Optimal. Leaf size=51 \[ \frac {\tan ^{-1}\left (\sqrt {2} \tanh (a+b x)+1\right )}{\sqrt {2} b}-\frac {\tan ^{-1}\left (1-\sqrt {2} \tanh (a+b x)\right )}{\sqrt {2} b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1162, 617, 204} \[ \frac {\tan ^{-1}\left (\sqrt {2} \tanh (a+b x)+1\right )}{\sqrt {2} b}-\frac {\tan ^{-1}\left (1-\sqrt {2} \tanh (a+b x)\right )}{\sqrt {2} b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 617
Rule 1162
Rubi steps
\begin {align*} \int \frac {\cosh ^4(a+b x)-\sinh ^4(a+b x)}{\cosh ^4(a+b x)+\sinh ^4(a+b x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\tanh (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\tanh (a+b x)\right )}{2 b}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\tanh (a+b x)\right )}{2 b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \tanh (a+b x)\right )}{\sqrt {2} b}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \tanh (a+b x)\right )}{\sqrt {2} b}\\ &=-\frac {\tan ^{-1}\left (1-\sqrt {2} \tanh (a+b x)\right )}{\sqrt {2} b}+\frac {\tan ^{-1}\left (1+\sqrt {2} \tanh (a+b x)\right )}{\sqrt {2} b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 25, normalized size = 0.49 \[ \frac {\tan ^{-1}\left (\frac {\sinh (2 a+2 b x)}{\sqrt {2}}\right )}{\sqrt {2} b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.42, size = 192, normalized size = 3.76 \[ -\frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} \cosh \left (b x + a\right )^{3} + 3 \, \sqrt {2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sqrt {2} \sinh \left (b x + a\right )^{3} + {\left (3 \, \sqrt {2} \cosh \left (b x + a\right )^{2} - 7 \, \sqrt {2}\right )} \sinh \left (b x + a\right ) + 7 \, \sqrt {2} \cosh \left (b x + a\right )}{4 \, {\left (\cosh \left (b x + a\right )^{3} - 3 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right ) + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - \sinh \left (b x + a\right )^{3}\right )}}\right ) + \sqrt {2} \arctan \left (-\frac {\sqrt {2} \cosh \left (b x + a\right ) + \sqrt {2} \sinh \left (b x + a\right )}{4 \, {\left (\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.77, size = 34, normalized size = 0.67 \[ \frac {\sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (e^{\left (4 \, b x + 4 \, a\right )} - 1\right )} e^{\left (-2 \, b x - 2 \, a\right )}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.78, size = 138, normalized size = 2.71 \[ \frac {i \sqrt {2}\, \ln \left (-2 i \sqrt {2}\, \left (\tanh ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\tanh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )-2 i \sqrt {2}\, \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )-2 \left (\tanh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1\right )}{4 b}-\frac {i \sqrt {2}\, \ln \left (2 i \sqrt {2}\, \left (\tanh ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\tanh ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )+2 i \sqrt {2}\, \tanh \left (\frac {b x}{2}+\frac {a}{2}\right )-2 \left (\tanh ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1\right )}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ 2 \, \int \frac {{\left (e^{\left (-b x - a\right )} + e^{\left (-5 \, b x - 5 \, a\right )}\right )} e^{\left (-b x - a\right )}}{6 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-8 \, b x - 8 \, a\right )} + 1}\,{d x} + 2 \, \int \frac {e^{\left (6 \, b x + 6 \, a\right )}}{e^{\left (8 \, b x + 8 \, a\right )} + 6 \, e^{\left (4 \, b x + 4 \, a\right )} + 1}\,{d x} + 2 \, \int \frac {e^{\left (-6 \, b x - 6 \, a\right )}}{6 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-8 \, b x - 8 \, a\right )} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.25, size = 77, normalized size = 1.51 \[ \frac {\sqrt {2}\,\left (\mathrm {atan}\left (\frac {\sqrt {2}\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}\,\sqrt {b^2}}{4\,b}\right )+\mathrm {atan}\left (\frac {\sqrt {b^2}\,\left (\frac {56\,\sqrt {2}\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{b}+\frac {8\,\sqrt {2}\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{6\,b\,x}}{b}\right )}{32}\right )\right )}{2\,\sqrt {b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 22.71, size = 100, normalized size = 1.96 \[ \begin {cases} \frac {x \left (- \sinh ^{4}{\relax (a )} + \cosh ^{4}{\relax (a )}\right )}{\sinh ^{4}{\relax (a )} + \cosh ^{4}{\relax (a )}} & \text {for}\: b = 0 \\- x & \text {for}\: a = \log {\left (- i e^{- b x} \right )} \vee a = \log {\left (i e^{- b x} \right )} \\\frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sinh {\left (a + b x \right )}}{\cosh {\left (a + b x \right )}} - 1 \right )}}{2 b} + \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sinh {\left (a + b x \right )}}{\cosh {\left (a + b x \right )}} + 1 \right )}}{2 b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________