Optimal. Leaf size=375 \[ \frac{7 a^6 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^3 (a x+1)^3}+\frac{3 a^5 x^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{8 (1-a x)^3 (a x+1)^2}-\frac{19 a^4 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^3 (a x+1)}+\frac{2 a^3 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{3 (1-a x)^2 (a x+1)}-\frac{23 a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{120 (1-a x) (a x+1)}-\frac{a x^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{15 (a x+1)}+\frac{x (1-a x) \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{6 (a x+1)}-\frac{2 a^6 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} \sin ^{-1}(a x)}{(1-a x)^{7/2} (a x+1)^{7/2}}+\frac{25 a^6 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{16 (1-a x)^{7/2} (a x+1)^{7/2}} \]
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Rubi [A] time = 0.541287, antiderivative size = 375, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458, Rules used = {6167, 6159, 6129, 97, 149, 154, 157, 41, 216, 92, 208} \[ \frac{7 a^6 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^3 (a x+1)^3}+\frac{3 a^5 x^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{8 (1-a x)^3 (a x+1)^2}-\frac{19 a^4 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{16 (1-a x)^3 (a x+1)}+\frac{2 a^3 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{3 (1-a x)^2 (a x+1)}-\frac{23 a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{120 (1-a x) (a x+1)}-\frac{a x^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{15 (a x+1)}+\frac{x (1-a x) \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}{6 (a x+1)}-\frac{2 a^6 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} \sin ^{-1}(a x)}{(1-a x)^{7/2} (a x+1)^{7/2}}+\frac{25 a^6 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{a x+1}\right )}{16 (1-a x)^{7/2} (a x+1)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6159
Rule 6129
Rule 97
Rule 149
Rule 154
Rule 157
Rule 41
Rule 216
Rule 92
Rule 208
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{7/2} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{7/2} \, dx\\ &=-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{e^{-2 \tanh ^{-1}(a x)} (1-a x)^{7/2} (1+a x)^{7/2}}{x^7} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{(1-a x)^{9/2} (1+a x)^{5/2}}{x^7} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{(1-a x)^{7/2} (1+a x)^{3/2} \left (-2 a-7 a^2 x\right )}{x^6} \, dx}{6 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{(1-a x)^{5/2} (1+a x)^{3/2} \left (-23 a^2+37 a^3 x\right )}{x^5} \, dx}{30 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}-\frac{23 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{(1-a x)^{3/2} (1+a x)^{3/2} \left (240 a^3-125 a^4 x\right )}{x^4} \, dx}{120 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}+\frac{2 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}-\frac{23 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{\sqrt{1-a x} (1+a x)^{3/2} \left (-855 a^4+135 a^5 x\right )}{x^3} \, dx}{360 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}-\frac{19 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}+\frac{2 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}-\frac{23 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{(1+a x)^{3/2} \left (270 a^5+585 a^6 x\right )}{x^2 \sqrt{1-a x}} \, dx}{720 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac{3 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^6}{8 (1-a x)^3 (1+a x)^2}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}-\frac{19 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}+\frac{2 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}-\frac{23 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{\sqrt{1+a x} \left (1125 a^6+315 a^7 x\right )}{x \sqrt{1-a x}} \, dx}{720 (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac{7 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}+\frac{3 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^6}{8 (1-a x)^3 (1+a x)^2}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}-\frac{19 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}+\frac{2 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}-\frac{23 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}+\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{-1125 a^7-1440 a^8 x}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{720 a (1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac{7 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}+\frac{3 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^6}{8 (1-a x)^3 (1+a x)^2}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}-\frac{19 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}+\frac{2 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}-\frac{23 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}-\frac{\left (25 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{1}{x \sqrt{1-a x} \sqrt{1+a x}} \, dx}{16 (1-a x)^{7/2} (1+a x)^{7/2}}-\frac{\left (2 a^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac{7 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}+\frac{3 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^6}{8 (1-a x)^3 (1+a x)^2}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}-\frac{19 