Optimal. Leaf size=123 \[ \frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac{7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac{7}{8} c^4 x \sqrt{1-a^2 x^2}+\frac{7 c^4 \sin ^{-1}(a x)}{8 a} \]
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Rubi [A] time = 0.0804764, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {6127, 671, 641, 195, 216} \[ \frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac{7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac{7}{8} c^4 x \sqrt{1-a^2 x^2}+\frac{7 c^4 \sin ^{-1}(a x)}{8 a} \]
Antiderivative was successfully verified.
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Rule 6127
Rule 671
Rule 641
Rule 195
Rule 216
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} (c-a c x)^4 \, dx &=c \int (c-a c x)^3 \sqrt{1-a^2 x^2} \, dx\\ &=\frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{1}{5} \left (7 c^2\right ) \int (c-a c x)^2 \sqrt{1-a^2 x^2} \, dx\\ &=\frac{7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{1}{4} \left (7 c^3\right ) \int (c-a c x) \sqrt{1-a^2 x^2} \, dx\\ &=\frac{7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac{7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{1}{4} \left (7 c^4\right ) \int \sqrt{1-a^2 x^2} \, dx\\ &=\frac{7}{8} c^4 x \sqrt{1-a^2 x^2}+\frac{7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac{7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{1}{8} \left (7 c^4\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{7}{8} c^4 x \sqrt{1-a^2 x^2}+\frac{7 c^4 \left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac{7 c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac{c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}{5 a}+\frac{7 c^4 \sin ^{-1}(a x)}{8 a}\\ \end{align*}
Mathematica [A] time = 0.106434, size = 75, normalized size = 0.61 \[ -\frac{c^4 \left (\sqrt{1-a^2 x^2} \left (24 a^4 x^4-90 a^3 x^3+112 a^2 x^2-15 a x-136\right )+210 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{120 a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.045, size = 137, normalized size = 1.1 \begin{align*} -{\frac{{c}^{4}{a}^{3}{x}^{4}}{5}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{14\,{c}^{4}a{x}^{2}}{15}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{17\,{c}^{4}}{15\,a}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{3\,{a}^{2}{c}^{4}{x}^{3}}{4}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{{c}^{4}x}{8}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{7\,{c}^{4}}{8}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43655, size = 171, normalized size = 1.39 \begin{align*} -\frac{1}{5} \, \sqrt{-a^{2} x^{2} + 1} a^{3} c^{4} x^{4} + \frac{3}{4} \, \sqrt{-a^{2} x^{2} + 1} a^{2} c^{4} x^{3} - \frac{14}{15} \, \sqrt{-a^{2} x^{2} + 1} a c^{4} x^{2} + \frac{1}{8} \, \sqrt{-a^{2} x^{2} + 1} c^{4} x + \frac{7 \, c^{4} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{8 \, \sqrt{a^{2}}} + \frac{17 \, \sqrt{-a^{2} x^{2} + 1} c^{4}}{15 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57251, size = 209, normalized size = 1.7 \begin{align*} -\frac{210 \, c^{4} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (24 \, a^{4} c^{4} x^{4} - 90 \, a^{3} c^{4} x^{3} + 112 \, a^{2} c^{4} x^{2} - 15 \, a c^{4} x - 136 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1}}{120 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.06879, size = 226, normalized size = 1.84 \begin{align*} \begin{cases} \frac{3 c^{4} \sqrt{- a^{2} x^{2} + 1} + 2 c^{4} \left (\begin{cases} - \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (a x \right )}}{2} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right ) + 2 c^{4} \left (\begin{cases} \frac{\left (- a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{3} - \sqrt{- a^{2} x^{2} + 1} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right ) - 3 c^{4} \left (\begin{cases} \frac{a x \left (- 2 a^{2} x^{2} + 1\right ) \sqrt{- a^{2} x^{2} + 1}}{8} - \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{3 \operatorname{asin}{\left (a x \right )}}{8} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right ) + c^{4} \left (\begin{cases} - \frac{\left (- a^{2} x^{2} + 1\right )^{\frac{5}{2}}}{5} + \frac{2 \left (- a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{3} - \sqrt{- a^{2} x^{2} + 1} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right ) + c^{4} \operatorname{asin}{\left (a x \right )}}{a} & \text{for}\: a \neq 0 \\c^{4} x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23868, size = 105, normalized size = 0.85 \begin{align*} \frac{7 \, c^{4} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{8 \,{\left | a \right |}} + \frac{1}{120} \, \sqrt{-a^{2} x^{2} + 1}{\left (\frac{136 \, c^{4}}{a} +{\left (15 \, c^{4} - 2 \,{\left (56 \, a c^{4} + 3 \,{\left (4 \, a^{3} c^{4} x - 15 \, a^{2} c^{4}\right )} x\right )} x\right )} x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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