cos (c+d x)}{12 x^3}+\frac{a^2 d^3 \cos (c+d x)}{24 x}-\frac{b^2 x \cos (c+d x)}{d}+2 a b d \cos (c) \text{Ci}(d x)+\frac{b^2 \sin (c+d x)}{d^2}-\frac{a^2 \sin (c+d x)}{4 x^4}+\frac{a^2 d^2 \sin (c+d x)}{24 x^2}-\frac{2 a b \sin (c+d x)}{x}-2 a b d \sin (c) \text{Si}(d x)+\frac{1}{24} \left (a^2 d^4\right ) \int \frac{\sin (c+d x)}{x} \, dx\\ &=-\frac{a^2 d \cos (c+d x)}{12 x^3}+\frac{a^2 d^3 \cos (c+d x)}{24 x}-\frac{b^2 x \cos (c+d x)}{d}+2 a b d \cos (c) \text{Ci}(d x)+\frac{b^2 \sin (c+d x)}{d^2}-\frac{a^2 \sin (c+d x)}{4 x^4}+\frac{a^2 d^2 \sin (c+d x)}{24 x^2}-\frac{2 a b \sin (c+d x)}{x}-2 a b d \sin (c) \text{Si}(d x)+\frac{1}{24} \left (a^2 d^4 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx+\frac{1}{24} \left (a^2 d^4 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx\\ &=-\frac{a^2 d \cos (c+d x)}{12 x^3}+\frac{a^2 d^3 \cos (c+d x)}{24 x}-\frac{b^2 x \cos (c+d x)}{d}+2 a b d \cos (c) \text{Ci}(d x)+\frac{1}{24} a^2 d^4 \text{Ci}(d x) \sin (c)+\frac{b^2 \sin (c+d x)}{d^2}-\frac{a^2 \sin (c+d x)}{4 x^4}+\frac{a^2 d^2 \sin (c+d x)}{24 x^2}-\frac{2 a b \sin (c+d x)}{x}+\frac{1}{24} a^2 d^4 \cos (c) \text{Si}(d x)-2 a b d \sin (c) \text{Si}(d x)\\ \end{align*}
Mathematica [A] time = 0.572035, size = 148, normalized size = 0.89 \[ \frac{1}{24} \left (\frac{a^2 d^2 \sin (c+d x)}{x^2}+\frac{a^2 d^3 \cos (c+d x)}{x}-\frac{6 a^2 \sin (c+d x)}{x^4}-\frac{2 a^2 d \cos (c+d x)}{x^3}+a d \text{CosIntegral}(d x) \left (a d^3 \sin (c)+48 b \cos (c)\right )+a d \text{Si}(d x) \left (a d^3 \cos (c)-48 b \sin (c)\right )-\frac{48 a b \sin (c+d x)}{x}+\frac{24 b^2 \sin (c+d x)}{d^2}-\frac{24 b^2 x \cos (c+d x)}{d}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 167, normalized size = 1. \begin{align*}{d}^{4} \left ({\frac{ \left ( 1+5\,c \right ){b}^{2} \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{{d}^{6}}}+6\,{\frac{c{b}^{2}\cos \left ( dx+c \right ) }{{d}^{6}}}+2\,{\frac{ab}{{d}^{3}} \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 ) }{4\,{x}^{4}{d}^{4}}}-{\frac{\cos \left ( dx+c \right ) }{12\,{d}^{3}{x}^{3}}}+{\frac{\sin \left ( dx+c \right ) }{24\,{d}^{2}{x}^{2}}}+{\frac{\cos \left ( dx+c \right ) }{24\,dx}}+{\frac{{\it Si} \left ( dx \right ) \cos \left ( c \right ) }{24}}+{\frac{{\it Ci} \left ( dx \right ) \sin \left ( c \right ) }{24}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 37.3766, size = 224, normalized size = 1.34 \begin{align*} \frac{{\left ({\left (a^{2}{\left (-i \, \Gamma \left (-4, i \, d x\right ) + i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) - a^{2}{\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{7} -{\left (48 \, a b{\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) - a b{\left (48 i \, \Gamma \left (-4, i \, d x\right ) - 48 i \, \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{4}\right )} x^{4} - 2 \,{\left (b^{2} d^{2} x^{5} + 2 \, a b d^{2} x^{2} - 12 \, a b\right )} \cos \left (d x + c\right ) + 2 \,{\left (b^{2} d x^{4} - 4 \, a b d x\right )} \sin \left (d x + c\right )}{2 \, d^{3} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14147, size = 502, normalized size = 3.01 \begin{align*} \frac{2 \,{\left (a^{2} d^{5} x^{3} - 24 \, b^{2} d x^{5} - 2 \, a^{2} d^{3} x\right )} \cos \left (d x + c\right ) + 2 \,{\left (a^{2} d^{6} x^{4} \operatorname{Si}\left (d x\right ) + 24 \, a b d^{3} x^{4} \operatorname{Ci}\left (d x\right ) + 24 \, a b d^{3} x^{4} \operatorname{Ci}\left (-d x\right )\right )} \cos \left (c\right ) + 2 \,{\left (a^{2} d^{4} x^{2} - 48 \, a b d^{2} x^{3} + 24 \, b^{2} x^{4} - 6 \, a^{2} d^{2}\right )} \sin \left (d x + c\right ) +{\left (a^{2} d^{6} x^{4} \operatorname{Ci}\left (d x\right ) + a^{2} d^{6} x^{4} \operatorname{Ci}\left (-d x\right ) - 96 \, a b d^{3} x^{4} \operatorname{Si}\left (d x\right )\right )} \sin \left (c\right )}{48 \, d^{2} x^{4}} \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^{5}}\, 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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