Optimal. Leaf size=89 \[ -\frac {1}{2} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \coth ^2(x)}{\sqrt {a+b \coth ^4(x)}}\right )+\frac {1}{2} \sqrt {a+b} \tanh ^{-1}\left (\frac {a+b \coth ^2(x)}{\sqrt {a+b} \sqrt {a+b \coth ^4(x)}}\right )-\frac {1}{2} \sqrt {a+b \coth ^4(x)} \]
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Rubi [A]
time = 0.10, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3751, 1262,
749, 858, 223, 212, 739} \begin {gather*} -\frac {1}{2} \sqrt {a+b \coth ^4(x)}-\frac {1}{2} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \coth ^2(x)}{\sqrt {a+b \coth ^4(x)}}\right )+\frac {1}{2} \sqrt {a+b} \tanh ^{-1}\left (\frac {a+b \coth ^2(x)}{\sqrt {a+b} \sqrt {a+b \coth ^4(x)}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 739
Rule 749
Rule 858
Rule 1262
Rule 3751
Rubi steps
\begin {align*} \int \coth (x) \sqrt {a+b \coth ^4(x)} \, dx &=\text {Subst}\left (\int \frac {x \sqrt {a+b x^4}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{1-x} \, dx,x,\coth ^2(x)\right )\\ &=-\frac {1}{2} \sqrt {a+b \coth ^4(x)}-\frac {1}{2} \text {Subst}\left (\int \frac {-a-b x}{(1-x) \sqrt {a+b x^2}} \, dx,x,\coth ^2(x)\right )\\ &=-\frac {1}{2} \sqrt {a+b \coth ^4(x)}-\frac {1}{2} (-a-b) \text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a+b x^2}} \, dx,x,\coth ^2(x)\right )-\frac {1}{2} b \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\coth ^2(x)\right )\\ &=-\frac {1}{2} \sqrt {a+b \coth ^4(x)}-\frac {1}{2} b \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\coth ^2(x)}{\sqrt {a+b \coth ^4(x)}}\right )-\frac {1}{2} (a+b) \text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\frac {-a-b \coth ^2(x)}{\sqrt {a+b \coth ^4(x)}}\right )\\ &=-\frac {1}{2} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \coth ^2(x)}{\sqrt {a+b \coth ^4(x)}}\right )+\frac {1}{2} \sqrt {a+b} \tanh ^{-1}\left (\frac {a+b \coth ^2(x)}{\sqrt {a+b} \sqrt {a+b \coth ^4(x)}}\right )-\frac {1}{2} \sqrt {a+b \coth ^4(x)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 86, normalized size = 0.97 \begin {gather*} \frac {1}{2} \left (-\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \coth ^2(x)}{\sqrt {a+b \coth ^4(x)}}\right )+\sqrt {a+b} \tanh ^{-1}\left (\frac {a+b \coth ^2(x)}{\sqrt {a+b} \sqrt {a+b \coth ^4(x)}}\right )-\sqrt {a+b \coth ^4(x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 3.25, size = 333, normalized size = 3.74
method | result | size |
derivativedivides | \(-\frac {\sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}{2}-\frac {\sqrt {b}\, \ln \left (2 \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )+2 \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}\right )}{2}-\frac {\left (a +b \right ) \left (-\frac {\arctanh \left (\frac {2 b \left (\coth ^{2}\left (x \right )\right )+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2 \sqrt {a +b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \EllipticPi \left (\coth \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2}-\frac {\left (a +b \right ) \left (-\frac {\arctanh \left (\frac {2 b \left (\coth ^{2}\left (x \right )\right )+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2 \sqrt {a +b}}-\frac {\sqrt {1-\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \EllipticPi \left (\coth \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2}\) | \(333\) |
default | \(-\frac {\sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}{2}-\frac {\sqrt {b}\, \ln \left (2 \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )+2 \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}\right )}{2}-\frac {\left (a +b \right ) \left (-\frac {\arctanh \left (\frac {2 b \left (\coth ^{2}\left (x \right )\right )+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2 \sqrt {a +b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \EllipticPi \left (\coth \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2}-\frac {\left (a +b \right ) \left (-\frac {\arctanh \left (\frac {2 b \left (\coth ^{2}\left (x \right )\right )+2 a}{2 \sqrt {a +b}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2 \sqrt {a +b}}-\frac {\sqrt {1-\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \left (\coth ^{2}\left (x \right )\right )}{\sqrt {a}}}\, \EllipticPi \left (\coth \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \left (\coth ^{4}\left (x \right )\right )}}\right )}{2}\) | \(333\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1053 vs.
\(2 (69) = 138\).
time = 0.52, size = 5172, normalized size = 58.11 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b \coth ^{4}{\left (x \right )}} \coth {\left (x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {coth}\left (x\right )\,\sqrt {b\,{\mathrm {coth}\left (x\right )}^4+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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