Optimal. Leaf size=97 \[ \frac{e^{2 i a} 2^{-m-6} x^m (-i b x)^{-m} \text{Gamma}(m+4,-2 i b x)}{b^4}+\frac{e^{-2 i a} 2^{-m-6} x^m (i b x)^{-m} \text{Gamma}(m+4,2 i b x)}{b^4}+\frac{x^{m+4}}{2 (m+4)} \]
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Rubi [A] time = 0.161679, antiderivative size = 97, 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} \[ \frac{e^{2 i a} 2^{-m-6} x^m (-i b x)^{-m} \text{Gamma}(m+4,-2 i b x)}{b^4}+\frac{e^{-2 i a} 2^{-m-6} x^m (i b x)^{-m} \text{Gamma}(m+4,2 i b x)}{b^4}+\frac{x^{m+4}}{2 (m+4)} \]
Antiderivative was successfully verified.
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Rule 3312
Rule 3307
Rule 2181
Rubi steps
\begin{align*} \int x^{3+m} \sin ^2(a+b x) \, dx &=\int \left (\frac{x^{3+m}}{2}-\frac{1}{2} x^{3+m} \cos (2 a+2 b x)\right ) \, dx\\ &=\frac{x^{4+m}}{2 (4+m)}-\frac{1}{2} \int x^{3+m} \cos (2 a+2 b x) \, dx\\ &=\frac{x^{4+m}}{2 (4+m)}-\frac{1}{4} \int e^{-i (2 a+2 b x)} x^{3+m} \, dx-\frac{1}{4} \int e^{i (2 a+2 b x)} x^{3+m} \, dx\\ &=\frac{x^{4+m}}{2 (4+m)}+\frac{2^{-6-m} e^{2 i a} x^m (-i b x)^{-m} \Gamma (4+m,-2 i b x)}{b^4}+\frac{2^{-6-m} e^{-2 i a} x^m (i b x)^{-m} \Gamma (4+m,2 i b x)}{b^4}\\ \end{align*}
Mathematica [A] time = 0.328603, size = 118, normalized size = 1.22 \[ \frac{2^{-m-6} x^m \left (b^2 x^2\right )^{-m} \left ((m+4) (\cos (a)-i \sin (a))^2 (-i b x)^m \text{Gamma}(m+4,2 i b x)+(m+4) (\cos (a)+i \sin (a))^2 (i b x)^m \text{Gamma}(m+4,-2 i b x)+b^4 2^{m+5} x^4 \left (b^2 x^2\right )^m\right )}{b^4 (m+4)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{x}^{3+m} \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(-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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Fricas [A] time = 1.83243, size = 235, normalized size = 2.42 \begin{align*} \frac{4 \, b x x^{m + 3} +{\left (-i \, m - 4 i\right )} e^{\left (-{\left (m + 3\right )} \log \left (2 i \, b\right ) - 2 i \, a\right )} \Gamma \left (m + 4, 2 i \, b x\right ) +{\left (i \, m + 4 i\right )} e^{\left (-{\left (m + 3\right )} \log \left (-2 i \, b\right ) + 2 i \, a\right )} \Gamma \left (m + 4, -2 i \, b x\right )}{8 \,{\left (b m + 4 \, 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 + 3} \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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