Optimal. Leaf size=101 \[ -i e^{2 i a} b 2^{-m-1} x^m (-i b x)^{-m} \text{Gamma}(m-1,-2 i b x)+i e^{-2 i a} b 2^{-m-1} x^m (i b x)^{-m} \text{Gamma}(m-1,2 i b x)-\frac{x^{m-1}}{2 (1-m)} \]
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Rubi [A] time = 0.137095, antiderivative size = 101, normalized size of antiderivative = 1., 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} \[ -i e^{2 i a} b 2^{-m-1} x^m (-i b x)^{-m} \text{Gamma}(m-1,-2 i b x)+i e^{-2 i a} b 2^{-m-1} x^m (i b x)^{-m} \text{Gamma}(m-1,2 i b x)-\frac{x^{m-1}}{2 (1-m)} \]
Antiderivative was successfully verified.
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Rule 3312
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int x^{-2+m} \sin ^2(a+b x) \, dx &=\int \left (\frac{x^{-2+m}}{2}-\frac{1}{2} x^{-2+m} \cos (2 a+2 b x)\right ) \, dx\\ &=-\frac{x^{-1+m}}{2 (1-m)}-\frac{1}{2} \int x^{-2+m} \cos (2 a+2 b x) \, dx\\ &=-\frac{x^{-1+m}}{2 (1-m)}-\frac{1}{4} \int e^{-i (2 a+2 b x)} x^{-2+m} \, dx-\frac{1}{4} \int e^{i (2 a+2 b x)} x^{-2+m} \, dx\\ &=-\frac{x^{-1+m}}{2 (1-m)}-i 2^{-1-m} b e^{2 i a} x^m (-i b x)^{-m} \Gamma (-1+m,-2 i b x)+i 2^{-1-m} b e^{-2 i a} x^m (i b x)^{-m} \Gamma (-1+m,2 i b x)\\ \end{align*}
Mathematica [A] time = 0.31142, size = 117, normalized size = 1.16 \[ \frac{2^{-m-1} x^{m-1} \left (b^2 x^2\right )^{-m} \left (b (m-1) x (\sin (2 a)+i \cos (2 a)) (-i b x)^m \text{Gamma}(m-1,2 i b x)+b (m-1) x (\sin (2 a)-i \cos (2 a)) (i b x)^m \text{Gamma}(m-1,-2 i b x)+2^m \left (b^2 x^2\right )^m\right )}{m-1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{x}^{m-2} \left ( \sin \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (m - 1\right )} x \int \frac{x^{m} \cos \left (2 \, b x + 2 \, a\right )}{x^{2}}\,{d x} - x^{m}}{2 \,{\left (m - 1\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73722, size = 227, normalized size = 2.25 \begin{align*} \frac{4 \, b x x^{m - 2} +{\left (-i \, m + i\right )} e^{\left (-{\left (m - 2\right )} \log \left (2 i \, b\right ) - 2 i \, a\right )} \Gamma \left (m - 1, 2 i \, b x\right ) +{\left (i \, m - i\right )} e^{\left (-{\left (m - 2\right )} \log \left (-2 i \, b\right ) + 2 i \, a\right )} \Gamma \left (m - 1, -2 i \, b x\right )}{8 \,{\left (b m - b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m - 2} \sin \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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