Optimal. Leaf size=237 \[ \frac {3 i e^{i a} x^{1+m} \left (-i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-i b x^n\right )}{8 n}-\frac {3 i e^{-i a} x^{1+m} \left (i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},i b x^n\right )}{8 n}-\frac {i 3^{-\frac {1+m}{n}} e^{3 i a} x^{1+m} \left (-i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-3 i b x^n\right )}{8 n}+\frac {i 3^{-\frac {1+m}{n}} e^{-3 i a} x^{1+m} \left (i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},3 i b x^n\right )}{8 n} \]
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Rubi [A]
time = 0.14, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3506, 3504,
2250} \begin {gather*} \frac {3 i e^{i a} x^{m+1} \left (-i b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},-i b x^n\right )}{8 n}-\frac {3 i e^{-i a} x^{m+1} \left (i b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},i b x^n\right )}{8 n}-\frac {i e^{3 i a} 3^{-\frac {m+1}{n}} x^{m+1} \left (-i b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},-3 i b x^n\right )}{8 n}+\frac {i e^{-3 i a} 3^{-\frac {m+1}{n}} x^{m+1} \left (i b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},3 i b x^n\right )}{8 n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2250
Rule 3504
Rule 3506
Rubi steps
\begin {align*} \int x^m \sin ^3\left (a+b x^n\right ) \, dx &=\int \left (\frac {3}{4} x^m \sin \left (a+b x^n\right )-\frac {1}{4} x^m \sin \left (3 a+3 b x^n\right )\right ) \, dx\\ &=-\left (\frac {1}{4} \int x^m \sin \left (3 a+3 b x^n\right ) \, dx\right )+\frac {3}{4} \int x^m \sin \left (a+b x^n\right ) \, dx\\ &=-\left (\frac {1}{8} i \int e^{-3 i a-3 i b x^n} x^m \, dx\right )+\frac {1}{8} i \int e^{3 i a+3 i b x^n} x^m \, dx+\frac {3}{8} i \int e^{-i a-i b x^n} x^m \, dx-\frac {3}{8} i \int e^{i a+i b x^n} x^m \, dx\\ &=\frac {3 i e^{i a} x^{1+m} \left (-i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-i b x^n\right )}{8 n}-\frac {3 i e^{-i a} x^{1+m} \left (i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},i b x^n\right )}{8 n}-\frac {i 3^{-\frac {1+m}{n}} e^{3 i a} x^{1+m} \left (-i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-3 i b x^n\right )}{8 n}+\frac {i 3^{-\frac {1+m}{n}} e^{-3 i a} x^{1+m} \left (i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},3 i b x^n\right )}{8 n}\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 225, normalized size = 0.95 \begin {gather*} \frac {i 3^{-\frac {1+m}{n}} e^{-3 i a} x^{1+m} \left (b^2 x^{2 n}\right )^{-\frac {1+m}{n}} \left (3^{\frac {1+m+n}{n}} e^{4 i a} \left (i b x^n\right )^{\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-i b x^n\right )-3^{\frac {1+m+n}{n}} e^{2 i a} \left (-i b x^n\right )^{\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},i b x^n\right )-e^{6 i a} \left (i b x^n\right )^{\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-3 i b x^n\right )+\left (-i b x^n\right )^{\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},3 i b x^n\right )\right )}{8 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int x^{m} \left (\sin ^{3}\left (a +b \,x^{n}\right )\right )\, dx\]
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{m} \sin ^{3}{\left (a + b x^{n} \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 = -0.00 \begin {gather*} \int x^m\,{\sin \left (a+b\,x^n\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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