Optimal. Leaf size=200 \[ -\frac{e^{3 a} 3^{-\frac{m+1}{n}} x^{m+1} \left (-b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-3 b x^n\right )}{8 n}-\frac{3 e^a x^{m+1} \left (-b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-b x^n\right )}{8 n}-\frac{3 e^{-a} x^{m+1} \left (b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},b x^n\right )}{8 n}-\frac{e^{-3 a} 3^{-\frac{m+1}{n}} x^{m+1} \left (b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},3 b x^n\right )}{8 n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.205599, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5363, 5361, 2218} \[ -\frac{e^{3 a} 3^{-\frac{m+1}{n}} x^{m+1} \left (-b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-3 b x^n\right )}{8 n}-\frac{3 e^a x^{m+1} \left (-b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-b x^n\right )}{8 n}-\frac{3 e^{-a} x^{m+1} \left (b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},b x^n\right )}{8 n}-\frac{e^{-3 a} 3^{-\frac{m+1}{n}} x^{m+1} \left (b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},3 b x^n\right )}{8 n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5363
Rule 5361
Rule 2218
Rubi steps
\begin{align*} \int x^m \cosh ^3\left (a+b x^n\right ) \, dx &=\int \left (\frac{3}{4} x^m \cosh \left (a+b x^n\right )+\frac{1}{4} x^m \cosh \left (3 a+3 b x^n\right )\right ) \, dx\\ &=\frac{1}{4} \int x^m \cosh \left (3 a+3 b x^n\right ) \, dx+\frac{3}{4} \int x^m \cosh \left (a+b x^n\right ) \, dx\\ &=\frac{1}{8} \int e^{-3 a-3 b x^n} x^m \, dx+\frac{1}{8} \int e^{3 a+3 b x^n} x^m \, dx+\frac{3}{8} \int e^{-a-b x^n} x^m \, dx+\frac{3}{8} \int e^{a+b x^n} x^m \, dx\\ &=-\frac{3^{-\frac{1+m}{n}} e^{3 a} x^{1+m} \left (-b x^n\right )^{-\frac{1+m}{n}} \Gamma \left (\frac{1+m}{n},-3 b x^n\right )}{8 n}-\frac{3 e^a x^{1+m} \left (-b x^n\right )^{-\frac{1+m}{n}} \Gamma \left (\frac{1+m}{n},-b x^n\right )}{8 n}-\frac{3 e^{-a} x^{1+m} \left (b x^n\right )^{-\frac{1+m}{n}} \Gamma \left (\frac{1+m}{n},b x^n\right )}{8 n}-\frac{3^{-\frac{1+m}{n}} e^{-3 a} x^{1+m} \left (b x^n\right )^{-\frac{1+m}{n}} \Gamma \left (\frac{1+m}{n},3 b x^n\right )}{8 n}\\ \end{align*}
Mathematica [A] time = 0.955639, size = 182, normalized size = 0.91 \[ -\frac{e^{-3 a} 3^{-\frac{m+1}{n}} x^{m+1} \left (-b^2 x^{2 n}\right )^{-\frac{m+1}{n}} \left (\left (-b x^n\right )^{\frac{m+1}{n}} \left (e^{2 a} 3^{\frac{m+n+1}{n}} \text{Gamma}\left (\frac{m+1}{n},b x^n\right )+\text{Gamma}\left (\frac{m+1}{n},3 b x^n\right )\right )+e^{6 a} \left (b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-3 b x^n\right )+e^{4 a} 3^{\frac{m+n+1}{n}} \left (b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-b x^n\right )\right )}{8 n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.11, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \cosh \left ( a+b{x}^{n} \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.31915, size = 234, normalized size = 1.17 \begin{align*} -\frac{x^{m + 1} e^{\left (-3 \, a\right )} \Gamma \left (\frac{m + 1}{n}, 3 \, b x^{n}\right )}{8 \, \left (3 \, b x^{n}\right )^{\frac{m + 1}{n}} n} - \frac{3 \, x^{m + 1} e^{\left (-a\right )} \Gamma \left (\frac{m + 1}{n}, b x^{n}\right )}{8 \, \left (b x^{n}\right )^{\frac{m + 1}{n}} n} - \frac{3 \, x^{m + 1} e^{a} \Gamma \left (\frac{m + 1}{n}, -b x^{n}\right )}{8 \, \left (-b x^{n}\right )^{\frac{m + 1}{n}} n} - \frac{x^{m + 1} e^{\left (3 \, a\right )} \Gamma \left (\frac{m + 1}{n}, -3 \, b x^{n}\right )}{8 \, \left (-3 \, b x^{n}\right )^{\frac{m + 1}{n}} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \cosh \left (b x^{n} + a\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \cosh ^{3}{\left (a + b x^{n} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \cosh \left (b x^{n} + a\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]