Optimal. Leaf size=138 \[ \frac{86 \left (1-a^2 x^2\right )^{3/2}}{105 a^4 c^4 (1-a x)^3}-\frac{19 \left (1-a^2 x^2\right )^{3/2}}{35 a^4 c^4 (1-a x)^4}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^4 c^4 (1-a x)^5}-\frac{2 \sqrt{1-a^2 x^2}}{a^4 c^4 (1-a x)}+\frac{\sin ^{-1}(a x)}{a^4 c^4} \]
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Rubi [A] time = 0.271601, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {6128, 1637, 659, 651, 663, 216} \[ \frac{86 \left (1-a^2 x^2\right )^{3/2}}{105 a^4 c^4 (1-a x)^3}-\frac{19 \left (1-a^2 x^2\right )^{3/2}}{35 a^4 c^4 (1-a x)^4}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^4 c^4 (1-a x)^5}-\frac{2 \sqrt{1-a^2 x^2}}{a^4 c^4 (1-a x)}+\frac{\sin ^{-1}(a x)}{a^4 c^4} \]
Antiderivative was successfully verified.
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Rule 6128
Rule 1637
Rule 659
Rule 651
Rule 663
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^3}{(c-a c x)^4} \, dx &=c \int \frac{x^3 \sqrt{1-a^2 x^2}}{(c-a c x)^5} \, dx\\ &=c \int \left (-\frac{\sqrt{1-a^2 x^2}}{a^3 c^5 (-1+a x)^5}-\frac{3 \sqrt{1-a^2 x^2}}{a^3 c^5 (-1+a x)^4}-\frac{3 \sqrt{1-a^2 x^2}}{a^3 c^5 (-1+a x)^3}-\frac{\sqrt{1-a^2 x^2}}{a^3 c^5 (-1+a x)^2}\right ) \, dx\\ &=-\frac{\int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^5} \, dx}{a^3 c^4}-\frac{\int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^2} \, dx}{a^3 c^4}-\frac{3 \int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^4} \, dx}{a^3 c^4}-\frac{3 \int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^3} \, dx}{a^3 c^4}\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{a^4 c^4 (1-a x)}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^4 c^4 (1-a x)^5}-\frac{3 \left (1-a^2 x^2\right )^{3/2}}{5 a^4 c^4 (1-a x)^4}+\frac{\left (1-a^2 x^2\right )^{3/2}}{a^4 c^4 (1-a x)^3}+\frac{2 \int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^4} \, dx}{7 a^3 c^4}+\frac{3 \int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^3} \, dx}{5 a^3 c^4}+\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a^3 c^4}\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{a^4 c^4 (1-a x)}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^4 c^4 (1-a x)^5}-\frac{19 \left (1-a^2 x^2\right )^{3/2}}{35 a^4 c^4 (1-a x)^4}+\frac{4 \left (1-a^2 x^2\right )^{3/2}}{5 a^4 c^4 (1-a x)^3}+\frac{\sin ^{-1}(a x)}{a^4 c^4}-\frac{2 \int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^3} \, dx}{35 a^3 c^4}\\ &=-\frac{2 \sqrt{1-a^2 x^2}}{a^4 c^4 (1-a x)}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a^4 c^4 (1-a x)^5}-\frac{19 \left (1-a^2 x^2\right )^{3/2}}{35 a^4 c^4 (1-a x)^4}+\frac{86 \left (1-a^2 x^2\right )^{3/2}}{105 a^4 c^4 (1-a x)^3}+\frac{\sin ^{-1}(a x)}{a^4 c^4}\\ \end{align*}
Mathematica [A] time = 0.233386, size = 94, normalized size = 0.68 \[ \frac{\sqrt{a x+1} \left (\sqrt{1-a^2 x^2} \left (296 a^3 x^3-659 a^2 x^2+559 a x-166\right )+105 (a x-1)^4 \sin ^{-1}(a x)\right )}{105 a^4 c^4 (1-a x)^{7/2} \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.065, size = 210, normalized size = 1.5 \begin{align*}{\frac{1}{{c}^{4}{a}^{3}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{43}{35\,{c}^{4}{a}^{7}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-3}}+{\frac{229}{105\,{a}^{6}{c}^{4}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}+{\frac{296}{105\,{c}^{4}{a}^{5}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}}+{\frac{2}{7\,{c}^{4}{a}^{8}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64324, size = 394, normalized size = 2.86 \begin{align*} -\frac{166 \, a^{4} x^{4} - 664 \, a^{3} x^{3} + 996 \, a^{2} x^{2} - 664 \, a x + 210 \,{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) -{\left (296 \, a^{3} x^{3} - 659 \, a^{2} x^{2} + 559 \, a x - 166\right )} \sqrt{-a^{2} x^{2} + 1} + 166}{105 \,{\left (a^{8} c^{4} x^{4} - 4 \, a^{7} c^{4} x^{3} + 6 \, a^{6} c^{4} x^{2} - 4 \, a^{5} c^{4} x + a^{4} c^{4}\right )}} \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*} \frac{\int \frac{x^{3}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{4}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22032, size = 297, normalized size = 2.15 \begin{align*} \frac{\arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{a^{3} c^{4}{\left | a \right |}} + \frac{2 \,{\left (\frac{1057 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{2751 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac{3640 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac{2170 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac{735 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac{105 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} - 166\right )}}{105 \, a^{3} c^{4}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{7}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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