Optimal. Leaf size=25 \[ b \text{Unintegrable}\left (x^4 \text{sech}\left (c+d x^2\right ),x\right )+\frac{a x^5}{5} \]
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Rubi [A] time = 0.0176152, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x^4 \left (a+b \text{sech}\left (c+d x^2\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x^4 \left (a+b \text{sech}\left (c+d x^2\right )\right ) \, dx &=\int \left (a x^4+b x^4 \text{sech}\left (c+d x^2\right )\right ) \, dx\\ &=\frac{a x^5}{5}+b \int x^4 \text{sech}\left (c+d x^2\right ) \, dx\\ \end{align*}
Mathematica [A] time = 3.8074, size = 0, normalized size = 0. \[ \int x^4 \left (a+b \text{sech}\left (c+d x^2\right )\right ) \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{x}^{4} \left ( a+b{\rm sech} \left (d{x}^{2}+c\right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{5} \, a x^{5} + 2 \, b \int \frac{x^{4}}{e^{\left (d x^{2} + c\right )} + e^{\left (-d x^{2} - c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b x^{4} \operatorname{sech}\left (d x^{2} + c\right ) + a x^{4}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \left (a + b \operatorname{sech}{\left (c + d x^{2} \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{sech}\left (d x^{2} + c\right ) + a\right )} x^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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