Optimal. Leaf size=105 \[ -\frac {e^{i a} x^{m+1} \left (-i b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-i b x^n\right )}{2 n}-\frac {e^{-i a} x^{m+1} \left (i b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},i b x^n\right )}{2 n} \]
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Rubi [A] time = 0.07, antiderivative size = 105, 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 = {3424, 2218} \[ -\frac {e^{i a} x^{m+1} \left (-i b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},-i b x^n\right )}{2 n}-\frac {e^{-i a} x^{m+1} \left (i b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},i b x^n\right )}{2 n} \]
Antiderivative was successfully verified.
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Rule 2218
Rule 3424
Rubi steps
\begin {align*} \int x^m \cos \left (a+b x^n\right ) \, dx &=\frac {1}{2} \int e^{-i a-i b x^n} x^m \, dx+\frac {1}{2} \int e^{i a+i b x^n} x^m \, dx\\ &=-\frac {e^{i a} x^{1+m} \left (-i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-i b x^n\right )}{2 n}-\frac {e^{-i a} x^{1+m} \left (i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},i b x^n\right )}{2 n}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 115, normalized size = 1.10 \[ -\frac {x^{m+1} \left (b^2 x^{2 n}\right )^{-\frac {m+1}{n}} \left ((\cos (a)-i \sin (a)) \left (-i b x^n\right )^{\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},i b x^n\right )+(\cos (a)+i \sin (a)) \left (i b x^n\right )^{\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-i b x^n\right )\right )}{2 n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \cos \left (b x^{n} + a\right ), x\right ) \]
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 \[ \int x^{m} \cos \left (b x^{n} + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 111, normalized size = 1.06 \[ \frac {x^{1+m} \hypergeom \left (\left [\frac {m}{2 n}+\frac {1}{2 n}\right ], \left [\frac {1}{2}, 1+\frac {m}{2 n}+\frac {1}{2 n}\right ], -\frac {x^{2 n} b^{2}}{4}\right ) \cos \relax (a )}{1+m}-\frac {b \,x^{1+m +n} \hypergeom \left (\left [\frac {1}{2}+\frac {m}{2 n}+\frac {1}{2 n}\right ], \left [\frac {3}{2}, \frac {3}{2}+\frac {m}{2 n}+\frac {1}{2 n}\right ], -\frac {x^{2 n} b^{2}}{4}\right ) \sin \relax (a )}{1+m +n} \]
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 \[ \int x^{m} \cos \left (b x^{n} + a\right )\,{d x} \]
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 \[ \int x^m\,\cos \left (a+b\,x^n\right ) \,d x \]
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 \[ \int x^{m} \cos {\left (a + b x^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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