Optimal. Leaf size=419 \[ \frac{b^3 \cos (a) (d e-c f) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^6 f \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}-\frac{b^3 \sin (a) (d e-c f) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^2 \sqrt [3]{c+d x} (d e-c f) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^6 f \cos (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{b^4 f (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}-\frac{b^2 f (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac{b^5 f \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}-\frac{b^3 f (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac{(c+d x) (d e-c f) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}+\frac{b (c+d x)^{2/3} (d e-c f) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b f (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{10 d^2} \]
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Rubi [A] time = 0.504016, antiderivative size = 419, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3431, 3297, 3303, 3299, 3302} \[ \frac{b^3 \cos (a) (d e-c f) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^6 f \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}-\frac{b^3 \sin (a) (d e-c f) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^2 \sqrt [3]{c+d x} (d e-c f) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^6 f \cos (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{b^4 f (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}-\frac{b^2 f (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac{b^5 f \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}-\frac{b^3 f (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac{(c+d x) (d e-c f) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}+\frac{b (c+d x)^{2/3} (d e-c f) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b f (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{10 d^2} \]
Antiderivative was successfully verified.
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Rule 3431
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int (e+f x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) \, dx &=-\frac{3 \operatorname{Subst}\left (\int \left (\frac{f \sin (a+b x)}{d x^7}+\frac{(d e-c f) \sin (a+b x)}{d x^4}\right ) \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d}\\ &=-\frac{(3 f) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^7} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac{(3 (d e-c f)) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^4} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d^2}\\ &=\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{(b f) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^6} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{(b (d e-c f)) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^3} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d^2}\\ &=\frac{b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b f (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{\left (b^2 f\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^5} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac{\left (b^2 (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^2} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{2 d^2}\\ &=\frac{b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b f (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{10 d^2}-\frac{b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac{b^2 f (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{\left (b^3 f\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^4} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac{\left (b^3 (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{2 d^2}\\ &=\frac{b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^3 f (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac{b f (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{10 d^2}-\frac{b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac{b^2 f (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{\left (b^4 f\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^3} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac{\left (b^3 (d e-c f) \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{\left (b^3 (d e-c f) \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{2 d^2}\\ &=\frac{b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^3 f (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac{b f (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac{b^3 (d e-c f) \cos (a) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^4 f (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac{b^2 f (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^3 (d e-c f) \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{\left (b^5 f\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^2} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{240 d^2}\\ &=\frac{b^5 f \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^3 f (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac{b f (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac{b^3 (d e-c f) \cos (a) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^4 f (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac{b^2 f (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^3 (d e-c f) \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{\left (b^6 f\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{240 d^2}\\ &=\frac{b^5 f \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^3 f (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac{b f (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac{b^3 (d e-c f) \cos (a) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^4 f (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac{b^2 f (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^3 (d e-c f) \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{\left (b^6 f \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{\left (b^6 f \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{240 d^2}\\ &=\frac{b^5 f \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{b (d e-c f) (c+d