Optimal. Leaf size=107 \[ -\frac {i e^{i a} \sqrt [3]{-\frac {i b}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {i b}{(c+d x)^3}\right )}{6 d}+\frac {i e^{-i a} \sqrt [3]{\frac {i b}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},\frac {i b}{(c+d x)^3}\right )}{6 d} \]
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Rubi [A]
time = 0.02, antiderivative size = 107, normalized size of antiderivative = 1.00, 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 = {3446, 2239}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2239
Rule 3446
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*}
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Mathematica [A]
time = 0.30, size = 203, normalized size = 1.90 \begin {gather*} \frac {b \cos (a) \left (\frac {\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 {\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 (c+d x)^3 \cos \left (\frac {b}{(c+d x)^3}\right ) \sin (a)+i b \left (\frac {\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 {\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 ) \sin (a)+2 (c+d x)^3 \cos (a) \sin \left (\frac {b}{(c+d x)^3}\right )}{2 d (c+d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \sin \left (a +\frac {b}{\left (d x +c \right )^{3}}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 175 vs. \(2 (77) = 154\).
time = 0.10, size = 175, normalized size = 1.64 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sin {\left (a + \frac {b}{\left (c + d x\right )^{3}} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^3}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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