Optimal. Leaf size=118 \[ \frac{a (a+2 b) \tanh (x)}{b^2 (a+b)^2 \sqrt{a+b \tanh ^2(x)}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right )}{b^{5/2}}+\frac{a \tanh ^3(x)}{3 b (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}} \]
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Rubi [A] time = 0.220451, antiderivative size = 118, 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, 470, 578, 523, 217, 206, 377} \[ \frac{a (a+2 b) \tanh (x)}{b^2 (a+b)^2 \sqrt{a+b \tanh ^2(x)}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right )}{b^{5/2}}+\frac{a \tanh ^3(x)}{3 b (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 470
Rule 578
Rule 523
Rule 217
Rule 206
Rule 377
Rubi steps
\begin{align*} \int \frac{\tanh ^6(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx &=\operatorname{Subst}\left (\int \frac{x^6}{\left (1-x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\tanh (x)\right )\\ &=\frac{a \tanh ^3(x)}{3 b (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (3 a-3 (a+b) x^2\right )}{\left (1-x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )}{3 b (a+b)}\\ &=\frac{a \tanh ^3(x)}{3 b (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac{a (a+2 b) \tanh (x)}{b^2 (a+b)^2 \sqrt{a+b \tanh ^2(x)}}-\frac{\operatorname{Subst}\left (\int \frac{3 a (a+2 b)-3 (a+b)^2 x^2}{\left (1-x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tanh (x)\right )}{3 b^2 (a+b)^2}\\ &=\frac{a \tanh ^3(x)}{3 b (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac{a (a+2 b) \tanh (x)}{b^2 (a+b)^2 \sqrt{a+b \tanh ^2(x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\tanh (x)\right )}{b^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{a+b x^2}} \, dx,x,\tanh (x)\right )}{(a+b)^2}\\ &=\frac{a \tanh ^3(x)}{3 b (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac{a (a+2 b) \tanh (x)}{b^2 (a+b)^2 \sqrt{a+b \tanh ^2(x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right )}{b^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-(a+b) x^2} \, dx,x,\frac{\tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right )}{(a+b)^2}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right )}{b^{5/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b} \tanh (x)}{\sqrt{a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}}+\frac{a \tanh ^3(x)}{3 b (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac{a (a+2 b) \tanh (x)}{b^2 (a+b)^2 \sqrt{a+b \tanh ^2(x)}}\\ \end{align*}
Mathematica [C] time = 1.86165, size = 231, normalized size = 1.96 \[ \frac{\sqrt{\text{sech}^2(x) ((a+b) \cosh (2 x)+a-b)} \left (\frac{a (a+b) \sinh (2 x) \left (\left (3 a^2+10 a b+7 b^2\right ) \cosh (2 x)+3 a^2+2 a b-7 b^2\right )}{((a+b) \cosh (2 x)+a-b)^2}-\frac{3 \sqrt{2} a \coth (x) \left (\left (a^2+3 a b+2 b^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}}}{\sqrt{2}}\right ),1\right )+b^2 \Pi \left (\frac{b}{a+b};\left .\sin ^{-1}\left (\frac{\sqrt{\frac{(a-b+(a+b) \cosh (2 x)) \text{csch}^2(x)}{b}}}{\sqrt{2}}\right )\right |1\right )\right )}{b \sqrt{\frac{\text{csch}^2(x) ((a+b) \cosh (2 x)+a-b)}{b}}}\right )}{3 \sqrt{2} b^2 (a+b)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.032, size = 549, normalized size = 4.7 \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 \frac{\tanh \left (x\right )^{6}}{{\left (b \tanh \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{6}{\left (x \right )}}{\left (a + b \tanh ^{2}{\left (x \right )}\right )^{\frac{5}{2}}}\, dx \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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