Optimal. Leaf size=83 \[ -\frac {e^{i a} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i b x^n\right )}{2 n}-\frac {e^{-i a} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i b x^n\right )}{2 n} \]
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Rubi [A] time = 0.02, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3366, 2208} \[ -\frac {e^{i a} x \left (-i b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-i b x^n\right )}{2 n}-\frac {e^{-i a} x \left (i b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},i b x^n\right )}{2 n} \]
Antiderivative was successfully verified.
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Rule 2208
Rule 3366
Rubi steps
\begin {align*} \int \cos \left (a+b x^n\right ) \, dx &=\frac {1}{2} \int e^{-i a-i b x^n} \, dx+\frac {1}{2} \int e^{i a+i b x^n} \, dx\\ &=-\frac {e^{i a} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i b x^n\right )}{2 n}-\frac {e^{-i a} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i b x^n\right )}{2 n}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 92, normalized size = 1.11 \[ -\frac {x \left (b^2 x^{2 n}\right )^{-1/n} \left ((\cos (a)-i \sin (a)) \left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},i b x^n\right )+(\cos (a)+i \sin (a)) \left (i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},-i b x^n\right )\right )}{2 n} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.14, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cos \left (b x^{n} + a\right ), x\right ) \]
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 \[ \int \cos \left (b x^{n} + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 75, normalized size = 0.90 \[ x \hypergeom \left (\left [\frac {1}{2 n}\right ], \left [\frac {1}{2}, 1+\frac {1}{2 n}\right ], -\frac {x^{2 n} b^{2}}{4}\right ) \cos \relax (a )-\frac {b \,x^{1+n} \hypergeom \left (\left [\frac {1}{2}+\frac {1}{2 n}\right ], \left [\frac {3}{2}, \frac {3}{2}+\frac {1}{2 n}\right ], -\frac {x^{2 n} b^{2}}{4}\right ) \sin \relax (a )}{1+n} \]
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 \[ \int \cos \left (b x^{n} + a\right )\,{d x} \]
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 \[ \int \cos \left (a+b\,x^n\right ) \,d x \]
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 \[ \int \cos {\left (a + b x^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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