Optimal. Leaf size=97 \[ \frac {x^{2+m}}{2 (2+m)}+\frac {2^{-4-m} e^{2 i a} x^m (-i b x)^{-m} \text {Gamma}(2+m,-2 i b x)}{b^2}+\frac {2^{-4-m} e^{-2 i a} x^m (i b x)^{-m} \text {Gamma}(2+m,2 i b x)}{b^2} \]
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Rubi [A]
time = 0.09, antiderivative size = 97, 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 = {3393, 3388,
2212} \begin {gather*} \frac {e^{2 i a} 2^{-m-4} x^m (-i b x)^{-m} \text {Gamma}(m+2,-2 i b x)}{b^2}+\frac {e^{-2 i a} 2^{-m-4} x^m (i b x)^{-m} \text {Gamma}(m+2,2 i b x)}{b^2}+\frac {x^{m+2}}{2 (m+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3388
Rule 3393
Rubi steps
\begin {align*} \int x^{1+m} \cos ^2(a+b x) \, dx &=\int \left (\frac {x^{1+m}}{2}+\frac {1}{2} x^{1+m} \cos (2 a+2 b x)\right ) \, dx\\ &=\frac {x^{2+m}}{2 (2+m)}+\frac {1}{2} \int x^{1+m} \cos (2 a+2 b x) \, dx\\ &=\frac {x^{2+m}}{2 (2+m)}+\frac {1}{4} \int e^{-i (2 a+2 b x)} x^{1+m} \, dx+\frac {1}{4} \int e^{i (2 a+2 b x)} x^{1+m} \, dx\\ &=\frac {x^{2+m}}{2 (2+m)}+\frac {2^{-4-m} e^{2 i a} x^m (-i b x)^{-m} \Gamma (2+m,-2 i b x)}{b^2}+\frac {2^{-4-m} e^{-2 i a} x^m (i b x)^{-m} \Gamma (2+m,2 i b x)}{b^2}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 90, normalized size = 0.93 \begin {gather*} \frac {1}{16} x^m \left (\frac {8 x^2}{2+m}+\frac {2^{-m} e^{2 i a} (-i b x)^{-m} \text {Gamma}(2+m,-2 i b x)}{b^2}+\frac {2^{-m} e^{-2 i a} (i b x)^{-m} \text {Gamma}(2+m,2 i b x)}{b^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int x^{1+m} \left (\cos ^{2}\left (b x +a \right )\right )\, dx\]
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 = 0.11, size = 77, normalized size = 0.79 \begin {gather*} \frac {4 \, b x x^{m + 1} + {\left (i \, m + 2 i\right )} e^{\left (-{\left (m + 1\right )} \log \left (2 i \, b\right ) - 2 i \, a\right )} \Gamma \left (m + 2, 2 i \, b x\right ) + {\left (-i \, m - 2 i\right )} e^{\left (-{\left (m + 1\right )} \log \left (-2 i \, b\right ) + 2 i \, a\right )} \Gamma \left (m + 2, -2 i \, b x\right )}{8 \, {\left (b m + 2 \, 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 + 1} \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+1}\,{\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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