Optimal. Leaf size=124 \[ \frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left (\frac{a+b \tanh ^2(x)}{\sqrt{a+b} \sqrt{a+b \tanh ^4(x)}}\right )-\frac{1}{6} \left (a+b \tanh ^4(x)\right )^{3/2}-\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh ^2(x)}{\sqrt{a+b \tanh ^4(x)}}\right )-\frac{1}{4} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt{a+b \tanh ^4(x)} \]
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Rubi [A] time = 0.241808, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {3670, 1248, 735, 815, 844, 217, 206, 725} \[ \frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left (\frac{a+b \tanh ^2(x)}{\sqrt{a+b} \sqrt{a+b \tanh ^4(x)}}\right )-\frac{1}{6} \left (a+b \tanh ^4(x)\right )^{3/2}-\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh ^2(x)}{\sqrt{a+b \tanh ^4(x)}}\right )-\frac{1}{4} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt{a+b \tanh ^4(x)} \]
Antiderivative was successfully verified.
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Rule 3670
Rule 1248
Rule 735
Rule 815
Rule 844
Rule 217
Rule 206
Rule 725
Rubi steps
\begin{align*} \int \tanh (x) \left (a+b \tanh ^4(x)\right )^{3/2} \, dx &=\operatorname{Subst}\left (\int \frac{x \left (a+b x^4\right )^{3/2}}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^{3/2}}{1-x} \, dx,x,\tanh ^2(x)\right )\\ &=-\frac{1}{6} \left (a+b \tanh ^4(x)\right )^{3/2}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{(-a-b x) \sqrt{a+b x^2}}{1-x} \, dx,x,\tanh ^2(x)\right )\\ &=-\frac{1}{4} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt{a+b \tanh ^4(x)}-\frac{1}{6} \left (a+b \tanh ^4(x)\right )^{3/2}-\frac{\operatorname{Subst}\left (\int \frac{-a b (2 a+b)-b^2 (3 a+2 b) x}{(1-x) \sqrt{a+b x^2}} \, dx,x,\tanh ^2(x)\right )}{4 b}\\ &=-\frac{1}{4} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt{a+b \tanh ^4(x)}-\frac{1}{6} \left (a+b \tanh ^4(x)\right )^{3/2}+\frac{1}{2} (a+b)^2 \operatorname{Subst}\left (\int \frac{1}{(1-x) \sqrt{a+b x^2}} \, dx,x,\tanh ^2(x)\right )-\frac{1}{4} (b (3 a+2 b)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\tanh ^2(x)\right )\\ &=-\frac{1}{4} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt{a+b \tanh ^4(x)}-\frac{1}{6} \left (a+b \tanh ^4(x)\right )^{3/2}-\frac{1}{2} (a+b)^2 \operatorname{Subst}\left (\int \frac{1}{a+b-x^2} \, dx,x,\frac{-a-b \tanh ^2(x)}{\sqrt{a+b \tanh ^4(x)}}\right )-\frac{1}{4} (b (3 a+2 b)) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\tanh ^2(x)}{\sqrt{a+b \tanh ^4(x)}}\right )\\ &=-\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh ^2(x)}{\sqrt{a+b \tanh ^4(x)}}\right )+\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left (\frac{a+b \tanh ^2(x)}{\sqrt{a+b} \sqrt{a+b \tanh ^4(x)}}\right )-\frac{1}{4} \left (2 (a+b)+b \tanh ^2(x)\right ) \sqrt{a+b \tanh ^4(x)}-\frac{1}{6} \left (a+b \tanh ^4(x)\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 4.39531, size = 166, normalized size = 1.34 \[ \frac{1}{12} \left (6 (a+b)^{3/2} \tanh ^{-1}\left (\frac{a+b \tanh ^2(x)}{\sqrt{a+b} \sqrt{a+b \tanh ^4(x)}}\right )-6 \sqrt{b} (a+b) \tanh ^{-1}\left (\frac{\sqrt{b} \tanh ^2(x)}{\sqrt{a+b \tanh ^4(x)}}\right )-\sqrt{a+b \tanh ^4(x)} \left (8 a+2 b \tanh ^4(x)+3 b \tanh ^2(x)+6 b\right )-\frac{3 \sqrt{a} \sqrt{b} \sqrt{a+b \tanh ^4(x)} \sinh ^{-1}\left (\frac{\sqrt{b} \tanh ^2(x)}{\sqrt{a}}\right )}{\sqrt{\frac{b \tanh ^4(x)}{a}+1}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.082, size = 620, normalized size = 5. \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 \tanh \left (x\right )^{4} + a\right )}^{\frac{3}{2}} \tanh \left (x\right )\,{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 \left (a + b \tanh ^{4}{\left (x \right )}\right )^{\frac{3}{2}} \tanh{\left (x \right )}\, dx \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{\left (b \tanh \left (x\right )^{4} + a\right )}^{\frac{3}{2}} \tanh \left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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