Optimal. Leaf size=107 \[ \frac {i e^{-i a} (c+d x) \sqrt [3]{\frac {i b}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},\frac {i b}{(c+d x)^3}\right )}{6 d}-\frac {i e^{i a} (c+d x) \sqrt [3]{-\frac {i b}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {i b}{(c+d x)^3}\right )}{6 d} \]
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Rubi [A] time = 0.03, 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 = {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 2208
Rule 3365
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 = 0.51, size = 203, normalized size = 1.90 \[ \frac {2 \sin (a) (c+d x)^3 \cos \left (\frac {b}{(c+d x)^3}\right )+2 \cos (a) (c+d x)^3 \sin \left (\frac {b}{(c+d x)^3}\right )+b \cos (a) \left (\frac {\Gamma \left (\frac {2}{3},-\frac {i b}{(c+d x)^3}\right )}{\left (-\frac {i b}{(c+d x)^3}\right )^{2/3}}+\frac {\Gamma \left (\frac {2}{3},\frac {i b}{(c+d x)^3}\right )}{\left (\frac {i b}{(c+d x)^3}\right )^{2/3}}\right )+i b \sin (a) \left (\frac {\Gamma \left (\frac {2}{3},-\frac {i b}{(c+d x)^3}\right )}{\left (-\frac {i b}{(c+d x)^3}\right )^{2/3}}-\frac {\Gamma \left (\frac {2}{3},\frac {i b}{(c+d x)^3}\right )}{\left (\frac {i b}{(c+d x)^3}\right )^{2/3}}\right )}{2 d (c+d x)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.79, size = 175, normalized size = 1.64 \[ \frac {-i \, d \left (\frac {i \, b}{d^{3}}\right )^{\frac {1}{3}} e^{\left (-i \, a\right )} \Gamma \left (\frac {2}{3}, \frac {i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + i \, d \left (-\frac {i \, b}{d^{3}}\right )^{\frac {1}{3}} e^{\left (i \, a\right )} \Gamma \left (\frac {2}{3}, -\frac {i \, b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + 2 \, {\left (d x + c\right )} \sin \left (\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{2 \, d} \]
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 \[ \int \sin \left (a + \frac {b}{{\left (d x + c\right )}^{3}}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, 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 \[ 3 \, b d \int \frac {x \cos \left (\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{2 \, {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )}}\,{d x} + 3 \, b d \int \frac {x \cos \left (\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}{2 \, {\left ({\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )} \cos \left (\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )^{2} + {\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )} \sin \left (\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )^{2}\right )}}\,{d x} + x \sin \left (\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) \]
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 \[ \int \sin \left (a+\frac {b}{{\left (c+d\,x\right )}^3}\right ) \,d x \]
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 \[ \int \sin {\left (a + \frac {b}{\left (c + d x\right )^{3}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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