Optimal. Leaf size=115 \[ -\frac {i e^{i a} \left (-\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x) \Gamma \left (-\frac {2}{3},-\frac {i b}{(c+d x)^{3/2}}\right )}{3 d}+\frac {i e^{-i a} \left (\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x) \Gamma \left (-\frac {2}{3},\frac {i b}{(c+d x)^{3/2}}\right )}{3 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3444, 3504,
2250} \begin {gather*} \frac {i e^{-i a} (c+d x) \left (\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} \text {Gamma}\left (-\frac {2}{3},\frac {i b}{(c+d x)^{3/2}}\right )}{3 d}-\frac {i e^{i a} (c+d x) \left (-\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} \text {Gamma}\left (-\frac {2}{3},-\frac {i b}{(c+d x)^{3/2}}\right )}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rule 3444
Rule 3504
Rubi steps
\begin {align*} \int \sin \left (a+\frac {b}{(c+d x)^{3/2}}\right ) \, dx &=\frac {2 \text {Subst}\left (\int x \sin \left (a+\frac {b}{x^3}\right ) \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=\frac {i \text {Subst}\left (\int e^{-i a-\frac {i b}{x^3}} x \, dx,x,\sqrt {c+d x}\right )}{d}-\frac {i \text {Subst}\left (\int e^{i a+\frac {i b}{x^3}} x \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=-\frac {i e^{i a} \left (-\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x) \Gamma \left (-\frac {2}{3},-\frac {i b}{(c+d x)^{3/2}}\right )}{3 d}+\frac {i e^{-i a} \left (\frac {i b}{(c+d x)^{3/2}}\right )^{2/3} (c+d x) \Gamma \left (-\frac {2}{3},\frac {i b}{(c+d x)^{3/2}}\right )}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 166, normalized size = 1.44 \begin {gather*} \frac {b \sqrt [3]{-\frac {i b}{(c+d x)^{3/2}}} \Gamma \left (\frac {1}{3},\frac {i b}{(c+d x)^{3/2}}\right ) (\cos (a)-i \sin (a))+b \sqrt [3]{\frac {i b}{(c+d x)^{3/2}}} \Gamma \left (\frac {1}{3},-\frac {i b}{(c+d x)^{3/2}}\right ) (\cos (a)+i \sin (a))+2 \sqrt [3]{\frac {b^2}{(c+d x)^3}} (c+d x)^{3/2} \sin \left (a+\frac {b}{(c+d x)^{3/2}}\right )}{2 d \sqrt [3]{\frac {b^2}{(c+d x)^3}} \sqrt {c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \sin \left (a +\frac {b}{\left (d x +c \right )^{\frac {3}{2}}}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 151, normalized size = 1.31 \begin {gather*} \frac {4 \, {\left (d x + c\right )}^{\frac {3}{2}} \left (\frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )^{\frac {1}{3}} \sin \left (\frac {{\left (d x + c\right )}^{\frac {3}{2}} a + b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left ({\left ({\left (\sqrt {3} - i\right )} \Gamma \left (\frac {1}{3}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left (\sqrt {3} + i\right )} \Gamma \left (\frac {1}{3}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )\right )} \cos \left (a\right ) + {\left ({\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, \frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right ) + {\left (i \, \sqrt {3} - 1\right )} \Gamma \left (\frac {1}{3}, -\frac {i \, b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )\right )} \sin \left (a\right )\right )} b}{4 \, \sqrt {d x + c} d \left (\frac {b}{{\left (d x + c\right )}^{\frac {3}{2}}}\right )^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.12, size = 144, normalized size = 1.25 \begin {gather*} \frac {-i \, \left (i \, b\right )^{\frac {2}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac {1}{3}, \frac {i \, \sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + i \, \left (-i \, b\right )^{\frac {2}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac {1}{3}, -\frac {i \, \sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 2 \, {\left (d x + c\right )} \sin \left (\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + \sqrt {d x + c} b}{d^{2} x^{2} + 2 \, c d x + c^{2}}\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 )^{\frac {3}{2}}} \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 )}^{3/2}}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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