Optimal. Leaf size=107 \[ \frac{i e^{-i a} (c+d x) \sqrt [3]{\frac{i b}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},\frac{i b}{(c+d x)^3}\right )}{6 d}-\frac{i e^{i a} (c+d x) \sqrt [3]{-\frac{i b}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{i b}{(c+d x)^3}\right )}{6 d} \]
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Rubi [A] time = 0.0267101, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3365, 2208} \[ \frac{i e^{-i a} (c+d x) \sqrt [3]{\frac{i b}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},\frac{i b}{(c+d x)^3}\right )}{6 d}-\frac{i e^{i a} (c+d x) \sqrt [3]{-\frac{i b}{(c+d x)^3}} \text{Gamma}\left (-\frac{1}{3},-\frac{i b}{(c+d x)^3}\right )}{6 d} \]
Antiderivative was successfully verified.
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Rule 3365
Rule 2208
Rubi steps
\begin{align*} \int \sin \left (a+\frac{b}{(c+d x)^3}\right ) \, dx &=\frac{1}{2} i \int e^{-i a-\frac{i b}{(c+d x)^3}} \, dx-\frac{1}{2} i \int e^{i a+\frac{i b}{(c+d x)^3}} \, dx\\ &=-\frac{i e^{i a} \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}+\frac{i e^{-i a} \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}\\ \end{align*}
Mathematica [A] time = 0.487147, size = 203, normalized size = 1.9 \[ \frac{b \cos (a) \left (\frac{\text{Gamma}\left (\frac{2}{3},-\frac{i b}{(c+d x)^3}\right )}{\left (-\frac{i b}{(c+d x)^3}\right )^{2/3}}+\frac{\text{Gamma}\left (\frac{2}{3},\frac{i b}{(c+d x)^3}\right )}{\left (\frac{i b}{(c+d x)^3}\right )^{2/3}}\right )+i b \sin (a) \left (\frac{\text{Gamma}\left (\frac{2}{3},-\frac{i b}{(c+d x)^3}\right )}{\left (-\frac{i b}{(c+d x)^3}\right )^{2/3}}-\frac{\text{Gamma}\left (\frac{2}{3},\frac{i b}{(c+d x)^3}\right )}{\left (\frac{i b}{(c+d x)^3}\right )^{2/3}}\right )+2 \sin (a) (c+d x)^3 \cos \left (\frac{b}{(c+d x)^3}\right )+2 \cos (a) (c+d x)^3 \sin \left (\frac{b}{(c+d x)^3}\right )}{2 d (c+d x)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.101, size = 0, normalized size = 0. \begin{align*} \int \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*} 3 \, b d \int \frac{x \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 )}}\,{d x} + 3 \, b d \int \frac{x \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 ({\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} + x \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.86678, size = 408, normalized size = 3.81 \begin{align*} \frac{-i \, d \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 ) + i \, d \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 x + c\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 \, d} \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 \sin{\left (a + \frac{b}{\left (c + d x\right )^{3}} \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 \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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