Optimal. Leaf size=41 \[ -\frac{\cosh (4 a+4 b x)}{128 b^2}+\frac{x \sinh (4 a+4 b x)}{32 b}-\frac{x^2}{16} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0504448, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5448, 3296, 2638} \[ -\frac{\cosh (4 a+4 b x)}{128 b^2}+\frac{x \sinh (4 a+4 b x)}{32 b}-\frac{x^2}{16} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5448
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac{x}{8}+\frac{1}{8} x \cosh (4 a+4 b x)\right ) \, dx\\ &=-\frac{x^2}{16}+\frac{1}{8} \int x \cosh (4 a+4 b x) \, dx\\ &=-\frac{x^2}{16}+\frac{x \sinh (4 a+4 b x)}{32 b}-\frac{\int \sinh (4 a+4 b x) \, dx}{32 b}\\ &=-\frac{x^2}{16}-\frac{\cosh (4 a+4 b x)}{128 b^2}+\frac{x \sinh (4 a+4 b x)}{32 b}\\ \end{align*}
Mathematica [A] time = 0.135183, size = 41, normalized size = 1. \[ -\frac{-8 a^2-4 b x \sinh (4 (a+b x))+\cosh (4 (a+b x))+8 b^2 x^2}{128 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.005, size = 114, normalized size = 2.8 \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{ \left ( bx+a \right ) \sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{3}}{4}}-{\frac{ \left ( bx+a \right ) \cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{8}}-{\frac{ \left ( bx+a \right ) ^{2}}{16}}-{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{16}}-a \left ({\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{3}\sinh \left ( bx+a \right ) }{4}}-{\frac{\cosh \left ( bx+a \right ) \sinh \left ( bx+a \right ) }{8}}-{\frac{bx}{8}}-{\frac{a}{8}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.10955, size = 69, normalized size = 1.68 \begin{align*} -\frac{1}{16} \, x^{2} + \frac{{\left (4 \, b x e^{\left (4 \, a\right )} - e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x\right )}}{256 \, b^{2}} - \frac{{\left (4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{256 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.83311, size = 234, normalized size = 5.71 \begin{align*} \frac{16 \, b x \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 16 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - 8 \, b^{2} x^{2} - \cosh \left (b x + a\right )^{4} - 6 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} - \sinh \left (b x + a\right )^{4}}{128 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.43515, size = 131, normalized size = 3.2 \begin{align*} \begin{cases} - \frac{x^{2} \sinh ^{4}{\left (a + b x \right )}}{16} + \frac{x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8} - \frac{x^{2} \cosh ^{4}{\left (a + b x \right )}}{16} + \frac{x \sinh ^{3}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{8 b} + \frac{x \sinh{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} - \frac{\sinh ^{4}{\left (a + b x \right )}}{32 b^{2}} - \frac{\cosh ^{4}{\left (a + b x \right )}}{32 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \sinh ^{2}{\left (a \right )} \cosh ^{2}{\left (a \right )}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.22371, size = 62, normalized size = 1.51 \begin{align*} -\frac{1}{16} \, x^{2} + \frac{{\left (4 \, b x - 1\right )} e^{\left (4 \, b x + 4 \, a\right )}}{256 \, b^{2}} - \frac{{\left (4 \, b x + 1\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{256 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]