Optimal. Leaf size=88 \[ -\frac {1}{2} \sqrt {b} (3 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )+(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )-\frac {1}{2} b \coth (x) \sqrt {a+b \coth ^2(x)} \]
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Rubi [A]
time = 0.06, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3742, 427, 537,
223, 212, 385} \begin {gather*} -\frac {1}{2} b \coth (x) \sqrt {a+b \coth ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )-\frac {1}{2} \sqrt {b} (3 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 385
Rule 427
Rule 537
Rule 3742
Rubi steps
\begin {align*} \int \left (a+b \coth ^2(x)\right )^{3/2} \, dx &=\text {Subst}\left (\int \frac {\left (a+b x^2\right )^{3/2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=-\frac {1}{2} b \coth (x) \sqrt {a+b \coth ^2(x)}-\frac {1}{2} \text {Subst}\left (\int \frac {-a (2 a+b)-b (3 a+2 b) x^2}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )\\ &=-\frac {1}{2} b \coth (x) \sqrt {a+b \coth ^2(x)}+(a+b)^2 \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\coth (x)\right )-\frac {1}{2} (b (3 a+2 b)) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\coth (x)\right )\\ &=-\frac {1}{2} b \coth (x) \sqrt {a+b \coth ^2(x)}+(a+b)^2 \text {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )-\frac {1}{2} (b (3 a+2 b)) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )\\ &=-\frac {1}{2} \sqrt {b} (3 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )+(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )-\frac {1}{2} b \coth (x) \sqrt {a+b \coth ^2(x)}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 111, normalized size = 1.26 \begin {gather*} \frac {1}{2} \left (-2 (-a-b)^{3/2} \text {ArcTan}\left (\frac {\coth (x) \sqrt {a+b \coth ^2(x)}-\sqrt {b} \text {csch}^2(x)}{\sqrt {-a-b}}\right )-b \coth (x) \sqrt {a+b \coth ^2(x)}+\sqrt {b} (3 a+2 b) \log \left (-\sqrt {b} \coth (x)+\sqrt {a+b \coth ^2(x)}\right )\right ) \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. \(472\) vs.
\(2(70)=140\).
time = 0.77, size = 473, normalized size = 5.38
method | result | size |
derivativedivides | \(\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}-\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}\) | \(473\) |
default | \(\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}-\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}\) | \(473\) |
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. 947 vs.
\(2 (70) = 140\).
time = 0.56, size = 5037, normalized size = 57.24 \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}}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (b\,{\mathrm {coth}\left (x\right )}^2+a\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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