Optimal. Leaf size=89 \[ -\frac {e^{2 a} 2^{-\frac {1}{n}-2} x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-2 b x^n\right )}{n}-\frac {e^{-2 a} 2^{-\frac {1}{n}-2} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},2 b x^n\right )}{n}+\frac {x}{2} \]
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Rubi [A] time = 0.07, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5309, 5307, 2208} \[ -\frac {e^{2 a} 2^{-\frac {1}{n}-2} x \left (-b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-2 b x^n\right )}{n}-\frac {e^{-2 a} 2^{-\frac {1}{n}-2} x \left (b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},2 b x^n\right )}{n}+\frac {x}{2} \]
Antiderivative was successfully verified.
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Rule 2208
Rule 5307
Rule 5309
Rubi steps
\begin {align*} \int \cosh ^2\left (a+b x^n\right ) \, dx &=\int \left (\frac {1}{2}+\frac {1}{2} \cosh \left (2 a+2 b x^n\right )\right ) \, dx\\ &=\frac {x}{2}+\frac {1}{2} \int \cosh \left (2 a+2 b x^n\right ) \, dx\\ &=\frac {x}{2}+\frac {1}{4} \int e^{-2 a-2 b x^n} \, dx+\frac {1}{4} \int e^{2 a+2 b x^n} \, dx\\ &=\frac {x}{2}-\frac {2^{-2-\frac {1}{n}} e^{2 a} x \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-2 b x^n\right )}{n}-\frac {2^{-2-\frac {1}{n}} e^{-2 a} x \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},2 b x^n\right )}{n}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 81, normalized size = 0.91 \[ -\frac {x \left (e^{2 a} 2^{-1/n} \left (-b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-2 b x^n\right )+e^{-2 a} 2^{-1/n} \left (b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},2 b x^n\right )-2 n\right )}{4 n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cosh \left (b x^{n} + a\right )^{2}, x\right ) \]
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 \[ \int \cosh \left (b x^{n} + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.26, size = 0, normalized size = 0.00 \[ \int \cosh ^{2}\left (a +b \,x^{n}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 68, normalized size = 0.76 \[ \frac {1}{2} \, x - \frac {x e^{\left (-2 \, a\right )} \Gamma \left (\frac {1}{n}, 2 \, b x^{n}\right )}{4 \, \left (2 \, b x^{n}\right )^{\left (\frac {1}{n}\right )} n} - \frac {x e^{\left (2 \, a\right )} \Gamma \left (\frac {1}{n}, -2 \, b x^{n}\right )}{4 \, \left (-2 \, b x^{n}\right )^{\left (\frac {1}{n}\right )} n} \]
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 \[ \int {\mathrm {cosh}\left (a+b\,x^n\right )}^2 \,d x \]
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 \[ \int \cosh ^{2}{\left (a + b x^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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