Optimal. Leaf size=85 \[ \frac {e^{2 a} 2^{-m-5} x^m (-b x)^{-m} \Gamma (m+3,-2 b x)}{b^3}-\frac {e^{-2 a} 2^{-m-5} x^m (b x)^{-m} \Gamma (m+3,2 b x)}{b^3}+\frac {x^{m+3}}{2 (m+3)} \]
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Rubi [A] time = 0.13, antiderivative size = 85, 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 = {3312, 3307, 2181} \[ \frac {e^{2 a} 2^{-m-5} x^m (-b x)^{-m} \text {Gamma}(m+3,-2 b x)}{b^3}-\frac {e^{-2 a} 2^{-m-5} x^m (b x)^{-m} \text {Gamma}(m+3,2 b x)}{b^3}+\frac {x^{m+3}}{2 (m+3)} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3307
Rule 3312
Rubi steps
\begin {align*} \int x^{2+m} \cosh ^2(a+b x) \, dx &=\int \left (\frac {x^{2+m}}{2}+\frac {1}{2} x^{2+m} \cosh (2 a+2 b x)\right ) \, dx\\ &=\frac {x^{3+m}}{2 (3+m)}+\frac {1}{2} \int x^{2+m} \cosh (2 a+2 b x) \, dx\\ &=\frac {x^{3+m}}{2 (3+m)}+\frac {1}{4} \int e^{-i (2 i a+2 i b x)} x^{2+m} \, dx+\frac {1}{4} \int e^{i (2 i a+2 i b x)} x^{2+m} \, dx\\ &=\frac {x^{3+m}}{2 (3+m)}+\frac {2^{-5-m} e^{2 a} x^m (-b x)^{-m} \Gamma (3+m,-2 b x)}{b^3}-\frac {2^{-5-m} e^{-2 a} x^m (b x)^{-m} \Gamma (3+m,2 b x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 78, normalized size = 0.92 \[ \frac {1}{32} x^m \left (\frac {e^{2 a} 2^{-m} (-b x)^{-m} \Gamma (m+3,-2 b x)}{b^3}-\frac {e^{-2 a} 2^{-m} (b x)^{-m} \Gamma (m+3,2 b x)}{b^3}+\frac {16 x^3}{m+3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 136, normalized size = 1.60 \[ \frac {4 \, b x \cosh \left ({\left (m + 2\right )} \log \relax (x)\right ) - {\left (m + 3\right )} \cosh \left ({\left (m + 2\right )} \log \left (2 \, b\right ) + 2 \, a\right ) \Gamma \left (m + 3, 2 \, b x\right ) + {\left (m + 3\right )} \cosh \left ({\left (m + 2\right )} \log \left (-2 \, b\right ) - 2 \, a\right ) \Gamma \left (m + 3, -2 \, b x\right ) + {\left (m + 3\right )} \Gamma \left (m + 3, 2 \, b x\right ) \sinh \left ({\left (m + 2\right )} \log \left (2 \, b\right ) + 2 \, a\right ) - {\left (m + 3\right )} \Gamma \left (m + 3, -2 \, b x\right ) \sinh \left ({\left (m + 2\right )} \log \left (-2 \, b\right ) - 2 \, a\right ) + 4 \, b x \sinh \left ({\left (m + 2\right )} \log \relax (x)\right )}{8 \, {\left (b m + 3 \, b\right )}} \]
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 \[ \int x^{m + 2} \cosh \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int x^{2+m} \left (\cosh ^{2}\left (b x +a \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 71, normalized size = 0.84 \[ -\frac {1}{4} \, \left (2 \, b x\right )^{-m - 3} x^{m + 3} e^{\left (-2 \, a\right )} \Gamma \left (m + 3, 2 \, b x\right ) - \frac {1}{4} \, \left (-2 \, b x\right )^{-m - 3} x^{m + 3} e^{\left (2 \, a\right )} \Gamma \left (m + 3, -2 \, b x\right ) + \frac {x^{m + 3}}{2 \, {\left (m + 3\right )}} \]
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 \[ \int x^{m+2}\,{\mathrm {cosh}\left (a+b\,x\right )}^2 \,d x \]
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 \[ \int x^{m + 2} \cosh ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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