Optimal. Leaf size=237 \[ \frac {3 i e^{i a} x^{m+1} \left (-i b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-i b x^n\right )}{8 n}-\frac {3 i e^{-i a} x^{m+1} \left (i b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},i b x^n\right )}{8 n}-\frac {i e^{3 i a} 3^{-\frac {m+1}{n}} x^{m+1} \left (-i b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-3 i b x^n\right )}{8 n}+\frac {i e^{-3 i a} 3^{-\frac {m+1}{n}} x^{m+1} \left (i b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},3 i b x^n\right )}{8 n} \]
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Rubi [A] time = 0.24, 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 = {3425, 3423, 2218} \[ \frac {3 i e^{i a} x^{m+1} \left (-i b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},-i b x^n\right )}{8 n}-\frac {3 i e^{-i a} x^{m+1} \left (i b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},i b x^n\right )}{8 n}-\frac {i e^{3 i a} 3^{-\frac {m+1}{n}} x^{m+1} \left (-i b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},-3 i b x^n\right )}{8 n}+\frac {i e^{-3 i a} 3^{-\frac {m+1}{n}} x^{m+1} \left (i b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},3 i b x^n\right )}{8 n} \]
Antiderivative was successfully verified.
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Rule 2218
Rule 3423
Rule 3425
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 = 0.60, size = 225, normalized size = 0.95 \[ \frac {i e^{-3 i a} 3^{-\frac {m+1}{n}} x^{m+1} \left (b^2 x^{2 n}\right )^{-\frac {m+1}{n}} \left (e^{2 i a} \left (-3^{\frac {m+n+1}{n}}\right ) \left (-i b x^n\right )^{\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},i b x^n\right )+e^{4 i a} 3^{\frac {m+n+1}{n}} \left (i b x^n\right )^{\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-i b x^n\right )-e^{6 i a} \left (i b x^n\right )^{\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-3 i b x^n\right )+\left (-i b x^n\right )^{\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},3 i b x^n\right )\right )}{8 n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (x^{m} \cos \left (b x^{n} + a\right )^{2} - x^{m}\right )} \sin \left (b x^{n} + a\right ), x\right ) \]
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 \[ \int x^{m} \sin \left (b x^{n} + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.48, 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 \[ \int x^{m} \sin \left (b x^{n} + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^m\,{\sin \left (a+b\,x^n\right )}^3 \,d x \]
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 \[ \int x^{m} \sin ^{3}{\left (a + b x^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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