Optimal. Leaf size=107 \[ \frac {i e^{i a} (c+d x) \Gamma \left (\frac {1}{3},-i b (c+d x)^3\right )}{6 d \sqrt [3]{-i b (c+d x)^3}}-\frac {i e^{-i a} (c+d x) \Gamma \left (\frac {1}{3},i b (c+d x)^3\right )}{6 d \sqrt [3]{i b (c+d x)^3}} \]
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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 = {3436, 2239}
\begin {gather*} \frac {i e^{i a} (c+d x) \text {Gamma}\left (\frac {1}{3},-i b (c+d x)^3\right )}{6 d \sqrt [3]{-i b (c+d x)^3}}-\frac {i e^{-i a} (c+d x) \text {Gamma}\left (\frac {1}{3},i b (c+d x)^3\right )}{6 d \sqrt [3]{i b (c+d x)^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2239
Rule 3436
Rubi steps
\begin {align*} \int \sin \left (a+b (c+d x)^3\right ) \, dx &=\frac {1}{2} i \int e^{-i a-i b (c+d x)^3} \, dx-\frac {1}{2} i \int e^{i a+i b (c+d x)^3} \, dx\\ &=\frac {i e^{i a} (c+d x) \Gamma \left (\frac {1}{3},-i b (c+d x)^3\right )}{6 d \sqrt [3]{-i b (c+d x)^3}}-\frac {i e^{-i a} (c+d x) \Gamma \left (\frac {1}{3},i b (c+d x)^3\right )}{6 d \sqrt [3]{i b (c+d x)^3}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 115, normalized size = 1.07 \begin {gather*} \frac {i (c+d x) \left (-\sqrt [3]{-i b (c+d x)^3} \Gamma \left (\frac {1}{3},i b (c+d x)^3\right ) (\cos (a)-i \sin (a))+\sqrt [3]{i b (c+d x)^3} \Gamma \left (\frac {1}{3},-i b (c+d x)^3\right ) (\cos (a)+i \sin (a))\right )}{6 d \sqrt [3]{b^2 (c+d x)^6}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \sin \left (a +b \left (d x +c \right )^{3}\right )\, dx\]
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 [A]
time = 0.09, size = 107, normalized size = 1.00 \begin {gather*} -\frac {\left (i \, b d^{3}\right )^{\frac {2}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac {1}{3}, i \, b d^{3} x^{3} + 3 i \, b c d^{2} x^{2} + 3 i \, b c^{2} d x + i \, b c^{3}\right ) + \left (-i \, b d^{3}\right )^{\frac {2}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac {1}{3}, -i \, b d^{3} x^{3} - 3 i \, b c d^{2} x^{2} - 3 i \, b c^{2} d x - i \, b c^{3}\right )}{6 \, b d^{3}} \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 + 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+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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