Optimal. Leaf size=191 \[ -\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac{c^4 (5 a x+24) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}+\frac{c^4 (5 a x+16) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac{3 c^4 (16-5 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}-\frac{15 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{16 a}-\frac{3 c^4 \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.366286, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {6157, 6148, 1807, 811, 813, 844, 216, 266, 63, 208} \[ -\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac{c^4 (5 a x+24) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}+\frac{c^4 (5 a x+16) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac{3 c^4 (16-5 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}-\frac{15 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{16 a}-\frac{3 c^4 \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 6157
Rule 6148
Rule 1807
Rule 811
Rule 813
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{3 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^4 \, dx &=\frac{c^4 \int \frac{e^{3 \tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^4}{x^8} \, dx}{a^8}\\ &=\frac{c^4 \int \frac{(1+a x)^3 \left (1-a^2 x^2\right )^{5/2}}{x^8} \, dx}{a^8}\\ &=-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac{c^4 \int \frac{\left (1-a^2 x^2\right )^{5/2} \left (-21 a-21 a^2 x-7 a^3 x^2\right )}{x^7} \, dx}{7 a^8}\\ &=-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}+\frac{c^4 \int \frac{\left (126 a^2+21 a^3 x\right ) \left (1-a^2 x^2\right )^{5/2}}{x^6} \, dx}{42 a^8}\\ &=-\frac{c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac{c^4 \int \frac{\left (1008 a^4+210 a^5 x\right ) \left (1-a^2 x^2\right )^{3/2}}{x^4} \, dx}{336 a^8}\\ &=\frac{c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac{c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}+\frac{c^4 \int \frac{\left (4032 a^6+1260 a^7 x\right ) \sqrt{1-a^2 x^2}}{x^2} \, dx}{1344 a^8}\\ &=-\frac{3 c^4 (16-5 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}+\frac{c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac{c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac{c^4 \int \frac{-2520 a^7+8064 a^8 x}{x \sqrt{1-a^2 x^2}} \, dx}{2688 a^8}\\ &=-\frac{3 c^4 (16-5 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}+\frac{c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac{c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\left (3 c^4\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx+\frac{\left (15 c^4\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{16 a}\\ &=-\frac{3 c^4 (16-5 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}+\frac{c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac{c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac{3 c^4 \sin ^{-1}(a x)}{a}+\frac{\left (15 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{32 a}\\ &=-\frac{3 c^4 (16-5 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}+\frac{c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac{c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac{3 c^4 \sin ^{-1}(a x)}{a}-\frac{\left (15 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{16 a^3}\\ &=-\frac{3 c^4 (16-5 a x) \sqrt{1-a^2 x^2}}{16 a^2 x}+\frac{c^4 (16+5 a x) \left (1-a^2 x^2\right )^{3/2}}{16 a^4 x^3}-\frac{c^4 (24+5 a x) \left (1-a^2 x^2\right )^{5/2}}{40 a^6 x^5}-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{7 a^8 x^7}-\frac{c^4 \left (1-a^2 x^2\right )^{7/2}}{2 a^7 x^6}-\frac{3 c^4 \sin ^{-1}(a x)}{a}-\frac{15 c^4 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{16 a}\\ \end{align*}
Mathematica [C] time = 0.209166, size = 191, normalized size = 1. \[ \frac{c^4 \left (-336 a^2 x^2 \text{Hypergeometric2F1}\left (-\frac{5}{2},-\frac{5}{2},-\frac{3}{2},a^2 x^2\right )-\frac{5 \left (16 a^7 x^7 \left (a^2 x^2-1\right )^4 \text{Hypergeometric2F1}\left (3,\frac{7}{2},\frac{9}{2},1-a^2 x^2\right )+16 a^8 x^8-231 a^7 x^7-64 a^6 x^6+413 a^5 x^5+96 a^4 x^4-238 a^3 x^3-64 a^2 x^2-105 a^7 x^7 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+56 a x+16\right )}{\sqrt{1-a^2 x^2}}\right )}{560 a^8 x^7} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.089, size = 273, normalized size = 1.4 \begin{align*} -{\frac{37\,{c}^{4}}{16\,{x}^{2}{a}^{3}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{218\,{c}^{4}}{35\,{a}^{2}x}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{{c}^{4}a{x}^{2}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-3\,{\frac{{c}^{4}}{\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{{c}^{4}}{2\,{a}^{7}{x}^{6}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{{c}^{4}}{35\,{a}^{6}{x}^{5}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{15\,{c}^{4}}{8\,{a}^{5}{x}^{4}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{68\,{c}^{4}}{35\,{a}^{4}{x}^{3}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{156\,{c}^{4}x}{35}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{15\,{c}^{4}}{16\,a}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{31\,{c}^{4}}{16\,a}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{{c}^{4}}{7\,{a}^{8}{x}^{7}}{\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 [B] time = 1.53111, size = 1022, normalized size = 5.35 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22048, size = 398, normalized size = 2.08 \begin{align*} \frac{3360 \, a^{7} c^{4} x^{7} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + 525 \, a^{7} c^{4} x^{7} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) + 560 \, a^{7} c^{4} x^{7} +{\left (560 \, a^{7} c^{4} x^{7} - 2496 \, a^{6} c^{4} x^{6} - 525 \, a^{5} c^{4} x^{5} + 992 \, a^{4} c^{4} x^{4} + 770 \, a^{3} c^{4} x^{3} - 96 \, a^{2} c^{4} x^{2} - 280 \, a c^{4} x - 80 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1}}{560 \, a^{8} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 40.3596, size = 935, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23801, size = 682, normalized size = 3.57 \begin{align*} \frac{{\left (5 \, c^{4} + \frac{35 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} c^{4}}{a^{2} x} + \frac{49 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{4}}{a^{4} x^{2}} - \frac{245 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{4}}{a^{6} x^{3}} - \frac{875 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} c^{4}}{a^{8} x^{4}} + \frac{455 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5} c^{4}}{a^{10} x^{5}} + \frac{9065 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6} c^{4}}{a^{12} x^{6}}\right )} a^{14} x^{7}}{4480 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{7}{\left | a \right |}} - \frac{3 \, c^{4} \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} - \frac{15 \, c^{4} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{16 \,{\left | a \right |}} + \frac{\sqrt{-a^{2} x^{2} + 1} c^{4}}{a} - \frac{\frac{9065 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{4} c^{4}}{x} + \frac{455 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} a^{2} c^{4}}{x^{2}} - \frac{875 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3} c^{4}}{x^{3}} - \frac{245 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4} c^{4}}{a^{2} x^{4}} + \frac{49 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5} c^{4}}{a^{4} x^{5}} + \frac{35 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6} c^{4}}{a^{6} x^{6}} + \frac{5 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{7} c^{4}}{a^{8} x^{7}}}{4480 \, a^{6}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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