Optimal. Leaf size=108 \[ \frac{b (2 a+b)}{a^2 (a+b)^2 \sqrt{a+b \coth ^2(x)}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right )}{(a+b)^{5/2}} \]
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Rubi [A] time = 0.204791, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {3670, 446, 85, 152, 156, 63, 208} \[ \frac{b (2 a+b)}{a^2 (a+b)^2 \sqrt{a+b \coth ^2(x)}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right )}{(a+b)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 446
Rule 85
Rule 152
Rule 156
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\tanh (x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x \left (1-x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\coth (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(1-x) x (a+b x)^{5/2}} \, dx,x,\coth ^2(x)\right )\\ &=\frac{b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{-a-b+b x}{(1-x) x (a+b x)^{3/2}} \, dx,x,\coth ^2(x)\right )}{2 a (a+b)}\\ &=\frac{b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac{b (2 a+b)}{a^2 (a+b)^2 \sqrt{a+b \coth ^2(x)}}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{2} (a+b)^2+\frac{1}{2} b (2 a+b) x}{(1-x) x \sqrt{a+b x}} \, dx,x,\coth ^2(x)\right )}{a^2 (a+b)^2}\\ &=\frac{b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac{b (2 a+b)}{a^2 (a+b)^2 \sqrt{a+b \coth ^2(x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\coth ^2(x)\right )}{2 a^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{(1-x) \sqrt{a+b x}} \, dx,x,\coth ^2(x)\right )}{2 (a+b)^2}\\ &=\frac{b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac{b (2 a+b)}{a^2 (a+b)^2 \sqrt{a+b \coth ^2(x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \coth ^2(x)}\right )}{a^2 b}+\frac{\operatorname{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)^2}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \coth ^2(x)}}{\sqrt{a+b}}\right )}{(a+b)^{5/2}}+\frac{b}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac{b (2 a+b)}{a^2 (a+b)^2 \sqrt{a+b \coth ^2(x)}}\\ \end{align*}
Mathematica [C] time = 0.0600422, size = 73, normalized size = 0.68 \[ \frac{(a+b) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{b \coth ^2(x)}{a}+1\right )-a \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{b \coth ^2(x)+a}{a+b}\right )}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.159, size = 0, normalized size = 0. \begin{align*} \int{\tanh \left ( x \right ) \left ( a+b \left ({\rm coth} \left (x\right ) \right ) ^{2} \right ) ^{-{\frac{5}{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 \frac{\tanh \left (x\right )}{{\left (b \coth \left (x\right )^{2} + a\right )}^{\frac{5}{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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