Optimal. Leaf size=141 \[ -\frac {e^{2 i a} 2^{-\frac {m+2 n+1}{n}} x^{m+1} \left (-i b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-2 i b x^n\right )}{n}-\frac {e^{-2 i a} 2^{-\frac {m+2 n+1}{n}} x^{m+1} \left (i b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},2 i b x^n\right )}{n}+\frac {x^{m+1}}{2 (m+1)} \]
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Rubi [A] time = 0.16, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3426, 3424, 2218} \[ -\frac {e^{2 i a} 2^{-\frac {m+2 n+1}{n}} x^{m+1} \left (-i b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},-2 i b x^n\right )}{n}-\frac {e^{-2 i a} 2^{-\frac {m+2 n+1}{n}} x^{m+1} \left (i b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},2 i b x^n\right )}{n}+\frac {x^{m+1}}{2 (m+1)} \]
Antiderivative was successfully verified.
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Rule 2218
Rule 3424
Rule 3426
Rubi steps
\begin {align*} \int x^m \cos ^2\left (a+b x^n\right ) \, dx &=\int \left (\frac {x^m}{2}+\frac {1}{2} x^m \cos \left (2 a+2 b x^n\right )\right ) \, dx\\ &=\frac {x^{1+m}}{2 (1+m)}+\frac {1}{2} \int x^m \cos \left (2 a+2 b x^n\right ) \, dx\\ &=\frac {x^{1+m}}{2 (1+m)}+\frac {1}{4} \int e^{-2 i a-2 i b x^n} x^m \, dx+\frac {1}{4} \int e^{2 i a+2 i b x^n} x^m \, dx\\ &=\frac {x^{1+m}}{2 (1+m)}-\frac {2^{-\frac {1+m+2 n}{n}} e^{2 i a} x^{1+m} \left (-i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-2 i b x^n\right )}{n}-\frac {2^{-\frac {1+m+2 n}{n}} e^{-2 i a} x^{1+m} \left (i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},2 i b x^n\right )}{n}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 129, normalized size = 0.91 \[ -\frac {x^{m+1} \left (e^{2 i a} (m+1) 2^{-\frac {m+1}{n}} \left (-i b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-2 i b x^n\right )+e^{-2 i a} (m+1) 2^{-\frac {m+1}{n}} \left (i b x^n\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},2 i b x^n\right )-2 n\right )}{4 (m+1) n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \cos \left (b x^{n} + a\right )^{2}, x\right ) \]
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 \[ \int x^{m} \cos \left (b x^{n} + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.19, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\cos ^{2}\left (a +b \,x^{n}\right )\right )\, dx \]
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 \[ \frac {x x^{m} + {\left (m + 1\right )} \int x^{m} \cos \left (2 \, b x^{n} + 2 \, a\right )\,{d x}}{2 \, {\left (m + 1\right )}} \]
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 \[ \int x^m\,{\cos \left (a+b\,x^n\right )}^2 \,d x \]
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 \[ \int x^{m} \cos ^{2}{\left (a + b x^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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