Optimal. Leaf size=89 \[ -\frac{e^{2 a} 2^{-\frac{1}{n}-2} x \left (-b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-2 b x^n\right )}{n}-\frac{e^{-2 a} 2^{-\frac{1}{n}-2} x \left (b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},2 b x^n\right )}{n}-\frac{x}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0649043, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5308, 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.
[In]
[Out]
Rule 5308
Rule 5307
Rule 2208
Rubi steps
\begin{align*} \int \sinh ^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*}
Mathematica [A] time = 1.07465, size = 81, normalized size = 0.91 \[ -\frac{x \left (e^{2 a} 2^{-1/n} \left (-b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-2 b x^n\right )+e^{-2 a} 2^{-1/n} \left (b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},2 b x^n\right )+2 n\right )}{4 n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int \left ( \sinh \left ( a+b{x}^{n} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.16296, size = 92, normalized size = 1.03 \begin{align*} -\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} \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 (\sinh \left (b x^{n} + a\right )^{2}, 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 \sinh ^{2}{\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 \sinh \left (b x^{n} + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]