Optimal. Leaf size=99 \[ -\frac {e^{2 i a} 2^{-m-4} x^m (-i b x)^{-m} \Gamma (m+2,-2 i b x)}{b^2}-\frac {e^{-2 i a} 2^{-m-4} x^m (i b x)^{-m} \Gamma (m+2,2 i b x)}{b^2}+\frac {x^{m+2}}{2 (m+2)} \]
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Rubi [A] time = 0.14, antiderivative size = 99, 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} \[ -\frac {e^{2 i a} 2^{-m-4} x^m (-i b x)^{-m} \text {Gamma}(m+2,-2 i b x)}{b^2}-\frac {e^{-2 i a} 2^{-m-4} x^m (i b x)^{-m} \text {Gamma}(m+2,2 i b x)}{b^2}+\frac {x^{m+2}}{2 (m+2)} \]
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^{2+m}}{2 (2+m)}-\frac {1}{2} \int x^{1+m} \cos (2 a+2 b x) \, dx\\ &=\frac {x^{2+m}}{2 (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^{2+m}}{2 (2+m)}-\frac {2^{-4-m} e^{2 i a} x^m (-i b x)^{-m} \Gamma (2+m,-2 i b x)}{b^2}-\frac {2^{-4-m} e^{-2 i a} x^m (i b x)^{-m} \Gamma (2+m,2 i b x)}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 116, normalized size = 1.17 \[ \frac {2^{-m-4} x^m \left (b^2 x^2\right )^{-m} \left (-\left ((m+2) (\cos (a)-i \sin (a))^2 (-i b x)^m \Gamma (m+2,2 i b x)\right )-(m+2) (\cos (a)+i \sin (a))^2 (i b x)^m \Gamma (m+2,-2 i b x)+2^{m+3} \left (b^2 x^2\right )^{m+1}\right )}{b^2 (m+2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 77, normalized size = 0.78 \[ \frac {4 \, b x x^{m + 1} + {\left (-i \, m - 2 i\right )} e^{\left (-{\left (m + 1\right )} \log \left (2 i \, b\right ) - 2 i \, a\right )} \Gamma \left (m + 2, 2 i \, b x\right ) + {\left (i \, m + 2 i\right )} e^{\left (-{\left (m + 1\right )} \log \left (-2 i \, b\right ) + 2 i \, a\right )} \Gamma \left (m + 2, -2 i \, b x\right )}{8 \, {\left (b m + 2 \, b\right )}} \]
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 \[ \int x^{m + 1} \sin \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.19, 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 \[ -\frac {{\left (m + 2\right )} \int x x^{m} \cos \left (2 \, b x + 2 \, a\right )\,{d x} - e^{\left (m \log \relax (x) + 2 \, \log \relax (x)\right )}}{2 \, {\left (m + 2\right )}} \]
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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