Optimal. Leaf size=31 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{\sqrt {a+b}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3742, 385, 212}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{\sqrt {a+b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 385
Rule 3742
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \coth ^2(x)}} \, dx &=\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )\\ &=\text {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{\sqrt {a+b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(31)=62\).
time = 0.09, size = 77, normalized size = 2.48 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {\frac {(a+b) \coth ^2(x)}{a}}}{\sqrt {1+\frac {b \coth ^2(x)}{a}}}\right ) \coth (x) \sqrt {1+\frac {b \coth ^2(x)}{a}}}{\sqrt {\frac {(a+b) \coth ^2(x)}{a}} \sqrt {a+b \coth ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(113\) vs.
\(2(25)=50\).
time = 0.86, size = 114, normalized size = 3.68
method | result | size |
derivativedivides | \(\frac {\ln \left (\frac {2 a +2 b +2 b \left (\coth \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{\coth \left (x \right )-1}\right )}{2 \sqrt {a +b}}-\frac {\ln \left (\frac {2 a +2 b -2 b \left (1+\coth \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{1+\coth \left (x \right )}\right )}{2 \sqrt {a +b}}\) | \(114\) |
default | \(\frac {\ln \left (\frac {2 a +2 b +2 b \left (\coth \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{\coth \left (x \right )-1}\right )}{2 \sqrt {a +b}}-\frac {\ln \left (\frac {2 a +2 b -2 b \left (1+\coth \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{1+\coth \left (x \right )}\right )}{2 \sqrt {a +b}}\) | \(114\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 399 vs.
\(2 (25) = 50\).
time = 0.41, size = 1357, normalized size = 43.77 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + b \coth ^{2}{\left (x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 196 vs.
\(2 (25) = 50\).
time = 0.49, size = 196, normalized size = 6.32 \begin {gather*} -\frac {\frac {\log \left ({\left | -{\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + 2 \, b e^{\left (2 \, x\right )} + a + b}\right )} {\left (a + b\right )} + \sqrt {a + b} {\left (a - b\right )} \right |}\right )}{\sqrt {a + b}} - \frac {\log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + 2 \, b e^{\left (2 \, x\right )} + a + b} + \sqrt {a + b} \right |}\right )}{\sqrt {a + b}} + \frac {\log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + 2 \, b e^{\left (2 \, x\right )} + a + b} - \sqrt {a + b} \right |}\right )}{\sqrt {a + b}}}{2 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.48, size = 25, normalized size = 0.81 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {\mathrm {coth}\left (x\right )\,\sqrt {a+b}}{\sqrt {b\,{\mathrm {coth}\left (x\right )}^2+a}}\right )}{\sqrt {a+b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________