Optimal. Leaf size=328 \[ \frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{5/2}}+\frac{128 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \sqrt{\frac{1}{a x}+1}}+\frac{163 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{262 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (\frac{1}{a x}+1\right )^{5/2}}-\frac{269}{21 a c^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}}-\frac{62}{21 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{5/2}}-\frac{11}{7 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{5/2}}-\frac{8}{7 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{5/2}}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^4} \]
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Rubi [A] time = 0.228436, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6194, 103, 152, 12, 92, 208} \[ \frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{5/2}}+\frac{128 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \sqrt{\frac{1}{a x}+1}}+\frac{163 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{262 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (\frac{1}{a x}+1\right )^{5/2}}-\frac{269}{21 a c^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}}-\frac{62}{21 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{5/2}}-\frac{11}{7 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{5/2}}-\frac{8}{7 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (\frac{1}{a x}+1\right )^{5/2}}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^4} \]
Antiderivative was successfully verified.
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Rule 6194
Rule 103
Rule 152
Rule 12
Rule 92
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^4} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-\frac{x}{a}\right )^{9/2} \left (1+\frac{x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{c^4}\\ &=\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{a}-\frac{7 x}{a^2}}{x \left (1-\frac{x}{a}\right )^{9/2} \left (1+\frac{x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{c^4}\\ &=-\frac{8}{7 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{a \operatorname{Subst}\left (\int \frac{\frac{7}{a^2}+\frac{48 x}{a^3}}{x \left (1-\frac{x}{a}\right )^{7/2} \left (1+\frac{x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{7 c^4}\\ &=-\frac{8}{7 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{11}{7 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{a^2 \operatorname{Subst}\left (\int \frac{-\frac{35}{a^3}-\frac{275 x}{a^4}}{x \left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{35 c^4}\\ &=-\frac{8}{7 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{11}{7 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{62}{21 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{a^3 \operatorname{Subst}\left (\int \frac{\frac{105}{a^4}+\frac{1240 x}{a^5}}{x \left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{105 c^4}\\ &=-\frac{8}{7 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{11}{7 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{62}{21 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{269}{21 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{a^4 \operatorname{Subst}\left (\int \frac{-\frac{105}{a^5}-\frac{4035 x}{a^6}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{105 c^4}\\ &=-\frac{8}{7 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{11}{7 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{62}{21 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{269}{21 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{262 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{a^5 \operatorname{Subst}\left (\int \frac{-\frac{525}{a^6}-\frac{7860 x}{a^7}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{525 c^4}\\ &=-\frac{8}{7 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{11}{7 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{62}{21 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{269}{21 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{262 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{163 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{a^6 \operatorname{Subst}\left (\int \frac{-\frac{1575}{a^7}-\frac{7335 x}{a^8}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{1575 c^4}\\ &=-\frac{8}{7 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{11}{7 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{62}{21 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{269}{21 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{262 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{163 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{128 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{a^7 \operatorname{Subst}\left (\int -\frac{1575}{a^8 x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{1575 c^4}\\ &=-\frac{8}{7 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{11}{7 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{62}{21 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{269}{21 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{262 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{163 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{128 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a c^4}\\ &=-\frac{8}{7 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{11}{7 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{62}{21 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{269}{21 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{262 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{163 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{128 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{1}{a}-\frac{x^2}{a}} \, dx,x,\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a^2 c^4}\\ &=-\frac{8}{7 a c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{11}{7 a c^4 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{62}{21 a c^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{269}{21 a c^4 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{262 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{163 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{128 \sqrt{1-\frac{1}{a x}}}{35 a c^4 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^4 \left (1-\frac{1}{a x}\right )^{7/2} \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a c^4}\\ \end{align*}
Mathematica [A] time = 0.26741, size = 115, normalized size = 0.35 \[ \frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (105 a^7 x^7-281 a^6 x^6-559 a^5 x^5+965 a^4 x^4+715 a^3 x^3-1065 a^2 x^2-279 a x+384\right )}{105 (a x-1)^4 (a x+1)^3}+\log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a c^4} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.156, size = 898, normalized size = 2.7 \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 [A] time = 1.01233, size = 311, normalized size = 0.95 \begin{align*} \frac{1}{6720} \, a{\left (\frac{5 \,{\left (\frac{39 \,{\left (a x - 1\right )}}{a x + 1} + \frac{287 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{2611 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac{5628 \,{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 3\right )}}{a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}} - a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}}} + \frac{7 \,{\left (3 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 50 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 705 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{2} c^{4}} + \frac{6720 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac{6720 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43686, size = 602, normalized size = 1.84 \begin{align*} \frac{105 \,{\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 105 \,{\left (a^{6} x^{6} - 2 \, a^{5} x^{5} - a^{4} x^{4} + 4 \, a^{3} x^{3} - a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (105 \, a^{7} x^{7} - 281 \, a^{6} x^{6} - 559 \, a^{5} x^{5} + 965 \, a^{4} x^{4} + 715 \, a^{3} x^{3} - 1065 \, a^{2} x^{2} - 279 \, a x + 384\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{105 \,{\left (a^{7} c^{4} x^{6} - 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} + 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15102, size = 386, normalized size = 1.18 \begin{align*} \frac{1}{6720} \, a{\left (\frac{6720 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac{6720 \, \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{4}} - \frac{5 \,{\left (a x + 1\right )}^{3}{\left (\frac{42 \,{\left (a x - 1\right )}}{a x + 1} + \frac{329 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{2940 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 3\right )}}{{\left (a x - 1\right )}^{3} a^{2} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}} - \frac{13440 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{4}{\left (\frac{a x - 1}{a x + 1} - 1\right )}} + \frac{7 \,{\left (\frac{50 \,{\left (a x - 1\right )} a^{8} c^{16} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} + \frac{3 \,{\left (a x - 1\right )}^{2} a^{8} c^{16} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + 705 \, a^{8} c^{16} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{10} c^{20}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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