Optimal. Leaf size=176 \[ -\frac {\sqrt {-\frac {1}{n^2}} n e^{3 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-1/n}}{16 x}+\frac {9 \sqrt {-\frac {1}{n^2}} n e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .-\frac {1}{3}\right /n}}{32 x}-\frac {9 \sqrt {-\frac {1}{n^2}} n e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .\frac {1}{3}\right /n}}{16 x}-\frac {\sqrt {-\frac {1}{n^2}} n e^{-3 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{\frac {1}{n}}}{8 x} \]
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Rubi [A] time = 0.13, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {4493, 4489} \[ -\frac {\sqrt {-\frac {1}{n^2}} n e^{3 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-1/n}}{16 x}+\frac {9 \sqrt {-\frac {1}{n^2}} n e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .-\frac {1}{3}\right /n}}{32 x}-\frac {9 \sqrt {-\frac {1}{n^2}} n e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .\frac {1}{3}\right /n}}{16 x}-\frac {\sqrt {-\frac {1}{n^2}} n e^{-3 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{\frac {1}{n}}}{8 x} \]
Antiderivative was successfully verified.
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Rule 4489
Rule 4493
Rubi steps
\begin {align*} \int \frac {\sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx &=\frac {\left (c x^n\right )^{\frac {1}{n}} \operatorname {Subst}\left (\int x^{-1-\frac {1}{n}} \sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n x}\\ &=-\frac {\left (\sqrt {-\frac {1}{n^2}} \left (c x^n\right )^{\frac {1}{n}}\right ) \operatorname {Subst}\left (\int \left (\frac {e^{-3 a \sqrt {-\frac {1}{n^2}} n}}{x}+3 e^{a \sqrt {-\frac {1}{n^2}} n} x^{-1-\frac {4}{3 n}}-3 e^{-a \sqrt {-\frac {1}{n^2}} n} x^{-1-\frac {2}{3 n}}-e^{3 a \sqrt {-\frac {1}{n^2}} n} x^{-\frac {2+n}{n}}\right ) \, dx,x,c x^n\right )}{8 x}\\ &=-\frac {e^{3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{-1/n}}{16 x}+\frac {9 e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\left .-\frac {1}{3}\right /n}}{32 x}-\frac {9 e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\left .\frac {1}{3}\right /n}}{16 x}-\frac {e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\frac {1}{n}} \log (x)}{8 x}\\ \end {align*}
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Mathematica [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [C] time = 0.45, size = 87, normalized size = 0.49 \[ \frac {{\left (-12 i \, x^{2} \log \left (x^{\frac {1}{3}}\right ) - 18 i \, x^{\frac {4}{3}} e^{\left (\frac {2 \, {\left (3 i \, a n - \log \relax (c)\right )}}{3 \, n}\right )} + 9 i \, x^{\frac {2}{3}} e^{\left (\frac {4 \, {\left (3 i \, a n - \log \relax (c)\right )}}{3 \, n}\right )} - 2 i \, e^{\left (\frac {2 \, {\left (3 i \, a n - \log \relax (c)\right )}}{n}\right )}\right )} e^{\left (-\frac {3 i \, a n - \log \relax (c)}{n}\right )}}{32 \, x^{2}} \]
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 \[ \int \frac {\sin \left (\frac {1}{3} \, \sqrt {-\frac {1}{n^{2}}} \log \left (c x^{n}\right ) + a\right )^{3}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{3}\left (a +\frac {\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{3}\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 122, normalized size = 0.69 \[ -\frac {{\left (4 \, c^{\frac {7}{3 \, n}} x e^{\left (\frac {\log \left (x^{n}\right )}{3 \, n} + 2 \, \log \relax (x)\right )} \log \relax (x) \sin \left (3 \, a\right ) - 2 \, c^{\frac {1}{3 \, n}} x {\left (x^{n}\right )}^{\frac {1}{3 \, n}} \sin \left (3 \, a\right ) + 9 \, c^{\left (\frac {1}{n}\right )} x^{2} \sin \relax (a) + 18 \, c^{\frac {5}{3 \, n}} e^{\left (\frac {2 \, \log \left (x^{n}\right )}{3 \, n} + 2 \, \log \relax (x)\right )} \sin \relax (a)\right )} e^{\left (-\frac {\log \left (x^{n}\right )}{3 \, n} - 2 \, \log \relax (x)\right )}}{32 \, c^{\frac {4}{3 \, n}} x} \]
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 \[ \int \frac {{\sin \left (a+\frac {\ln \left (c\,x^n\right )\,\sqrt {-\frac {1}{n^2}}}{3}\right )}^3}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 90.21, size = 316, normalized size = 1.80 \[ - \frac {i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} \cos {\left (3 a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{8 x} - \frac {9 i n \sqrt {\frac {1}{n^{2}}} \cos {\left (a + \frac {i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )}}{3} + \frac {i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )}}{3} \right )}}{32 x} - \frac {i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \cos {\left (3 a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{8 x} - \frac {\log {\relax (x )} \sin {\left (3 a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{8 x} - \frac {27 \sin {\left (a + \frac {i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )}}{3} + \frac {i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )}}{3} \right )}}{32 x} + \frac {\sin {\left (3 a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{8 x} - \frac {\log {\relax (c )} \sin {\left (3 a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{8 n x} \]
Verification of antiderivative is not currently implemented for this CAS.
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