Optimal. Leaf size=82 \[ (a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )-(a+b) \sqrt {a+b \coth ^2(x)}-\frac {1}{3} \left (a+b \coth ^2(x)\right )^{3/2}-\frac {\left (a+b \coth ^2(x)\right )^{5/2}}{5 b} \]
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Rubi [A]
time = 0.10, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3751, 457, 81,
52, 65, 214} \begin {gather*} -\frac {\left (a+b \coth ^2(x)\right )^{5/2}}{5 b}-\frac {1}{3} \left (a+b \coth ^2(x)\right )^{3/2}-(a+b) \sqrt {a+b \coth ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 214
Rule 457
Rule 3751
Rubi steps
\begin {align*} \int \coth ^3(x) \left (a+b \coth ^2(x)\right )^{3/2} \, dx &=\text {Subst}\left (\int \frac {x^3 \left (a+b x^2\right )^{3/2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {x (a+b x)^{3/2}}{1-x} \, dx,x,\coth ^2(x)\right )\\ &=-\frac {\left (a+b \coth ^2(x)\right )^{5/2}}{5 b}+\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{1-x} \, dx,x,\coth ^2(x)\right )\\ &=-\frac {1}{3} \left (a+b \coth ^2(x)\right )^{3/2}-\frac {\left (a+b \coth ^2(x)\right )^{5/2}}{5 b}+\frac {1}{2} (a+b) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{1-x} \, dx,x,\coth ^2(x)\right )\\ &=-(a+b) \sqrt {a+b \coth ^2(x)}-\frac {1}{3} \left (a+b \coth ^2(x)\right )^{3/2}-\frac {\left (a+b \coth ^2(x)\right )^{5/2}}{5 b}+\frac {1}{2} (a+b)^2 \text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a+b x}} \, dx,x,\coth ^2(x)\right )\\ &=-(a+b) \sqrt {a+b \coth ^2(x)}-\frac {1}{3} \left (a+b \coth ^2(x)\right )^{3/2}-\frac {\left (a+b \coth ^2(x)\right )^{5/2}}{5 b}+\frac {(a+b)^2 \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \coth ^2(x)}\right )}{b}\\ &=(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )-(a+b) \sqrt {a+b \coth ^2(x)}-\frac {1}{3} \left (a+b \coth ^2(x)\right )^{3/2}-\frac {\left (a+b \coth ^2(x)\right )^{5/2}}{5 b}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 86, normalized size = 1.05 \begin {gather*} (a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )-\frac {\sqrt {a+b \coth ^2(x)} \left (3 a^2+20 a b+15 b^2+b (6 a+5 b) \coth ^2(x)+3 b^2 \coth ^4(x)\right )}{15 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(487\) vs.
\(2(66)=132\).
time = 0.62, size = 488, normalized size = 5.95
method | result | size |
derivativedivides | \(-\frac {\left (a +b \left (\coth ^{2}\left (x \right )\right )\right )^{\frac {5}{2}}}{5 b}-\frac {\left (b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}{6}-\frac {b \left (\frac {\left (2 b \left (\coth \left (x \right )-1\right )+2 b \right ) \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{4 b}+\frac {\left (4 b \left (a +b \right )-4 b^{2}\right ) \ln \left (\frac {b \left (\coth \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}\right )}{8 b^{\frac {3}{2}}}\right )}{2}-\frac {\left (a +b \right ) \left (\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}+\sqrt {b}\, \ln \left (\frac {b \left (\coth \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}\right )-\sqrt {a +b}\, \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 )\right )}{2}-\frac {\left (b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}{6}+\frac {b \left (\frac {\left (2 b \left (1+\coth \left (x \right )\right )-2 b \right ) \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{4 b}+\frac {\left (4 b \left (a +b \right )-4 b^{2}\right ) \ln \left (\frac {b \left (1+\coth \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}\right )}{8 b^{\frac {3}{2}}}\right )}{2}-\frac {\left (a +b \right ) \left (\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}-\sqrt {b}\, \ln \left (\frac {b \left (1+\coth \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}\right )-\sqrt {a +b}\, \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 )\right )}{2}\) | \(488\) |
default | \(-\frac {\left (a +b \left (\coth ^{2}\left (x \right )\right )\right )^{\frac {5}{2}}}{5 b}-\frac {\left (b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}{6}-\frac {b \left (\frac {\left (2 b \left (\coth \left (x \right )-1\right )+2 b \right ) \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{4 b}+\frac {\left (4 b \left (a +b \right )-4 b^{2}\right ) \ln \left (\frac {b \left (\coth \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}\right )}{8 b^{\frac {3}{2}}}\right )}{2}-\frac {\left (a +b \right ) \left (\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}+\sqrt {b}\, \ln \left (\frac {b \left (\coth \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}\right )-\sqrt {a +b}\, \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 )\right )}{2}-\frac {\left (b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}{6}+\frac {b \left (\frac {\left (2 b \left (1+\coth \left (x \right )\right )-2 b \right ) \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{4 b}+\frac {\left (4 b \left (a +b \right )-4 b^{2}\right ) \ln \left (\frac {b \left (1+\coth \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}\right )}{8 b^{\frac {3}{2}}}\right )}{2}-\frac {\left (a +b \right ) \left (\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}-\sqrt {b}\, \ln \left (\frac {b \left (1+\coth \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}\right )-\sqrt {a +b}\, \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 )\right )}{2}\) | \(488\) |
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. 2154 vs.
\(2 (66) = 132\).
time = 0.60, size = 4940, normalized size = 60.24 \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 \left (a + b \coth ^{2}{\left (x \right )}\right )^{\frac {3}{2}} \coth ^{3}{\left (x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.04, size = 112, normalized size = 1.37 \begin {gather*} -\frac {{\left (b\,{\mathrm {coth}\left (x\right )}^2+a\right )}^{5/2}}{5\,b}-\left (\frac {a+b}{3\,b}-\frac {a}{3\,b}\right )\,{\left (b\,{\mathrm {coth}\left (x\right )}^2+a\right )}^{3/2}-\left (a+b\right )\,\left (\frac {a+b}{b}-\frac {a}{b}\right )\,\sqrt {b\,{\mathrm {coth}\left (x\right )}^2+a}-\mathrm {atan}\left (\frac {{\left (a+b\right )}^{3/2}\,\sqrt {b\,{\mathrm {coth}\left (x\right )}^2+a}\,1{}\mathrm {i}}{a^2+2\,a\,b+b^2}\right )\,{\left (a+b\right )}^{3/2}\,1{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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