Optimal. Leaf size=64 \[ \frac{\sinh ^2(a+b x)}{4 b^3}-\frac{x \sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac{x^2 \sinh ^2(a+b x)}{2 b}+\frac{x^2}{4 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0427648, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {5372, 3310, 30} \[ \frac{\sinh ^2(a+b x)}{4 b^3}-\frac{x \sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac{x^2 \sinh ^2(a+b x)}{2 b}+\frac{x^2}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5372
Rule 3310
Rule 30
Rubi steps
\begin{align*} \int x^2 \cosh (a+b x) \sinh (a+b x) \, dx &=\frac{x^2 \sinh ^2(a+b x)}{2 b}-\frac{\int x \sinh ^2(a+b x) \, dx}{b}\\ &=-\frac{x \cosh (a+b x) \sinh (a+b x)}{2 b^2}+\frac{\sinh ^2(a+b x)}{4 b^3}+\frac{x^2 \sinh ^2(a+b x)}{2 b}+\frac{\int x \, dx}{2 b}\\ &=\frac{x^2}{4 b}-\frac{x \cosh (a+b x) \sinh (a+b x)}{2 b^2}+\frac{\sinh ^2(a+b x)}{4 b^3}+\frac{x^2 \sinh ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0673341, size = 39, normalized size = 0.61 \[ \frac{\left (2 b^2 x^2+1\right ) \cosh (2 (a+b x))-2 b x \sinh (2 (a+b x))}{8 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.004, size = 114, normalized size = 1.8 \begin{align*}{\frac{1}{{b}^{3}} \left ({\frac{ \left ( bx+a \right ) ^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{2}}-{\frac{ \left ( bx+a \right ) \cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{2}}-{\frac{ \left ( bx+a \right ) ^{2}}{4}}+{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{4}}-2\,a \left ( 1/2\, \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}-1/4\,\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) -1/4\,bx-a/4 \right ) +{\frac{{a}^{2} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.08568, size = 86, normalized size = 1.34 \begin{align*} \frac{{\left (2 \, b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{16 \, b^{3}} + \frac{{\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.123, size = 154, normalized size = 2.41 \begin{align*} -\frac{4 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) -{\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} -{\left (2 \, b^{2} x^{2} + 1\right )} \sinh \left (b x + a\right )^{2}}{8 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.21291, size = 75, normalized size = 1.17 \begin{align*} \begin{cases} \frac{x^{2} \sinh ^{2}{\left (a + b x \right )}}{4 b} + \frac{x^{2} \cosh ^{2}{\left (a + b x \right )}}{4 b} - \frac{x \sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{2 b^{2}} + \frac{\sinh ^{2}{\left (a + b x \right )}}{4 b^{3}} & \text{for}\: b \neq 0 \\\frac{x^{3} \sinh{\left (a \right )} \cosh{\left (a \right )}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14793, size = 77, normalized size = 1.2 \begin{align*} \frac{{\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b^{3}} + \frac{{\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]