Optimal. Leaf size=229 \[ -\frac{3 e^{i a} x^{m+1} \left (-i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-i b x^n\right )}{8 n}-\frac{3 e^{-i a} x^{m+1} \left (i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},i b x^n\right )}{8 n}-\frac{e^{3 i a} 3^{-\frac{m+1}{n}} x^{m+1} \left (-i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-3 i b x^n\right )}{8 n}-\frac{e^{-3 i a} 3^{-\frac{m+1}{n}} x^{m+1} \left (i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},3 i b x^n\right )}{8 n} \]
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Rubi [A] time = 0.211757, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3426, 3424, 2218} \[ -\frac{3 e^{i a} x^{m+1} \left (-i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-i b x^n\right )}{8 n}-\frac{3 e^{-i a} x^{m+1} \left (i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},i b x^n\right )}{8 n}-\frac{e^{3 i a} 3^{-\frac{m+1}{n}} x^{m+1} \left (-i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-3 i b x^n\right )}{8 n}-\frac{e^{-3 i a} 3^{-\frac{m+1}{n}} x^{m+1} \left (i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},3 i b x^n\right )}{8 n} \]
Antiderivative was successfully verified.
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Rule 3426
Rule 3424
Rule 2218
Rubi steps
\begin{align*} \int x^m \cos ^3\left (a+b x^n\right ) \, dx &=\int \left (\frac{3}{4} x^m \cos \left (a+b x^n\right )+\frac{1}{4} x^m \cos \left (3 a+3 b x^n\right )\right ) \, dx\\ &=\frac{1}{4} \int x^m \cos \left (3 a+3 b x^n\right ) \, dx+\frac{3}{4} \int x^m \cos \left (a+b x^n\right ) \, dx\\ &=\frac{1}{8} \int e^{-3 i a-3 i b x^n} x^m \, dx+\frac{1}{8} \int e^{3 i a+3 i b x^n} x^m \, dx+\frac{3}{8} \int e^{-i a-i b x^n} x^m \, dx+\frac{3}{8} \int e^{i a+i b x^n} x^m \, dx\\ &=-\frac{3 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 )}{8 n}-\frac{3 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 )}{8 n}-\frac{3^{-\frac{1+m}{n}} e^{3 i a} x^{1+m} \left (-i b x^n\right )^{-\frac{1+m}{n}} \Gamma \left (\frac{1+m}{n},-3 i b x^n\right )}{8 n}-\frac{3^{-\frac{1+m}{n}} e^{-3 i a} x^{1+m} \left (i b x^n\right )^{-\frac{1+m}{n}} \Gamma \left (\frac{1+m}{n},3 i b x^n\right )}{8 n}\\ \end{align*}
Mathematica [A] time = 0.561795, size = 221, normalized size = 0.97 \[ -\frac{e^{-3 i a} 3^{-\frac{m+1}{n}} x^{m+1} \left (b^2 x^{2 n}\right )^{-\frac{m+1}{n}} \left (e^{2 i a} 3^{\frac{m+n+1}{n}} \left (-i b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},i b x^n\right )+e^{4 i a} 3^{\frac{m+n+1}{n}} \left (i b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-i b x^n\right )+e^{6 i a} \left (i b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-3 i b x^n\right )+\left (-i b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},3 i b x^n\right )\right )}{8 n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.178, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \cos \left ( a+b{x}^{n} \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \cos \left (b x^{n} + a\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \cos \left (b x^{n} + a\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \cos ^{3}{\left (a + b x^{n} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \cos \left (b x^{n} + a\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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