Optimal. Leaf size=72 \[ \frac {3}{8} x \left (a^2-b^2\right )^2+\frac {3}{8} \left (a^2-b^2\right ) (a \sinh (x)+b \cosh (x)) (a \cosh (x)+b \sinh (x))+\frac {1}{4} (a \sinh (x)+b \cosh (x)) (a \cosh (x)+b \sinh (x))^3 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3073, 8} \[ \frac {3}{8} x \left (a^2-b^2\right )^2+\frac {3}{8} \left (a^2-b^2\right ) (a \sinh (x)+b \cosh (x)) (a \cosh (x)+b \sinh (x))+\frac {1}{4} (a \sinh (x)+b \cosh (x)) (a \cosh (x)+b \sinh (x))^3 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3073
Rubi steps
\begin {align*} \int (a \cosh (x)+b \sinh (x))^4 \, dx &=\frac {1}{4} (b \cosh (x)+a \sinh (x)) (a \cosh (x)+b \sinh (x))^3+\frac {1}{4} \left (3 \left (a^2-b^2\right )\right ) \int (a \cosh (x)+b \sinh (x))^2 \, dx\\ &=\frac {3}{8} \left (a^2-b^2\right ) (b \cosh (x)+a \sinh (x)) (a \cosh (x)+b \sinh (x))+\frac {1}{4} (b \cosh (x)+a \sinh (x)) (a \cosh (x)+b \sinh (x))^3+\frac {1}{8} \left (3 \left (a^2-b^2\right )^2\right ) \int 1 \, dx\\ &=\frac {3}{8} \left (a^2-b^2\right )^2 x+\frac {3}{8} \left (a^2-b^2\right ) (b \cosh (x)+a \sinh (x)) (a \cosh (x)+b \sinh (x))+\frac {1}{4} (b \cosh (x)+a \sinh (x)) (a \cosh (x)+b \sinh (x))^3\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 87, normalized size = 1.21 \[ \frac {1}{32} \left (8 \left (a^4-b^4\right ) \sinh (2 x)+16 a b \left (a^2-b^2\right ) \cosh (2 x)+4 a b \left (a^2+b^2\right ) \cosh (4 x)+\left (a^4+6 a^2 b^2+b^4\right ) \sinh (4 x)+12 x (a-b)^2 (a+b)^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.43, size = 168, normalized size = 2.33 \[ \frac {1}{8} \, {\left (a^{3} b + a b^{3}\right )} \cosh \relax (x)^{4} + \frac {1}{8} \, {\left (a^{4} + 6 \, a^{2} b^{2} + b^{4}\right )} \cosh \relax (x) \sinh \relax (x)^{3} + \frac {1}{8} \, {\left (a^{3} b + a b^{3}\right )} \sinh \relax (x)^{4} + \frac {1}{2} \, {\left (a^{3} b - a b^{3}\right )} \cosh \relax (x)^{2} + \frac {1}{4} \, {\left (2 \, a^{3} b - 2 \, a b^{3} + 3 \, {\left (a^{3} b + a b^{3}\right )} \cosh \relax (x)^{2}\right )} \sinh \relax (x)^{2} + \frac {3}{8} \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} x + \frac {1}{8} \, {\left ({\left (a^{4} + 6 \, a^{2} b^{2} + b^{4}\right )} \cosh \relax (x)^{3} + 4 \, {\left (a^{4} - b^{4}\right )} \cosh \relax (x)\right )} \sinh \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.12, size = 208, normalized size = 2.89 \[ \frac {1}{64} \, a^{4} e^{\left (4 \, x\right )} + \frac {1}{16} \, a^{3} b e^{\left (4 \, x\right )} + \frac {3}{32} \, a^{2} b^{2} e^{\left (4 \, x\right )} + \frac {1}{16} \, a b^{3} e^{\left (4 \, x\right )} + \frac {1}{64} \, b^{4} e^{\left (4 \, x\right )} + \frac {1}{8} \, a^{4} e^{\left (2 \, x\right )} + \frac {1}{4} \, a^{3} b e^{\left (2 \, x\right )} - \frac {1}{4} \, a b^{3} e^{\left (2 \, x\right )} - \frac {1}{8} \, b^{4} e^{\left (2 \, x\right )} + \frac {3}{8} \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} x - \frac {1}{64} \, {\left (18 \, a^{4} e^{\left (4 \, x\right )} - 36 \, a^{2} b^{2} e^{\left (4 \, x\right )} + 18 \, b^{4} e^{\left (4 \, x\right )} + 8 \, a^{4} e^{\left (2 \, x\right )} - 16 \, a^{3} b e^{\left (2 \, x\right )} + 16 \, a b^{3} e^{\left (2 \, x\right )} - 