Integrand size = 12, antiderivative size = 85 \[ \int x^m \cosh ^2(a+b x) \, dx=\frac {x^{1+m}}{2 (1+m)}+\frac {2^{-3-m} e^{2 a} x^m (-b x)^{-m} \Gamma (1+m,-2 b x)}{b}-\frac {2^{-3-m} e^{-2 a} x^m (b x)^{-m} \Gamma (1+m,2 b x)}{b} \]
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Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3393, 3388, 2212} \[ \int x^m \cosh ^2(a+b x) \, dx=\frac {e^{2 a} 2^{-m-3} x^m (-b x)^{-m} \Gamma (m+1,-2 b x)}{b}-\frac {e^{-2 a} 2^{-m-3} x^m (b x)^{-m} \Gamma (m+1,2 b x)}{b}+\frac {x^{m+1}}{2 (m+1)} \]
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Rule 2212
Rule 3388
Rule 3393
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x^m}{2}+\frac {1}{2} x^m \cosh (2 a+2 b x)\right ) \, dx \\ & = \frac {x^{1+m}}{2 (1+m)}+\frac {1}{2} \int x^m \cosh (2 a+2 b x) \, dx \\ & = \frac {x^{1+m}}{2 (1+m)}+\frac {1}{4} \int e^{-i (2 i a+2 i b x)} x^m \, dx+\frac {1}{4} \int e^{i (2 i a+2 i b x)} x^m \, dx \\ & = \frac {x^{1+m}}{2 (1+m)}+\frac {2^{-3-m} e^{2 a} x^m (-b x)^{-m} \Gamma (1+m,-2 b x)}{b}-\frac {2^{-3-m} e^{-2 a} x^m (b x)^{-m} \Gamma (1+m,2 b x)}{b} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.89 \[ \int x^m \cosh ^2(a+b x) \, dx=\frac {1}{8} x^m \left (\frac {4 x}{1+m}+\frac {2^{-m} e^{2 a} (-b x)^{-m} \Gamma (1+m,-2 b x)}{b}-\frac {2^{-m} e^{-2 a} (b x)^{-m} \Gamma (1+m,2 b x)}{b}\right ) \]
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\[\int x^{m} \cosh \left (b x +a \right )^{2}d x\]
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none
Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.44 \[ \int x^m \cosh ^2(a+b x) \, dx=\frac {4 \, b x \cosh \left (m \log \left (x\right )\right ) - {\left (m + 1\right )} \cosh \left (m \log \left (2 \, b\right ) + 2 \, a\right ) \Gamma \left (m + 1, 2 \, b x\right ) + {\left (m + 1\right )} \cosh \left (m \log \left (-2 \, b\right ) - 2 \, a\right ) \Gamma \left (m + 1, -2 \, b x\right ) + {\left (m + 1\right )} \Gamma \left (m + 1, 2 \, b x\right ) \sinh \left (m \log \left (2 \, b\right ) + 2 \, a\right ) - {\left (m + 1\right )} \Gamma \left (m + 1, -2 \, b x\right ) \sinh \left (m \log \left (-2 \, b\right ) - 2 \, a\right ) + 4 \, b x \sinh \left (m \log \left (x\right )\right )}{8 \, {\left (b m + b\right )}} \]
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\[ \int x^m \cosh ^2(a+b x) \, dx=\int x^{m} \cosh ^{2}{\left (a + b x \right )}\, dx \]
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none
Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.84 \[ \int x^m \cosh ^2(a+b x) \, dx=-\frac {1}{4} \, \left (2 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (-2 \, a\right )} \Gamma \left (m + 1, 2 \, b x\right ) - \frac {1}{4} \, \left (-2 \, b x\right )^{-m - 1} x^{m + 1} e^{\left (2 \, a\right )} \Gamma \left (m + 1, -2 \, b x\right ) + \frac {x^{m + 1}}{2 \, {\left (m + 1\right )}} \]
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\[ \int x^m \cosh ^2(a+b x) \, dx=\int { x^{m} \cosh \left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int x^m \cosh ^2(a+b x) \, dx=\int x^m\,{\mathrm {cosh}\left (a+b\,x\right )}^2 \,d x \]
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