Optimal. Leaf size=179 \[ -\frac {3 e^{i a} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i b x^n\right )}{8 n}-\frac {e^{3 i a} 3^{-1/n} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-3 i b x^n\right )}{8 n}-\frac {3 e^{-i a} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i b x^n\right )}{8 n}-\frac {e^{-3 i a} 3^{-1/n} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},3 i b x^n\right )}{8 n} \]
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Rubi [A] time = 0.08, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3368, 3366, 2208} \[ -\frac {3 e^{i a} x \left (-i b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-i b x^n\right )}{8 n}-\frac {e^{3 i a} 3^{-1/n} x \left (-i b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-3 i b x^n\right )}{8 n}-\frac {3 e^{-i a} x \left (i b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},i b x^n\right )}{8 n}-\frac {e^{-3 i a} 3^{-1/n} x \left (i b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},3 i b x^n\right )}{8 n} \]
Antiderivative was successfully verified.
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Rule 2208
Rule 3366
Rule 3368
Rubi steps
\begin {align*} \int \cos ^3\left (a+b x^n\right ) \, dx &=\int \left (\frac {3}{4} \cos \left (a+b x^n\right )+\frac {1}{4} \cos \left (3 a+3 b x^n\right )\right ) \, dx\\ &=\frac {1}{4} \int \cos \left (3 a+3 b x^n\right ) \, dx+\frac {3}{4} \int \cos \left (a+b x^n\right ) \, dx\\ &=\frac {1}{8} \int e^{-3 i a-3 i b x^n} \, dx+\frac {1}{8} \int e^{3 i a+3 i b x^n} \, dx+\frac {3}{8} \int e^{-i a-i b x^n} \, dx+\frac {3}{8} \int e^{i a+i b x^n} \, dx\\ &=-\frac {3 e^{i a} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i b x^n\right )}{8 n}-\frac {3 e^{-i a} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i b x^n\right )}{8 n}-\frac {3^{-1/n} e^{3 i a} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-3 i b x^n\right )}{8 n}-\frac {3^{-1/n} e^{-3 i a} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},3 i b x^n\right )}{8 n}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 173, normalized size = 0.97 \[ -\frac {e^{-3 i a} 3^{-1/n} x \left (b^2 x^{2 n}\right )^{-1/n} \left (e^{2 i a} 3^{\frac {1}{n}+1} \left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},i b x^n\right )+e^{4 i a} 3^{\frac {1}{n}+1} \left (i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},-i b x^n\right )+e^{6 i a} \left (i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},-3 i b x^n\right )+\left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},3 i b x^n\right )\right )}{8 n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cos \left (b x^{n} + a\right )^{3}, x\right ) \]
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 \[ \int \cos \left (b x^{n} + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \cos ^{3}\left (a +b \,x^{n}\right )\, dx \]
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 \[ \int \cos \left (b x^{n} + a\right )^{3}\,{d x} \]
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 \[ \int {\cos \left (a+b\,x^n\right )}^3 \,d x \]
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 \[ \int \cos ^{3}{\left (a + b x^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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