x)^{2/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}-\frac{b^3 f (c+d x) \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{120 d^2}+\frac{b f (c+d x)^{5/3} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{10 d^2}+\frac{b^3 (d e-c f) \cos (a) \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^6 f \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{240 d^2}-\frac{b^2 (d e-c f) \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^4 f (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}+\frac{(d e-c f) (c+d x) \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{d^2}-\frac{b^2 f (c+d x)^{4/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{40 d^2}+\frac{f (c+d x)^2 \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}+\frac{b^6 f \cos (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{240 d^2}-\frac{b^3 (d e-c f) \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{2 d^2}\\ \end{align*}
Mathematica [A] time = 0.855827, size = 540, normalized size = 1.29 \[ \frac{b^3 f \left (b^3 \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )+b^3 \cos (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )-120 c \cos (a) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )+120 c \sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )\right )}{240 d^2}+\frac{b^3 e \left (\cos (a) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )-\sin (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )\right )}{2 d}+\frac{f \sqrt [3]{c+d x} \cos \left (\frac{b}{\sqrt [3]{c+d x}}\right ) \left (b^4 \sin (a) \sqrt [3]{c+d x}-6 b^2 \sin (a) (c+d x)-2 b^3 \cos (a) (c+d x)^{2/3}+120 b^2 c \sin (a)+b^5 \cos (a)+24 b \cos (a) (c+d x)^{4/3}-120 b c \cos (a) \sqrt [3]{c+d x}+120 \sin (a) (c+d x)^{5/3}-240 c \sin (a) (c+d x)^{2/3}\right )}{240 d^2}+\frac{f \sqrt [3]{c+d x} \sin \left (\frac{b}{\sqrt [3]{c+d x}}\right ) \left (2 b^3 \sin (a) (c+d x)^{2/3}+b^4 \cos (a) \sqrt [3]{c+d x}-6 b^2 \cos (a) (c+d x)+120 b^2 c \cos (a)+b^5 (-\sin (a))-24 b \sin (a) (c+d x)^{4/3}+120 b c \sin (a) \sqrt [3]{c+d x}+120 \cos (a) (c+d x)^{5/3}-240 c \cos (a) (c+d x)^{2/3}\right )}{240 d^2}+\frac{e \sqrt [3]{c+d x} \cos \left (\frac{b}{\sqrt [3]{c+d x}}\right ) \left (b^2 (-\sin (a))+b \cos (a) \sqrt [3]{c+d x}+2 \sin (a) (c+d x)^{2/3}\right )}{2 d}+\frac{e \sqrt [3]{c+d x} \sin \left (\frac{b}{\sqrt [3]{c+d x}}\right ) \left (b^2 (-\cos (a))-b \sin (a) \sqrt [3]{c+d x}+2 \cos (a) (c+d x)^{2/3}\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 391, normalized size = 0.9 \begin{align*} -3\,{\frac{{b}^{3}}{{d}^{2}} \left ( -cf \left ( -1/3\,{\frac{dx+c}{{b}^{3}}\sin \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }-1/6\,{\frac{ \left ( dx+c \right ) ^{2/3}}{{b}^{2}}\cos \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }+1/6\,{\frac{\sqrt [3]{dx+c}}{b}\sin \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }+1/6\,{\it Si} \left ({\frac{b}{\sqrt [3]{dx+c}}} \right ) \sin \left ( a \right ) -1/6\,{\it Ci} \left ({\frac{b}{\sqrt [3]{dx+c}}} \right ) \cos \left ( a \right ) \right ) +de \left ( -1/3\,{\frac{dx+c}{{b}^{3}}\sin \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }-1/6\,{\frac{ \left ( dx+c \right ) ^{2/3}}{{b}^{2}}\cos \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }+1/6\,{\frac{\sqrt [3]{dx+c}}{b}\sin \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }+1/6\,{\it Si} \left ({\frac{b}{\sqrt [3]{dx+c}}} \right ) \sin \left ( a \right ) -1/6\,{\it Ci} \left ({\frac{b}{\sqrt [3]{dx+c}}} \right ) \cos \left ( a \right ) \right ) +{b}^{3}f \left ( -1/6\,{\frac{ \left ( dx+c \right ) ^{2}}{{b}^{6}}\sin \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }-1/30\,{\frac{ \left ( dx+c \right ) ^{5/3}}{{b}^{5}}\cos \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }+{\frac{ \left ( dx+c \right ) ^{4/3}}{120\,{b}^{4}}\sin \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }+{\frac{dx+c}{360\,{b}^{3}}\cos \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }-{\frac{ \left ( dx+c \right ) ^{2/3}}{720\,{b}^{2}}\sin \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }-{\frac{\sqrt [3]{dx+c}}{720\,b}\cos \left ( a+{\frac{b}{\sqrt [3]{dx+c}}} \right ) }-{\frac{\cos \left ( a \right ) }{720}{\it Si} \left ({\frac{b}{\sqrt [3]{dx+c}}} \right ) }-{\frac{\sin \left ( a \right ) }{720}{\it Ci} \left ({\frac{b}{\sqrt [3]{dx+c}}} \right ) } \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.57241, size = 618, normalized size = 1.47 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03869, size = 814, normalized size = 1.94 \begin{align*} \frac{2 \,{\left ({\left (d x + c\right )}^{\frac{1}{3}} b^{5} f - 2 \, b^{3} d f x - 2 \, b^{3} c f + 24 \,{\left (b d f x + 5 \, b d e - 4 \, b c f\right )}{\left (d x + c\right )}^{\frac{2}{3}}\right )} \cos \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{2}{3}} b}{d x + c}\right ) +{\left (b^{6} f \sin \left (a\right ) + 120 \,{\left (b^{3} d e - b^{3} c f\right )} \cos \left (a\right )\right )} \operatorname{Ci}\left (\frac{b}{{\left (d x + c\right )}^{\frac{1}{3}}}\right ) +{\left (b^{6} f \sin \left (a\right ) + 120 \,{\left (b^{3} d e - b^{3} c f\right )} \cos \left (a\right )\right )} \operatorname{Ci}\left (-\frac{b}{{\left (d x + c\right )}^{\frac{1}{3}}}\right ) + 2 \,{\left ({\left (d x + c\right )}^{\frac{2}{3}} b^{4} f + 120 \, d^{2} f x^{2} + 240 \, d^{2} e x + 240 \, c d e - 120 \, c^{2} f - 6 \,{\left (b^{2} d f x + 20 \, b^{2} d e - 19 \, b^{2} c f\right )}{\left (d x + c\right )}^{\frac{1}{3}}\right )} \sin \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{2}{3}} b}{d x + c}\right ) + 2 \,{\left (b^{6} f \cos \left (a\right ) - 120 \,{\left (b^{3} d e - b^{3} c f\right )} \sin \left (a\right )\right )} \operatorname{Si}\left (\frac{b}{{\left (d x + c\right )}^{\frac{1}{3}}}\right )}{480 \, d^{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 \left (e + f x\right ) \sin{\left (a + \frac{b}{\sqrt [3]{c + d x}} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )} \sin \left (a + \frac{b}{{\left (d x + c\right )}^{\frac{1}{3}}}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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