Optimal. Leaf size=65 \[ -\frac {1}{2} e^{i a} x^m (-i b x)^{-m} \Gamma (m,-i b x)-\frac {1}{2} e^{-i a} x^m (i b x)^{-m} \Gamma (m,i b x) \]
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Rubi [A] time = 0.07, antiderivative size = 65, 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 = {3307, 2181} \[ -\frac {1}{2} e^{i a} x^m (-i b x)^{-m} \text {Gamma}(m,-i b x)-\frac {1}{2} e^{-i a} x^m (i b x)^{-m} \text {Gamma}(m,i b x) \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3307
Rubi steps
\begin {align*} \int x^{-1+m} \cos (a+b x) \, dx &=\frac {1}{2} \int e^{-i (a+b x)} x^{-1+m} \, dx+\frac {1}{2} \int e^{i (a+b x)} x^{-1+m} \, dx\\ &=-\frac {1}{2} e^{i a} x^m (-i b x)^{-m} \Gamma (m,-i b x)-\frac {1}{2} e^{-i a} x^m (i b x)^{-m} \Gamma (m,i b x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 62, normalized size = 0.95 \[ \frac {1}{2} e^{-i a} x^m \left (-e^{2 i a} (-i b x)^{-m} \Gamma (m,-i b x)-(i b x)^{-m} \Gamma (m,i b x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 50, normalized size = 0.77 \[ \frac {i \, e^{\left (-{\left (m - 1\right )} \log \left (i \, b\right ) - i \, a\right )} \Gamma \left (m, i \, b x\right ) - i \, e^{\left (-{\left (m - 1\right )} \log \left (-i \, b\right ) + i \, a\right )} \Gamma \left (m, -i \, b x\right )}{2 \, b} \]
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 \[ \int x^{m - 1} \cos \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.11, size = 427, normalized size = 6.57 \[ 2^{-1+m} \left (b^{2}\right )^{-\frac {m}{2}} \sqrt {\pi }\, \left (\frac {3 x^{-1+m} 2^{-m} \left (b^{2}\right )^{\frac {m}{2}} \left (2 x^{2} b^{2}+2 m +4\right ) \sin \left (b x \right )}{\sqrt {\pi }\, m \left (6+3 m \right ) b}+\frac {2^{1-m} x^{-1+m} \left (b^{2}\right )^{\frac {m}{2}} \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right )}{\sqrt {\pi }\, m b}-\frac {3 x^{2+m} 2^{1-m} \left (b^{2}\right )^{\frac {m}{2}} b^{2} \left (b x \right )^{-\frac {3}{2}-m} \LommelS 1 \left (m +\frac {3}{2}, \frac {3}{2}, b x \right ) \sin \left (b x \right )}{\sqrt {\pi }\, m \left (6+3 m \right )}-\frac {x^{2+m} 2^{1-m} \left (b^{2}\right )^{\frac {m}{2}} b^{2} \left (b x \right )^{-\frac {5}{2}-m} \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right ) \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, b x \right )}{\sqrt {\pi }\, m}\right ) \cos \relax (a )-2^{-1+m} b^{-m} \sqrt {\pi }\, \left (\frac {2^{1-m} x^{m} b^{m} \sin \left (b x \right )}{\sqrt {\pi }\, \left (1+m \right )}-\frac {2^{1-m} x^{m} b^{m} \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right )}{\sqrt {\pi }\, \left (1+m \right ) m}-\frac {x^{2+m} b^{2+m} 2^{1-m} \left (b x \right )^{-\frac {3}{2}-m} \LommelS 1 \left (m +\frac {1}{2}, \frac {3}{2}, b x \right ) \sin \left (b x \right )}{\sqrt {\pi }\, \left (1+m \right )}+\frac {x^{2+m} b^{2+m} 2^{1-m} \left (b x \right )^{-\frac {5}{2}-m} \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right ) \LommelS 1 \left (m +\frac {3}{2}, \frac {1}{2}, b x \right )}{\sqrt {\pi }\, \left (1+m \right ) m}\right ) \sin \relax (a ) \]
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 \[ \int x^{m - 1} \cos \left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^{m-1}\,\cos \left (a+b\,x\right ) \,d x \]
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 \[ \int x^{m - 1} \cos {\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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