Optimal. Leaf size=123 \[ -\frac{\left (3 a^2+12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right )}{8 \sqrt{b}}-\frac{1}{4} b \coth ^3(x) \sqrt{a+b \coth ^2(x)}-\frac{1}{8} (5 a+4 b) \coth (x) \sqrt{a+b \coth ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right ) \]
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Rubi [A] time = 0.244035, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {3670, 477, 582, 523, 217, 206, 377} \[ -\frac{\left (3 a^2+12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right )}{8 \sqrt{b}}-\frac{1}{4} b \coth ^3(x) \sqrt{a+b \coth ^2(x)}-\frac{1}{8} (5 a+4 b) \coth (x) \sqrt{a+b \coth ^2(x)}+(a+b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right ) \]
Antiderivative was successfully verified.
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Rule 3670
Rule 477
Rule 582
Rule 523
Rule 217
Rule 206
Rule 377
Rubi steps
\begin{align*} \int \coth ^2(x) \left (a+b \coth ^2(x)\right )^{3/2} \, dx &=\operatorname{Subst}\left (\int \frac{x^2 \left (a+b x^2\right )^{3/2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=-\frac{1}{4} b \coth ^3(x) \sqrt{a+b \coth ^2(x)}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2 \left (-a (4 a+3 b)-b (5 a+4 b) x^2\right )}{\left (1-x^2\right ) \sqrt{a+b x^2}} \, dx,x,\coth (x)\right )\\ &=-\frac{1}{8} (5 a+4 b) \coth (x) \sqrt{a+b \coth ^2(x)}-\frac{1}{4} b \coth ^3(x) \sqrt{a+b \coth ^2(x)}-\frac{\operatorname{Subst}\left (\int \frac{-a b (5 a+4 b)-b \left (3 a^2+12 a b+8 b^2\right ) x^2}{\left (1-x^2\right ) \sqrt{a+b x^2}} \, dx,x,\coth (x)\right )}{8 b}\\ &=-\frac{1}{8} (5 a+4 b) \coth (x) \sqrt{a+b \coth ^2(x)}-\frac{1}{4} b \coth ^3(x) \sqrt{a+b \coth ^2(x)}+(a+b)^2 \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{a+b x^2}} \, dx,x,\coth (x)\right )-\frac{1}{8} \left (3 a^2+12 a b+8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\coth (x)\right )\\ &=-\frac{1}{8} (5 a+4 b) \coth (x) \sqrt{a+b \coth ^2(x)}-\frac{1}{4} b \coth ^3(x) \sqrt{a+b \coth ^2(x)}+(a+b)^2 \operatorname{Subst}\left (\int \frac{1}{1-(a+b) x^2} \, dx,x,\frac{\coth (x)}{\sqrt{a+b \coth ^2(x)}}\right )-\frac{1}{8} \left (3 a^2+12 a b+8 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\coth (x)}{\sqrt{a+b \coth ^2(x)}}\right )\\ &=-\frac{\left (3 a^2+12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right )}{8 \sqrt{b}}+(a+b)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b} \coth (x)}{\sqrt{a+b \coth ^2(x)}}\right )-\frac{1}{8} (5 a+4 b) \coth (x) \sqrt{a+b \coth ^2(x)}-\frac{1}{4} b \coth ^3(x) \sqrt{a+b \coth ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.880448, size = 219, normalized size = 1.78 \[ \frac{\sinh (x) \sqrt{\text{csch}^2(x) ((a+b) \cosh (2 x)-a+b)} \left (\sqrt{b} \left (8 \sqrt{2} (a+b)^2 \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a+b} \cosh (x)}{\sqrt{(a+b) \cosh (2 x)-a+b}}\right )-\sqrt{a+b} \coth (x) \text{csch}(x) \sqrt{(a+b) \cosh (2 x)-a+b} \left (5 a+2 b \text{csch}^2(x)+6 b\right )\right )-\sqrt{2} \sqrt{a+b} \left (3 a^2+12 a b+8 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{b} \cosh (x)}{\sqrt{(a+b) \cosh (2 x)-a+b}}\right )\right )}{8 \sqrt{2} \sqrt{b} \sqrt{a+b} \sqrt{(a+b) \cosh (2 x)-a+b}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 633, normalized size = 5.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \coth \left (x\right )^{2} + a\right )}^{\frac{3}{2}} \coth \left (x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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