Optimal. Leaf size=167 \[ -\frac{\left (6 a^2+18 a+11\right ) \sqrt{a+b x+1} (-a-b x+1)^{3/2}}{6 b^3}-\frac{\left (6 a^2+18 a+11\right ) \sqrt{a+b x+1} \sqrt{-a-b x+1}}{2 b^3}-\frac{\left (6 a^2+18 a+11\right ) \sin ^{-1}(a+b x)}{2 b^3}-\frac{\sqrt{a+b x+1} (-a-b x+1)^{5/2}}{3 b^3}-\frac{(a+1)^2 (-a-b x+1)^{5/2}}{b^3 \sqrt{a+b x+1}} \]
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Rubi [A] time = 0.200624, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6163, 89, 80, 50, 53, 619, 216} \[ -\frac{\left (6 a^2+18 a+11\right ) \sqrt{a+b x+1} (-a-b x+1)^{3/2}}{6 b^3}-\frac{\left (6 a^2+18 a+11\right ) \sqrt{a+b x+1} \sqrt{-a-b x+1}}{2 b^3}-\frac{\left (6 a^2+18 a+11\right ) \sin ^{-1}(a+b x)}{2 b^3}-\frac{\sqrt{a+b x+1} (-a-b x+1)^{5/2}}{3 b^3}-\frac{(a+1)^2 (-a-b x+1)^{5/2}}{b^3 \sqrt{a+b x+1}} \]
Antiderivative was successfully verified.
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Rule 6163
Rule 89
Rule 80
Rule 50
Rule 53
Rule 619
Rule 216
Rubi steps
\begin{align*} \int e^{-3 \tanh ^{-1}(a+b x)} x^2 \, dx &=\int \frac{x^2 (1-a-b x)^{3/2}}{(1+a+b x)^{3/2}} \, dx\\ &=-\frac{(1+a)^2 (1-a-b x)^{5/2}}{b^3 \sqrt{1+a+b x}}+\frac{\int \frac{(1-a-b x)^{3/2} \left (-(1+a) (3+2 a) b+b^2 x\right )}{\sqrt{1+a+b x}} \, dx}{b^3}\\ &=-\frac{(1+a)^2 (1-a-b x)^{5/2}}{b^3 \sqrt{1+a+b x}}-\frac{(1-a-b x)^{5/2} \sqrt{1+a+b x}}{3 b^3}-\frac{\left (11+18 a+6 a^2\right ) \int \frac{(1-a-b x)^{3/2}}{\sqrt{1+a+b x}} \, dx}{3 b^2}\\ &=-\frac{(1+a)^2 (1-a-b x)^{5/2}}{b^3 \sqrt{1+a+b x}}-\frac{\left (11+18 a+6 a^2\right ) (1-a-b x)^{3/2} \sqrt{1+a+b x}}{6 b^3}-\frac{(1-a-b x)^{5/2} \sqrt{1+a+b x}}{3 b^3}-\frac{\left (11+18 a+6 a^2\right ) \int \frac{\sqrt{1-a-b x}}{\sqrt{1+a+b x}} \, dx}{2 b^2}\\ &=-\frac{(1+a)^2 (1-a-b x)^{5/2}}{b^3 \sqrt{1+a+b x}}-\frac{\left (11+18 a+6 a^2\right ) \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 b^3}-\frac{\left (11+18 a+6 a^2\right ) (1-a-b x)^{3/2} \sqrt{1+a+b x}}{6 b^3}-\frac{(1-a-b x)^{5/2} \sqrt{1+a+b x}}{3 b^3}-\frac{\left (11+18 a+6 a^2\right ) \int \frac{1}{\sqrt{1-a-b x} \sqrt{1+a+b x}} \, dx}{2 b^2}\\ &=-\frac{(1+a)^2 (1-a-b x)^{5/2}}{b^3 \sqrt{1+a+b x}}-\frac{\left (11+18 a+6 a^2\right ) \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 b^3}-\frac{\left (11+18 a+6 a^2\right ) (1-a-b x)^{3/2} \sqrt{1+a+b x}}{6 b^3}-\frac{(1-a-b x)^{5/2} \sqrt{1+a+b x}}{3 b^3}-\frac{\left (11+18 a+6 a^2\right ) \int \frac{1}{\sqrt{(1-a) (1+a)-2 a b x-b^2 x^2}} \, dx}{2 b^2}\\ &=-\frac{(1+a)^2 (1-a-b x)^{5/2}}{b^3 \sqrt{1+a+b x}}-\frac{\left (11+18 a+6 a^2\right ) \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 b^3}-\frac{\left (11+18 a+6 a^2\right ) (1-a-b x)^{3/2} \sqrt{1+a+b x}}{6 b^3}-\frac{(1-a-b x)^{5/2} \sqrt{1+a+b x}}{3 b^3}+\frac{\left (11+18 a+6 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{4 b^2}}} \, dx,x,-2 a b-2 b^2 x\right )}{4 b^4}\\ &=-\frac{(1+a)^2 (1-a-b x)^{5/2}}{b^3 \sqrt{1+a+b x}}-\frac{\left (11+18 a+6 a^2\right ) \sqrt{1-a-b x} \sqrt{1+a+b x}}{2 b^3}-\frac{\left (11+18 a+6 a^2\right ) (1-a-b x)^{3/2} \sqrt{1+a+b x}}{6 b^3}-\frac{(1-a-b x)^{5/2} \sqrt{1+a+b x}}{3 b^3}-\frac{\left (11+18 a+6 a^2\right ) \sin ^{-1}(a+b x)}{2 b^3}\\ \end{align*}
