Optimal. Leaf size=136 \[ -\frac{1}{6} a c x^5 \sqrt{1-a^2 x^2}-\frac{3}{5} c x^4 \sqrt{1-a^2 x^2}-\frac{23 c x^3 \sqrt{1-a^2 x^2}}{24 a}-\frac{17 c x^2 \sqrt{1-a^2 x^2}}{15 a^2}-\frac{c (345 a x+544) \sqrt{1-a^2 x^2}}{240 a^4}+\frac{23 c \sin ^{-1}(a x)}{16 a^4} \]
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Rubi [A] time = 0.27617, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {6148, 1809, 833, 780, 216} \[ -\frac{1}{6} a c x^5 \sqrt{1-a^2 x^2}-\frac{3}{5} c x^4 \sqrt{1-a^2 x^2}-\frac{23 c x^3 \sqrt{1-a^2 x^2}}{24 a}-\frac{17 c x^2 \sqrt{1-a^2 x^2}}{15 a^2}-\frac{c (345 a x+544) \sqrt{1-a^2 x^2}}{240 a^4}+\frac{23 c \sin ^{-1}(a x)}{16 a^4} \]
Antiderivative was successfully verified.
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Rule 6148
Rule 1809
Rule 833
Rule 780
Rule 216
Rubi steps
\begin{align*} \int e^{3 \tanh ^{-1}(a x)} x^3 \left (c-a^2 c x^2\right ) \, dx &=c \int \frac{x^3 (1+a x)^3}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{1}{6} a c x^5 \sqrt{1-a^2 x^2}-\frac{c \int \frac{x^3 \left (-6 a^2-23 a^3 x-18 a^4 x^2\right )}{\sqrt{1-a^2 x^2}} \, dx}{6 a^2}\\ &=-\frac{3}{5} c x^4 \sqrt{1-a^2 x^2}-\frac{1}{6} a c x^5 \sqrt{1-a^2 x^2}+\frac{c \int \frac{x^3 \left (102 a^4+115 a^5 x\right )}{\sqrt{1-a^2 x^2}} \, dx}{30 a^4}\\ &=-\frac{23 c x^3 \sqrt{1-a^2 x^2}}{24 a}-\frac{3}{5} c x^4 \sqrt{1-a^2 x^2}-\frac{1}{6} a c x^5 \sqrt{1-a^2 x^2}-\frac{c \int \frac{x^2 \left (-345 a^5-408 a^6 x\right )}{\sqrt{1-a^2 x^2}} \, dx}{120 a^6}\\ &=-\frac{17 c x^2 \sqrt{1-a^2 x^2}}{15 a^2}-\frac{23 c x^3 \sqrt{1-a^2 x^2}}{24 a}-\frac{3}{5} c x^4 \sqrt{1-a^2 x^2}-\frac{1}{6} a c x^5 \sqrt{1-a^2 x^2}+\frac{c \int \frac{x \left (816 a^6+1035 a^7 x\right )}{\sqrt{1-a^2 x^2}} \, dx}{360 a^8}\\ &=-\frac{17 c x^2 \sqrt{1-a^2 x^2}}{15 a^2}-\frac{23 c x^3 \sqrt{1-a^2 x^2}}{24 a}-\frac{3}{5} c x^4 \sqrt{1-a^2 x^2}-\frac{1}{6} a c x^5 \sqrt{1-a^2 x^2}-\frac{c (544+345 a x) \sqrt{1-a^2 x^2}}{240 a^4}+\frac{(23 c) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{16 a^3}\\ &=-\frac{17 c x^2 \sqrt{1-a^2 x^2}}{15 a^2}-\frac{23 c x^3 \sqrt{1-a^2 x^2}}{24 a}-\frac{3}{5} c x^4 \sqrt{1-a^2 x^2}-\frac{1}{6} a c x^5 \sqrt{1-a^2 x^2}-\frac{c (544+345 a x) \sqrt{1-a^2 x^2}}{240 a^4}+\frac{23 c \sin ^{-1}(a x)}{16 a^4}\\ \end{align*}
Mathematica [A] time = 0.107107, size = 70, normalized size = 0.51 \[ \frac{345 c \sin ^{-1}(a x)-c \sqrt{1-a^2 x^2} \left (40 a^5 x^5+144 a^4 x^4+230 a^3 x^3+272 a^2 x^2+345 a x+544\right )}{240 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.083, size = 191, normalized size = 1.4 \begin{align*}{\frac{{a}^{3}c{x}^{7}}{6}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{19\,ac{x}^{5}}{24}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{23\,c{x}^{3}}{48\,a}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{23\,cx}{16\,{a}^{3}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{23\,c}{16\,{a}^{3}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{3\,{a}^{2}c{x}^{6}}{5}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{8\,c{x}^{4}}{15}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{17\,c{x}^{2}}{15\,{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{34\,c}{15\,{a}^{4}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45072, size = 244, normalized size = 1.79 \begin{align*} \frac{a^{3} c x^{7}}{6 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{3 \, a^{2} c x^{6}}{5 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{19 \, a c x^{5}}{24 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{8 \, c x^{4}}{15 \, \sqrt{-a^{2} x^{2} + 1}} + \frac{23 \, c x^{3}}{48 \, \sqrt{-a^{2} x^{2} + 1} a} + \frac{17 \, c x^{2}}{15 \, \sqrt{-a^{2} x^{2} + 1} a^{2}} - \frac{23 \, c x}{16 \, \sqrt{-a^{2} x^{2} + 1} a^{3}} + \frac{23 \, c \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{16 \, \sqrt{a^{2}} a^{3}} - \frac{34 \, c}{15 \, \sqrt{-a^{2} x^{2} + 1} a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.7007, size = 220, normalized size = 1.62 \begin{align*} -\frac{690 \, c \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (40 \, a^{5} c x^{5} + 144 \, a^{4} c x^{4} + 230 \, a^{3} c x^{3} + 272 \, a^{2} c x^{2} + 345 \, a c x + 544 \, c\right )} \sqrt{-a^{2} x^{2} + 1}}{240 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 20.1568, size = 483, normalized size = 3.55 \begin{align*} a^{3} c \left (\begin{cases} - \frac{i x^{7}}{6 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{5}}{24 a^{2} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i x^{3}}{48 a^{4} \sqrt{a^{2} x^{2} - 1}} + \frac{5 i x}{16 a^{6} \sqrt{a^{2} x^{2} - 1}} - \frac{5 i \operatorname{acosh}{\left (a x \right )}}{16 a^{7}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{x^{7}}{6 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{5}}{24 a^{2} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 x^{3}}{48 a^{4} \sqrt{- a^{2} x^{2} + 1}} - \frac{5 x}{16 a^{6} \sqrt{- a^{2} x^{2} + 1}} + \frac{5 \operatorname{asin}{\left (a x \right )}}{16 a^{7}} & \text{otherwise} \end{cases}\right ) + 3 a^{2} c \left (\begin{cases} - \frac{x^{4} \sqrt{- a^{2} x^{2} + 1}}{5 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1}}{15 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1}}{15 a^{6}} & \text{for}\: a \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases}\right ) + 3 a c \left (\begin{cases} - \frac{i x^{5}}{4 \sqrt{a^{2} x^{2} - 1}} - \frac{i x^{3}}{8 a^{2} \sqrt{a^{2} x^{2} - 1}} + \frac{3 i x}{8 a^{4} \sqrt{a^{2} x^{2} - 1}} - \frac{3 i \operatorname{acosh}{\left (a x \right )}}{8 a^{5}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{x^{5}}{4 \sqrt{- a^{2} x^{2} + 1}} + \frac{x^{3}}{8 a^{2} \sqrt{- a^{2} x^{2} + 1}} - \frac{3 x}{8 a^{4} \sqrt{- a^{2} x^{2} + 1}} + \frac{3 \operatorname{asin}{\left (a x \right )}}{8 a^{5}} & \text{otherwise} \end{cases}\right ) + c \left (\begin{cases} - \frac{x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a^{2}} - \frac{2 \sqrt{- a^{2} x^{2} + 1}}{3 a^{4}} & \text{for}\: a \neq 0 \\\frac{x^{4}}{4} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16678, size = 105, normalized size = 0.77 \begin{align*} -\frac{1}{240} \, \sqrt{-a^{2} x^{2} + 1}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \, a c x + 18 \, c\right )} x + \frac{115 \, c}{a}\right )} x + \frac{136 \, c}{a^{2}}\right )} x + \frac{345 \, c}{a^{3}}\right )} x + \frac{544 \, c}{a^{4}}\right )} + \frac{23 \, c \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{16 \, a^{3}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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