Optimal. Leaf size=34 \[ \frac{x^2 \cosh \left (a+b x^2\right )}{2 b}-\frac{\sinh \left (a+b x^2\right )}{2 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0355861, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5320, 3296, 2637} \[ \frac{x^2 \cosh \left (a+b x^2\right )}{2 b}-\frac{\sinh \left (a+b x^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5320
Rule 3296
Rule 2637
Rubi steps
\begin{align*} \int x^3 \sinh \left (a+b x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x \sinh (a+b x) \, dx,x,x^2\right )\\ &=\frac{x^2 \cosh \left (a+b x^2\right )}{2 b}-\frac{\operatorname{Subst}\left (\int \cosh (a+b x) \, dx,x,x^2\right )}{2 b}\\ &=\frac{x^2 \cosh \left (a+b x^2\right )}{2 b}-\frac{\sinh \left (a+b x^2\right )}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0344656, size = 31, normalized size = 0.91 \[ \frac{b x^2 \cosh \left (a+b x^2\right )-\sinh \left (a+b x^2\right )}{2 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.013, size = 45, normalized size = 1.3 \begin{align*}{\frac{ \left ( b{x}^{2}-1 \right ){{\rm e}^{b{x}^{2}+a}}}{4\,{b}^{2}}}+{\frac{ \left ( b{x}^{2}+1 \right ){{\rm e}^{-b{x}^{2}-a}}}{4\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.03451, size = 109, normalized size = 3.21 \begin{align*} \frac{1}{4} \, x^{4} \sinh \left (b x^{2} + a\right ) - \frac{1}{8} \, b{\left (\frac{{\left (b^{2} x^{4} e^{a} - 2 \, b x^{2} e^{a} + 2 \, e^{a}\right )} e^{\left (b x^{2}\right )}}{b^{3}} - \frac{{\left (b^{2} x^{4} + 2 \, b x^{2} + 2\right )} e^{\left (-b x^{2} - a\right )}}{b^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.98473, size = 69, normalized size = 2.03 \begin{align*} \frac{b x^{2} \cosh \left (b x^{2} + a\right ) - \sinh \left (b x^{2} + a\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.12747, size = 36, normalized size = 1.06 \begin{align*} \begin{cases} \frac{x^{2} \cosh{\left (a + b x^{2} \right )}}{2 b} - \frac{\sinh{\left (a + b x^{2} \right )}}{2 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{4} \sinh{\left (a \right )}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.39141, size = 63, normalized size = 1.85 \begin{align*} \frac{\frac{{\left (b x^{2} - 1\right )} e^{\left (b x^{2} + a\right )}}{b} + \frac{{\left (b x^{2} + 1\right )} e^{\left (-b x^{2} - a\right )}}{b}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]