Mathematica [A] time = 0.169403, size = 190, normalized size = 1.14 \[ -\frac{\sqrt{-b} \left (a^3 (2 b x+51)+a^2 (69 b x+50)+2 a^4+a \left (2 b^3 x^3+9 b^2 x^2+106 b x-51\right )+2 b^4 x^4-9 b^3 x^3+26 b^2 x^2+33 b x-52\right )+6 \left (6 a^2+18 a+11\right ) \sqrt{b} \sqrt{-a^2-2 a b x-b^2 x^2+1} \sinh ^{-1}\left (\frac{\sqrt{-b} \sqrt{-a-b x+1}}{\sqrt{2} \sqrt{b}}\right )}{6 (-b)^{7/2} \sqrt{-(a+b x-1) (a+b x+1)}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.048, size = 830, 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 [C] time = 1.49084, size = 840, normalized size = 5.03 \begin{align*} \frac{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac{3}{2}} a^{2}}{b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3} + 2 \, b^{4} x + 2 \, a b^{3} + b^{3}} + \frac{2 \,{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac{3}{2}} a}{b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3} + 2 \, b^{4} x + 2 \, a b^{3} + b^{3}} - \frac{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac{3}{2}} a}{b^{4} x + a b^{3} + b^{3}} - \frac{6 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a^{2}}{b^{4} x + a b^{3} + b^{3}} + \frac{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac{3}{2}}}{b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3} + 2 \, b^{4} x + 2 \, a b^{3} + b^{3}} - \frac{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac{3}{2}}}{b^{4} x + a b^{3} + b^{3}} - \frac{12 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{4} x + a b^{3} + b^{3}} - \frac{3 \, a^{2} \arcsin \left (b x + a\right )}{b^{3}} - \frac{6 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{4} x + a b^{3} + b^{3}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 4 \, b x + 4 \, a + 3} x}{2 \, b^{2}} - \frac{9 \, a \arcsin \left (b x + a\right )}{b^{3}} + \frac{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac{3}{2}}}{3 \, b^{3}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 4 \, b x + 4 \, a + 3} a}{2 \, b^{3}} - \frac{3 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} a}{b^{3}} - \frac{i \, \arcsin \left (b x + a + 2\right )}{2 \, b^{3}} - \frac{6 \, \arcsin \left (b x + a\right )}{b^{3}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 4 \, b x + 4 \, a + 3}}{b^{3}} - \frac{3 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01453, size = 375, normalized size = 2.25 \begin{align*} \frac{3 \,{\left (6 \, a^{3} +{\left (6 \, a^{2} + 18 \, a + 11\right )} b x + 24 \, a^{2} + 29 \, a + 11\right )} \arctan \left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x + a\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) -{\left (2 \, b^{3} x^{3} - 7 \, b^{2} x^{2} + 2 \, a^{3} +{\left (16 \, a + 19\right )} b x + 53 \, a^{2} + 103 \, a + 52\right )} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{6 \,{\left (b^{4} x +{\left (a + 1\right )} b^{3}\right )}} \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 [A] time = 1.19749, size = 224, normalized size = 1.34 \begin{align*} -\frac{1}{6} \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (x{\left (\frac{2 \, x}{b} - \frac{2 \, a b^{6} + 9 \, b^{6}}{b^{8}}\right )} + \frac{2 \, a^{2} b^{5} + 27 \, a b^{5} + 28 \, b^{5}}{b^{8}}\right )} + \frac{{\left (6 \, a^{2} + 18 \, a + 11\right )} \arcsin \left (-b x - a\right ) \mathrm{sgn}\left (b\right )}{2 \, b^{2}{\left | b \right |}} + \frac{8 \,{\left (a^{2} + 2 \, a + 1\right )}}{b^{2}{\left (\frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left | b \right |} + b}{b^{2} x + a b} + 1\right )}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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