Optimal. Leaf size=83 \[ e^{2 i a} 2^{-m-2} x^m (-i b x)^{-m} \Gamma (m,-2 i b x)+e^{-2 i a} 2^{-m-2} x^m (i b x)^{-m} \Gamma (m,2 i b x)+\frac {x^m}{2 m} \]
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Rubi [A] time = 0.13, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3312, 3307, 2181} \[ e^{2 i a} 2^{-m-2} x^m (-i b x)^{-m} \text {Gamma}(m,-2 i b x)+e^{-2 i a} 2^{-m-2} x^m (i b x)^{-m} \text {Gamma}(m,2 i b x)+\frac {x^m}{2 m} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3307
Rule 3312
Rubi steps
\begin {align*} \int x^{-1+m} \sin ^2(a+b x) \, dx &=\int \left (\frac {x^{-1+m}}{2}-\frac {1}{2} x^{-1+m} \cos (2 a+2 b x)\right ) \, dx\\ &=\frac {x^m}{2 m}-\frac {1}{2} \int x^{-1+m} \cos (2 a+2 b x) \, dx\\ &=\frac {x^m}{2 m}-\frac {1}{4} \int e^{-i (2 a+2 b x)} x^{-1+m} \, dx-\frac {1}{4} \int e^{i (2 a+2 b x)} x^{-1+m} \, dx\\ &=\frac {x^m}{2 m}+2^{-2-m} e^{2 i a} x^m (-i b x)^{-m} \Gamma (m,-2 i b x)+2^{-2-m} e^{-2 i a} x^m (i b x)^{-m} \Gamma (m,2 i b x)\\ \end {align*}
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Mathematica [A] time = 0.25, size = 99, normalized size = 1.19 \[ \frac {2^{-m-2} x^m \left (b^2 x^2\right )^{-m} \left (m (\cos (a)-i \sin (a))^2 (-i b x)^m \Gamma (m,2 i b x)+m (\cos (a)+i \sin (a))^2 (i b x)^m \Gamma (m,-2 i b x)+2^{m+1} \left (b^2 x^2\right )^m\right )}{m} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 64, normalized size = 0.77 \[ \frac {4 \, b x x^{m - 1} - i \, m e^{\left (-{\left (m - 1\right )} \log \left (2 i \, b\right ) - 2 i \, a\right )} \Gamma \left (m, 2 i \, b x\right ) + i \, m e^{\left (-{\left (m - 1\right )} \log \left (-2 i \, b\right ) + 2 i \, a\right )} \Gamma \left (m, -2 i \, b x\right )}{8 \, b m} \]
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 \[ \int x^{m - 1} \sin \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int x^{-1+m} \left (\sin ^{2}\left (b x +a \right )\right )\, dx \]
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 \[ -\frac {m \int \frac {x^{m} \cos \left (2 \, b x + 2 \, a\right )}{x}\,{d x} - x^{m}}{2 \, m} \]
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 \[ \int x^{m-1}\,{\sin \left (a+b\,x\right )}^2 \,d x \]
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 \[ \int x^{m - 1} \sin ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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