Optimal. Leaf size=235 \[ -\frac{i e^{i a} (c+d x) \sqrt [3]{-\frac{i b}{(c+d x)^3}} (d e-c f) \text{Gamma}\left (-\frac{1}{3},-\frac{i b}{(c+d x)^3}\right )}{6 d^2}+\frac{i e^{-i a} (c+d x) \sqrt [3]{\frac{i b}{(c+d x)^3}} (d e-c f) \text{Gamma}\left (-\frac{1}{3},\frac{i b}{(c+d x)^3}\right )}{6 d^2}-\frac{i e^{i a} f (c+d x)^2 \left (-\frac{i b}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{i b}{(c+d x)^3}\right )}{6 d^2}+\frac{i e^{-i a} f (c+d x)^2 \left (\frac{i b}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},\frac{i b}{(c+d x)^3}\right )}{6 d^2} \]
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Rubi [A] time = 0.14581, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {3433, 3365, 2208, 3423, 2218} \[ -\frac{i e^{i a} (c+d x) \sqrt [3]{-\frac{i b}{(c+d x)^3}} (d e-c f) \text{Gamma}\left (-\frac{1}{3},-\frac{i b}{(c+d x)^3}\right )}{6 d^2}+\frac{i e^{-i a} (c+d x) \sqrt [3]{\frac{i b}{(c+d x)^3}} (d e-c f) \text{Gamma}\left (-\frac{1}{3},\frac{i b}{(c+d x)^3}\right )}{6 d^2}-\frac{i e^{i a} f (c+d x)^2 \left (-\frac{i b}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{i b}{(c+d x)^3}\right )}{6 d^2}+\frac{i e^{-i a} f (c+d x)^2 \left (\frac{i b}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},\frac{i b}{(c+d x)^3}\right )}{6 d^2} \]
Antiderivative was successfully verified.
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Rule 3433
Rule 3365
Rule 2208
Rule 3423
Rule 2218
Rubi steps
\begin{align*} \int (e+f x) \sin \left (a+\frac{b}{(c+d x)^3}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (d e \left (1-\frac{c f}{d e}\right ) \sin \left (a+\frac{b}{x^3}\right )+f x \sin \left (a+\frac{b}{x^3}\right )\right ) \, dx,x,c+d x\right )}{d^2}\\ &=\frac{f \operatorname{Subst}\left (\int x \sin \left (a+\frac{b}{x^3}\right ) \, dx,x,c+d x\right )}{d^2}+\frac{(d e-c f) \operatorname{Subst}\left (\int \sin \left (a+\frac{b}{x^3}\right ) \, dx,x,c+d x\right )}{d^2}\\ &=\frac{(i f) \operatorname{Subst}\left (\int e^{-i a-\frac{i b}{x^3}} x \, dx,x,c+d x\right )}{2 d^2}-\frac{(i f) \operatorname{Subst}\left (\int e^{i a+\frac{i b}{x^3}} x \, dx,x,c+d x\right )}{2 d^2}+\frac{(i (d e-c f)) \operatorname{Subst}\left (\int e^{-i a-\frac{i b}{x^3}} \, dx,x,c+d x\right )}{2 d^2}-\frac{(i (d e-c f)) \operatorname{Subst}\left (\int e^{i a+\frac{i b}{x^3}} \, dx,x,c+d x\right )}{2 d^2}\\ &=-\frac{i e^{i a} f \left (-\frac{i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac{2}{3},-\frac{i b}{(c+d x)^3}\right )}{6 d^2}+\frac{i e^{-i a} f \left (\frac{i b}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac{2}{3},\frac{i b}{(c+d x)^3}\right )}{6 d^2}-\frac{i e^{i a} (d e-c f) \sqrt [3]{-\frac{i b}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac{1}{3},-\frac{i b}{(c+d x)^3}\right )}{6 d^2}+\frac{i e^{-i a} (d e-c f) \sqrt [3]{\frac{i b}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac{1}{3},\frac{i b}{(c+d x)^3}\right )}{6 d^2}\\ \end{align*}
Mathematica [B] time = 2.27234, size = 700, normalized size = 2.98 \[ \frac{3 b f \left (\frac{1}{2} \cos (a) \left (\frac{\text{Gamma}\left (\frac{1}{3},-\frac{i b}{(c+d x)^3}\right )}{3 (c+d x) \sqrt [3]{-\frac{i b}{(c+d x)^3}}}+\frac{\text{Gamma}\left (\frac{1}{3},\frac{i b}{(c+d x)^3}\right )}{3 (c+d x) \sqrt [3]{\frac{i b}{(c+d x)^3}}}\right )+\frac{1}{2} i \sin (a) \left (\frac{\text{Gamma}\left (\frac{1}{3},-\frac{i b}{(c+d x)^3}\right )}{3 (c+d x) \sqrt [3]{-\frac{i b}{(c+d x)^3}}}-\frac{\text{Gamma}\left (\frac{1}{3},\frac{i b}{(c+d x)^3}\right )}{3 (c+d x) \sqrt [3]{\frac{i b}{(c+d x)^3}}}\right )\right )}{2 d^2}-\frac{3 b c f \left (\frac{1}{2} \cos (a) \left (\frac{\text{Gamma}\left (\frac{2}{3},-\frac{i b}{(c+d x)^3}\right )}{3 (c+d x)^2 \left (-\frac{i b}{(c+d x)^3}\right )^{2/3}}+\frac{\text{Gamma}\left (\frac{2}{3},\frac{i b}{(c+d x)^3}\right )}{3 (c+d x)^2 \left (\frac{i b}{(c+d x)^3}\right )^{2/3}}\right )+\frac{1}{2} i \sin (a) \left (\frac{\text{Gamma}\left (\frac{2}{3},-\frac{i b}{(c+d x)^3}\right )}{3 (c+d x)^2 \left (-\frac{i b}{(c+d x)^3}\right )^{2/3}}-\frac{\text{Gamma}\left (\frac{2}{3},\frac{i b}{(c+d x)^3}\right )}{3 (c+d x)^2 \left (\frac{i b}{(c+d x)^3}\right )^{2/3}}\right )\right )}{d^2}+\frac{3 b e \left (\frac{1}{2} \cos (a) \left (\frac{\text{Gamma}\left (\frac{2}{3},-\frac{i b}{(c+d x)^3}\right )}{3 (c+d x)^2 \left (-\frac{i b}{(c+d x)^3}\right )^{2/3}}+\frac{\text{Gamma}\left (\frac{2}{3},\frac{i b}{(c+d x)^3}\right )}{3 (c+d x)^2 \left (\frac{i b}{(c+d x)^3}\right )^{2/3}}\right )+\frac{1}{2} i \sin (a) \left (\frac{\text{Gamma}\left (\frac{2}{3},-\frac{i b}{(c+d x)^3}\right )}{3 (c+d x)^2 \left (-\frac{i b}{(c+d x)^3}\right )^{2/3}}-\frac{\text{Gamma}\left (\frac{2}{3},\frac{i b}{(c+d x)^3}\right )}{3 (c+d x)^2 \left (\frac{i b}{(c+d x)^3}\right )^{2/3}}\right )\right )}{d}+\frac{f \sin (a) (d x-c) (c+d x) \cos \left (\frac{b}{(c+d x)^3}\right )}{2 d^2}+\frac{f \cos (a) (d x-c) (c+d x) \sin \left (\frac{b}{(c+d x)^3}\right )}{2 d^2}+\frac{e \sin (a) (c+d x) \cos \left (\frac{b}{(c+d x)^3}\right )}{d}+\frac{e \cos (a) (c+d x) \sin \left (\frac{b}{(c+d x)^3}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.21, size = 0, normalized size = 0. \begin{align*} \int \left ( fx+e \right ) \sin \left ( a+{\frac{b}{ \left ( dx+c \right ) ^{3}}} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \,{\left (f x^{2} + 2 \, e x\right )} \sin \left (\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + \int \frac{3 \,{\left (b d f x^{2} + 2 \, b d e x\right )} \cos \left (\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{4 \,{\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )}}\,{d x} + \int \frac{3 \,{\left (b d f x^{2} + 2 \, b d e x\right )} \cos \left (\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{4 \,{\left ({\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )} \cos \left (\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )^{2} +{\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )} \sin \left (\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )^{2}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0135, size = 761, normalized size = 3.24 \begin{align*} \frac{-i \, d^{2} f \left (\frac{i \, b}{d^{3}}\right )^{\frac{2}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac{1}{3}, \frac{i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + i \, d^{2} f \left (-\frac{i \, b}{d^{3}}\right )^{\frac{2}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac{1}{3}, -\frac{i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) +{\left (-2 i \, d^{2} e + 2 i \, c d f\right )} \left (\frac{i \, b}{d^{3}}\right )^{\frac{1}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac{2}{3}, \frac{i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) +{\left (2 i \, d^{2} e - 2 i \, c d f\right )} \left (-\frac{i \, b}{d^{3}}\right )^{\frac{1}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac{2}{3}, -\frac{i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + 2 \,{\left (d^{2} f x^{2} + 2 \, d^{2} e x + 2 \, c d e - c^{2} f\right )} \sin \left (\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{4 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )}^{3}}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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