Optimal. Leaf size=79 \[ \frac {i e^{-i a} x^m (i b x)^{-m} \Gamma (m+2,i b x)}{2 b^2}-\frac {i e^{i a} x^m (-i b x)^{-m} \Gamma (m+2,-i b x)}{2 b^2} \]
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Rubi [A] time = 0.07, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3308, 2181} \[ \frac {i e^{-i a} x^m (i b x)^{-m} \text {Gamma}(m+2,i b x)}{2 b^2}-\frac {i e^{i a} x^m (-i b x)^{-m} \text {Gamma}(m+2,-i b x)}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3308
Rubi steps
\begin {align*} \int x^{1+m} \sin (a+b x) \, dx &=\frac {1}{2} i \int e^{-i (a+b x)} x^{1+m} \, dx-\frac {1}{2} i \int e^{i (a+b x)} x^{1+m} \, dx\\ &=-\frac {i e^{i a} x^m (-i b x)^{-m} \Gamma (2+m,-i b x)}{2 b^2}+\frac {i e^{-i a} x^m (i b x)^{-m} \Gamma (2+m,i b x)}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 79, normalized size = 1.00 \[ \frac {i e^{-i a} x^m (i b x)^{-m} \Gamma (m+2,i b x)}{2 b^2}-\frac {i e^{i a} x^m (-i b x)^{-m} \Gamma (m+2,-i b x)}{2 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 52, normalized size = 0.66 \[ -\frac {e^{\left (-{\left (m + 1\right )} \log \left (i \, b\right ) - i \, a\right )} \Gamma \left (m + 2, i \, b x\right ) + e^{\left (-{\left (m + 1\right )} \log \left (-i \, b\right ) + i \, a\right )} \Gamma \left (m + 2, -i \, b x\right )}{2 \, b} \]
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 \[ \int x^{m + 1} \sin \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.09, size = 290, normalized size = 3.67 \[ \frac {2^{1+m} \left (b^{2}\right )^{-\frac {m}{2}} \sqrt {\pi }\, \left (\frac {2^{-1-m} x^{1+m} b \left (b^{2}\right )^{\frac {m}{2}} \sin \left (b x \right )}{\sqrt {\pi }\, \left (2+m \right )}+\frac {3 \,2^{-2-m} x^{2+m} b^{2} \left (b^{2}\right )^{\frac {m}{2}} \left (\frac {2}{3}+\frac {2 m}{3}\right ) \left (b x \right )^{-\frac {3}{2}-m} \LommelS 1 \left (m +\frac {3}{2}, \frac {3}{2}, b x \right ) \sin \left (b x \right )}{\sqrt {\pi }\, \left (2+m \right )}+\frac {2^{-1-m} x^{2+m} b^{2} \left (b^{2}\right )^{\frac {m}{2}} \left (1+m \right ) \left (b x \right )^{-\frac {5}{2}-m} \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right ) \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, b x \right )}{\sqrt {\pi }}\right ) \sin \relax (a )}{b^{2}}+2^{1+m} b^{-2-m} \sqrt {\pi }\, \left (\frac {2^{-1-m} x^{2+m} b^{2+m} m \left (b x \right )^{-\frac {3}{2}-m} \LommelS 1 \left (m +\frac {1}{2}, \frac {3}{2}, b x \right ) \sin \left (b x \right )}{\sqrt {\pi }}-\frac {2^{-1-m} x^{2+m} b^{2+m} \left (b x \right )^{-\frac {5}{2}-m} \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right ) \LommelS 1 \left (m +\frac {3}{2}, \frac {1}{2}, b x \right )}{\sqrt {\pi }}\right ) \cos \relax (a ) \]
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 \[ \int x^{m + 1} \sin \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^{m+1}\,\sin \left (a+b\,x\right ) \,d x \]
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 \[ \int x^{m + 1} \sin {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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