Optimal. Leaf size=46 \[ \frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}-\frac {\sinh (a+b x) \cosh (a+b x)}{8 b}-\frac {x}{8} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ \frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}-\frac {\sinh (a+b x) \cosh (a+b x)}{8 b}-\frac {x}{8} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2568
Rule 2635
Rubi steps
\begin {align*} \int \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx &=\frac {\cosh ^3(a+b x) \sinh (a+b x)}{4 b}-\frac {1}{4} \int \cosh ^2(a+b x) \, dx\\ &=-\frac {\cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {\cosh ^3(a+b x) \sinh (a+b x)}{4 b}-\frac {\int 1 \, dx}{8}\\ &=-\frac {x}{8}-\frac {\cosh (a+b x) \sinh (a+b x)}{8 b}+\frac {\cosh ^3(a+b x) \sinh (a+b x)}{4 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 23, normalized size = 0.50 \[ \frac {\sinh (4 (a+b x))-4 (a+b x)}{32 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.38, size = 40, normalized size = 0.87 \[ \frac {\cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} - b x}{8 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.23, size = 32, normalized size = 0.70 \[ -\frac {1}{8} \, x + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{64 \, b} - \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{64 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.09, size = 43, normalized size = 0.93 \[ \frac {\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{4}-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {b x}{8}-\frac {a}{8}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 39, normalized size = 0.85 \[ -\frac {b x + a}{8 \, b} + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{64 \, b} - \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{64 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.11, size = 18, normalized size = 0.39 \[ \frac {\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{32\,b}-\frac {x}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.84, size = 92, normalized size = 2.00 \[ \begin {cases} - \frac {x \sinh ^{4}{\left (a + b x \right )}}{8} + \frac {x \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{4} - \frac {x \cosh ^{4}{\left (a + b x \right )}}{8} + \frac {\sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{8 b} + \frac {\sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} & \text {for}\: b \neq 0 \\x \sinh ^{2}{\relax (a )} \cosh ^{2}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________