Optimal. Leaf size=125 \[ -\frac{\left (8 a^2+12 a b+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a+b}}\right )}{8 (a+b)^{3/2}}-\frac{1}{4} \coth ^4(x) \sqrt{a+b \text{sech}^2(x)}-\frac{(4 a+3 b) \coth ^2(x) \sqrt{a+b \text{sech}^2(x)}}{8 (a+b)}+\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.220621, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {4139, 446, 99, 151, 156, 63, 208} \[ -\frac{\left (8 a^2+12 a b+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a+b}}\right )}{8 (a+b)^{3/2}}-\frac{1}{4} \coth ^4(x) \sqrt{a+b \text{sech}^2(x)}-\frac{(4 a+3 b) \coth ^2(x) \sqrt{a+b \text{sech}^2(x)}}{8 (a+b)}+\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Rule 4139
Rule 446
Rule 99
Rule 151
Rule 156
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \coth ^5(x) \sqrt{a+b \text{sech}^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2}}{x \left (-1+x^2\right )^3} \, dx,x,\text{sech}(x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{(-1+x)^3 x} \, dx,x,\text{sech}^2(x)\right )\\ &=-\frac{1}{4} \coth ^4(x) \sqrt{a+b \text{sech}^2(x)}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{-2 a-\frac{3 b x}{2}}{(-1+x)^2 x \sqrt{a+b x}} \, dx,x,\text{sech}^2(x)\right )\\ &=-\frac{(4 a+3 b) \coth ^2(x) \sqrt{a+b \text{sech}^2(x)}}{8 (a+b)}-\frac{1}{4} \coth ^4(x) \sqrt{a+b \text{sech}^2(x)}-\frac{\operatorname{Subst}\left (\int \frac{-2 a (a+b)-\frac{1}{4} b (4 a+3 b) x}{(-1+x) x \sqrt{a+b x}} \, dx,x,\text{sech}^2(x)\right )}{4 (a+b)}\\ &=-\frac{(4 a+3 b) \coth ^2(x) \sqrt{a+b \text{sech}^2(x)}}{8 (a+b)}-\frac{1}{4} \coth ^4(x) \sqrt{a+b \text{sech}^2(x)}-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\text{sech}^2(x)\right )+\frac{\left (8 a^2+12 a b+3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a+b x}} \, dx,x,\text{sech}^2(x)\right )}{16 (a+b)}\\ &=-\frac{(4 a+3 b) \coth ^2(x) \sqrt{a+b \text{sech}^2(x)}}{8 (a+b)}-\frac{1}{4} \coth ^4(x) \sqrt{a+b \text{sech}^2(x)}-\frac{a \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \text{sech}^2(x)}\right )}{b}+\frac{\left (8 a^2+12 a b+3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \text{sech}^2(x)}\right )}{8 b (a+b)}\\ &=\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a}}\right )-\frac{\left (8 a^2+12 a b+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}^2(x)}}{\sqrt{a+b}}\right )}{8 (a+b)^{3/2}}-\frac{(4 a+3 b) \coth ^2(x) \sqrt{a+b \text{sech}^2(x)}}{8 (a+b)}-\frac{1}{4} \coth ^4(x) \sqrt{a+b \text{sech}^2(x)}\\ \end{align*}
Mathematica [A] time = 0.909891, size = 191, normalized size = 1.53 \[ -\frac{\cosh (x) \sqrt{a+b \text{sech}^2(x)} \left (\sqrt{2} \left (8 a^2+12 a b+3 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a+b} \cosh (x)}{\sqrt{a \cosh (2 x)+a+2 b}}\right )+\sqrt{a+b} \left (\frac{1}{2} \coth (x) \text{csch}^3(x) \sqrt{a \cosh (2 x)+a+2 b} ((6 a+5 b) \cosh (2 x)-2 a-b)-8 \sqrt{2} \sqrt{a} (a+b) \log \left (\sqrt{a \cosh (2 x)+a+2 b}+\sqrt{2} \sqrt{a} \cosh (x)\right )\right )\right )}{8 (a+b)^{3/2} \sqrt{a \cosh (2 x)+a+2 b}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.128, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm coth} \left (x\right ) \right ) ^{5}\sqrt{a+b \left ({\rm sech} \left (x\right ) \right ) ^{2}}\, dx \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 \sqrt{b \operatorname{sech}\left (x\right )^{2} + a} \coth \left (x\right )^{5}\,{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \operatorname{sech}\left (x\right )^{2} + a} \coth \left (x\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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