8 \, b^{4} e^{\left (2 \, x\right )} + a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} e^{\left (-4 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.46, size = 90, normalized size = 1.25 \[ b^{4} \left (\left (\frac {\left (\sinh ^{3}\relax (x )\right )}{4}-\frac {3 \sinh \relax (x )}{8}\right ) \cosh \relax (x )+\frac {3 x}{8}\right )+a \,b^{3} \left (\sinh ^{4}\relax (x )\right )+6 a^{2} b^{2} \left (\frac {\sinh \relax (x ) \left (\cosh ^{3}\relax (x )\right )}{4}-\frac {\cosh \relax (x ) \sinh \relax (x )}{8}-\frac {x}{8}\right )+a^{3} b \left (\cosh ^{4}\relax (x )\right )+a^{4} \left (\left (\frac {\left (\cosh ^{3}\relax (x )\right )}{4}+\frac {3 \cosh \relax (x )}{8}\right ) \sinh \relax (x )+\frac {3 x}{8}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.57, size = 103, normalized size = 1.43 \[ a^{3} b \cosh \relax (x)^{4} + a b^{3} \sinh \relax (x)^{4} + \frac {1}{64} \, a^{4} {\left (24 \, x + e^{\left (4 \, x\right )} + 8 \, e^{\left (2 \, x\right )} - 8 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )}\right )} + \frac {1}{64} \, b^{4} {\left (24 \, x + e^{\left (4 \, x\right )} - 8 \, e^{\left (2 \, x\right )} + 8 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )}\right )} - \frac {3}{32} \, a^{2} b^{2} {\left (8 \, x - e^{\left (4 \, x\right )} + e^{\left (-4 \, x\right )}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.56, size = 143, normalized size = 1.99 \[ \mathrm {cosh}\relax (x)\,{\mathrm {sinh}\relax (x)}^3\,\left (-\frac {3\,a^4}{8}+\frac {3\,a^2\,b^2}{4}+\frac {5\,b^4}{8}\right )-{\mathrm {cosh}\relax (x)}^4\,\left (a\,b^3-a^3\,b\right )+{\mathrm {cosh}\relax (x)}^3\,\mathrm {sinh}\relax (x)\,\left (\frac {5\,a^4}{8}+\frac {3\,a^2\,b^2}{4}-\frac {3\,b^4}{8}\right )+\frac {3\,x\,{\mathrm {cosh}\relax (x)}^4\,{\left (a^2-b^2\right )}^2}{8}+\frac {3\,x\,{\mathrm {sinh}\relax (x)}^4\,{\left (a^2-b^2\right )}^2}{8}+2\,a\,b^3\,{\mathrm {cosh}\relax (x)}^2\,{\mathrm {sinh}\relax (x)}^2-\frac {3\,x\,{\mathrm {cosh}\relax (x)}^2\,{\mathrm {sinh}\relax (x)}^2\,{\left (a^2-b^2\right )}^2}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.69, size = 265, normalized size = 3.68 \[ \frac {3 a^{4} x \sinh ^{4}{\relax (x )}}{8} - \frac {3 a^{4} x \sinh ^{2}{\relax (x )} \cosh ^{2}{\relax (x )}}{4} + \frac {3 a^{4} x \cosh ^{4}{\relax (x )}}{8} - \frac {3 a^{4} \sinh ^{3}{\relax (x )} \cosh {\relax (x )}}{8} + \frac {5 a^{4} \sinh {\relax (x )} \cosh ^{3}{\relax (x )}}{8} + a^{3} b \cosh ^{4}{\relax (x )} - \frac {3 a^{2} b^{2} x \sinh ^{4}{\relax (x )}}{4} + \frac {3 a^{2} b^{2} x \sinh ^{2}{\relax (x )} \cosh ^{2}{\relax (x )}}{2} - \frac {3 a^{2} b^{2} x \cosh ^{4}{\relax (x )}}{4} + \frac {3 a^{2} b^{2} \sinh ^{3}{\relax (x )} \cosh {\relax (x )}}{4} + \frac {3 a^{2} b^{2} \sinh {\relax (x )} \cosh ^{3}{\relax (x )}}{4} + a b^{3} \sinh ^{4}{\relax (x )} + \frac {3 b^{4} x \sinh ^{4}{\relax (x )}}{8} - \frac {3 b^{4} x \sinh ^{2}{\relax (x )} \cosh ^{2}{\relax (x )}}{4} + \frac {3 b^{4} x \cosh ^{4}{\relax (x )}}{8} + \frac {5 b^{4} \sinh ^{3}{\relax (x )} \cosh {\relax (x )}}{8} - \frac {3 b^{4} \sinh {\relax (x )} \cosh ^{3}{\relax (x )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________