Optimal. Leaf size=103 \[ \frac {x^{1+m}}{2 (1+m)}-\frac {i 2^{-3-m} e^{2 i a} x^m (-i b x)^{-m} \text {Gamma}(1+m,-2 i b x)}{b}+\frac {i 2^{-3-m} e^{-2 i a} x^m (i b x)^{-m} \text {Gamma}(1+m,2 i b x)}{b} \]
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Rubi [A]
time = 0.09, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3393, 3388,
2212} \begin {gather*} -\frac {i e^{2 i a} 2^{-m-3} x^m (-i b x)^{-m} \text {Gamma}(m+1,-2 i b x)}{b}+\frac {i e^{-2 i a} 2^{-m-3} x^m (i b x)^{-m} \text {Gamma}(m+1,2 i b x)}{b}+\frac {x^{m+1}}{2 (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3388
Rule 3393
Rubi steps
\begin {align*} \int x^m \cos ^2(a+b x) \, dx &=\int \left (\frac {x^m}{2}+\frac {1}{2} x^m \cos (2 a+2 b x)\right ) \, dx\\ &=\frac {x^{1+m}}{2 (1+m)}+\frac {1}{2} \int x^m \cos (2 a+2 b x) \, dx\\ &=\frac {x^{1+m}}{2 (1+m)}+\frac {1}{4} \int e^{-i (2 a+2 b x)} x^m \, dx+\frac {1}{4} \int e^{i (2 a+2 b x)} x^m \, dx\\ &=\frac {x^{1+m}}{2 (1+m)}-\frac {i 2^{-3-m} e^{2 i a} x^m (-i b x)^{-m} \Gamma (1+m,-2 i b x)}{b}+\frac {i 2^{-3-m} e^{-2 i a} x^m (i b x)^{-m} \Gamma (1+m,2 i b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 90, normalized size = 0.87 \begin {gather*} \frac {1}{8} x^m \left (\frac {4 x}{1+m}-2^{-m} e^{2 i a} x (-i b x)^{-1-m} \text {Gamma}(1+m,-2 i b x)-2^{-m} e^{-2 i a} x (i b x)^{-1-m} \text {Gamma}(1+m,2 i b x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int x^{m} \left (\cos ^{2}\left (b x +a \right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.10, size = 69, normalized size = 0.67 \begin {gather*} \frac {4 \, b x x^{m} + {\left (i \, m + i\right )} e^{\left (-m \log \left (2 i \, b\right ) - 2 i \, a\right )} \Gamma \left (m + 1, 2 i \, b x\right ) + {\left (-i \, m - i\right )} e^{\left (-m \log \left (-2 i \, b\right ) + 2 i \, a\right )} \Gamma \left (m + 1, -2 i \, b x\right )}{8 \, {\left (b m + b\right )}} \end {gather*}
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 \begin {gather*} \int x^{m} \cos ^{2}{\left (a + b x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^m\,{\cos \left (a+b\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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