Optimal. Leaf size=40 \[ \frac {\left (a^2-b^2\right ) \log (a+b \cosh (x))}{b^3}-\frac {a \cosh (x)}{b^2}+\frac {\cosh ^2(x)}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2668, 697} \[ \frac {\left (a^2-b^2\right ) \log (a+b \cosh (x))}{b^3}-\frac {a \cosh (x)}{b^2}+\frac {\cosh ^2(x)}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 697
Rule 2668
Rubi steps
\begin {align*} \int \frac {\sinh ^3(x)}{a+b \cosh (x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {b^2-x^2}{a+x} \, dx,x,b \cosh (x)\right )}{b^3}\\ &=-\frac {\operatorname {Subst}\left (\int \left (a-x+\frac {-a^2+b^2}{a+x}\right ) \, dx,x,b \cosh (x)\right )}{b^3}\\ &=-\frac {a \cosh (x)}{b^2}+\frac {\cosh ^2(x)}{2 b}+\frac {\left (a^2-b^2\right ) \log (a+b \cosh (x))}{b^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 40, normalized size = 1.00 \[ \frac {\left (a^2-b^2\right ) \log (a+b \cosh (x))}{b^3}-\frac {a \cosh (x)}{b^2}+\frac {\cosh (2 x)}{4 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.42, size = 234, normalized size = 5.85 \[ \frac {b^{2} \cosh \relax (x)^{4} + b^{2} \sinh \relax (x)^{4} - 4 \, a b \cosh \relax (x)^{3} - 8 \, {\left (a^{2} - b^{2}\right )} x \cosh \relax (x)^{2} + 4 \, {\left (b^{2} \cosh \relax (x) - a b\right )} \sinh \relax (x)^{3} - 4 \, a b \cosh \relax (x) + 2 \, {\left (3 \, b^{2} \cosh \relax (x)^{2} - 6 \, a b \cosh \relax (x) - 4 \, {\left (a^{2} - b^{2}\right )} x\right )} \sinh \relax (x)^{2} + b^{2} + 8 \, {\left ({\left (a^{2} - b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a^{2} - b^{2}\right )} \sinh \relax (x)^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \relax (x) + a\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 4 \, {\left (b^{2} \cosh \relax (x)^{3} - 3 \, a b \cosh \relax (x)^{2} - 4 \, {\left (a^{2} - b^{2}\right )} x \cosh \relax (x) - a b\right )} \sinh \relax (x)}{8 \, {\left (b^{3} \cosh \relax (x)^{2} + 2 \, b^{3} \cosh \relax (x) \sinh \relax (x) + b^{3} \sinh \relax (x)^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.13, size = 56, normalized size = 1.40 \[ \frac {b {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4 \, a {\left (e^{\left (-x\right )} + e^{x}\right )}}{8 \, b^{2}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left ({\left | b {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, a \right |}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.06, size = 283, normalized size = 7.08 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right ) a^{2}}{b^{3}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b}+\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {a}{b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {a}{b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) a^{2}}{b^{3}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b}+\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b -a -b \right ) a^{3}}{b^{3} \left (a -b \right )}-\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b -a -b \right ) a^{2}}{b^{2} \left (a -b \right )}-\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b -a -b \right ) a}{b \left (a -b \right )}+\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b -a -b \right )}{a -b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.31, size = 84, normalized size = 2.10 \[ -\frac {{\left (4 \, a e^{\left (-x\right )} - b\right )} e^{\left (2 \, x\right )}}{8 \, b^{2}} - \frac {4 \, a e^{\left (-x\right )} - b e^{\left (-2 \, x\right )}}{8 \, b^{2}} + \frac {{\left (a^{2} - b^{2}\right )} x}{b^{3}} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} + b\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.04, size = 79, normalized size = 1.98 \[ \frac {{\mathrm {e}}^{-2\,x}}{8\,b}+\frac {{\mathrm {e}}^{2\,x}}{8\,b}+\frac {\ln \left (b+2\,a\,{\mathrm {e}}^x+b\,{\mathrm {e}}^{2\,x}\right )\,\left (a^2-b^2\right )}{b^3}-\frac {a\,{\mathrm {e}}^x}{2\,b^2}-\frac {a\,{\mathrm {e}}^{-x}}{2\,b^2}-\frac {x\,\left (a^2-b^2\right )}{b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________