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}+\frac{2 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}-\frac{23 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}+\frac{\left (25 a^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\right ) \operatorname{Subst}\left (\int \frac{1}{a-a x^2} \, dx,x,\sqrt{1-a x} \sqrt{1+a x}\right )}{16 (1-a x)^{7/2} (1+a x)^{7/2}}-\frac{\left (2 a^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{(1-a x)^{7/2} (1+a x)^{7/2}}\\ &=\frac{7 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}{16 (1-a x)^3 (1+a x)^3}+\frac{3 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^6}{8 (1-a x)^3 (1+a x)^2}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}{15 (1+a x)}-\frac{19 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^5}{16 (1-a x)^3 (1+a x)}+\frac{2 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}{3 (1-a x)^2 (1+a x)}-\frac{23 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}{120 (1-a x) (1+a x)}+\frac{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x (1-a x)}{6 (1+a x)}-\frac{2 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7 \sin ^{-1}(a x)}{(1-a x)^{7/2} (1+a x)^{7/2}}+\frac{25 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7 \tanh ^{-1}\left (\sqrt{1-a x} \sqrt{1+a x}\right )}{16 (1-a x)^{7/2} (1+a x)^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.171492, size = 150, normalized size = 0.4 \[ \frac{c^3 \sqrt{c-\frac{c}{a^2 x^2}} \left (\sqrt{a^2 x^2-1} \left (240 a^6 x^6+736 a^5 x^5+105 a^4 x^4-352 a^3 x^3+70 a^2 x^2+96 a x-40\right )-480 a^6 x^6 \log \left (\sqrt{a^2 x^2-1}+a x\right )+375 a^6 x^6 \tan ^{-1}\left (\frac{1}{\sqrt{a^2 x^2-1}}\right )\right )}{240 a^6 x^5 \sqrt{a^2 x^2-1}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.263, size = 795, normalized size = 2.1 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x - 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{7}{2}}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81637, size = 971, normalized size = 2.59 \begin{align*} \left [\frac{960 \, a^{5} \sqrt{-c} c^{3} x^{5} \arctan \left (\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 375 \, a^{5} \sqrt{-c} c^{3} x^{5} \log \left (-\frac{a^{2} c x^{2} - 2 \, a \sqrt{-c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{x^{2}}\right ) + 2 \,{\left (240 \, a^{6} c^{3} x^{6} + 736 \, a^{5} c^{3} x^{5} + 105 \, a^{4} c^{3} x^{4} - 352 \, a^{3} c^{3} x^{3} + 70 \, a^{2} c^{3} x^{2} + 96 \, a c^{3} x - 40 \, c^{3}\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{480 \, a^{6} x^{5}}, \frac{375 \, a^{5} c^{\frac{7}{2}} x^{5} \arctan \left (\frac{a \sqrt{c} x \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + 240 \, a^{5} c^{\frac{7}{2}} x^{5} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) +{\left (240 \, a^{6} c^{3} x^{6} + 736 \, a^{5} c^{3} x^{5} + 105 \, a^{4} c^{3} x^{4} - 352 \, a^{3} c^{3} x^{3} + 70 \, a^{2} c^{3} x^{2} + 96 \, a c^{3} x - 40 \, c^{3}\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{240 \, a^{6} x^{5}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 37.5511, size = 1059, normalized size = 2.82 \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 [A] time = 69.2752, size = 757, normalized size = 2.02 \begin{align*} -\frac{1}{120} \,{\left (\frac{375 \, c^{\frac{7}{2}} \arctan \left (-\frac{\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}}{\sqrt{c}}\right ) \mathrm{sgn}\left (x\right )}{a^{2}} - \frac{240 \, c^{\frac{7}{2}} \log \left ({\left | -\sqrt{a^{2} c} x + \sqrt{a^{2} c x^{2} - c} \right |}\right ) \mathrm{sgn}\left (x\right )}{a{\left | a \right |}} - \frac{120 \, \sqrt{a^{2} c x^{2} - c} c^{3} \mathrm{sgn}\left (x\right )}{a^{2}} + \frac{105 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{11} c^{4}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 1440 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{10} a c^{\frac{9}{2}} \mathrm{sgn}\left (x\right ) + 595 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{9} c^{5}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 4320 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{8} a c^{\frac{11}{2}} \mathrm{sgn}\left (x\right ) - 150 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{7} c^{6}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 7360 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{6} a c^{\frac{13}{2}} \mathrm{sgn}\left (x\right ) + 150 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{5} c^{7}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 6720 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{4} a c^{\frac{15}{2}} \mathrm{sgn}\left (x\right ) - 595 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{3} c^{8}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 2976 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} a c^{\frac{17}{2}} \mathrm{sgn}\left (x\right ) - 105 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )} c^{9}{\left | a \right |} \mathrm{sgn}\left (x\right ) - 736 \, a c^{\frac{19}{2}} \mathrm{sgn}\left (x\right )}{{\left ({\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - c}\right )}^{2} + c\right )}^{6} a^{2}{\left | a \right |}}\right )}{\left | a \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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