Optimal. Leaf size=113 \[ -\frac{2 a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}-\frac{2 a^3 \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}-4 a^3 \sqrt{c-\frac{c}{a x}}+4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.416017, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6167, 6133, 25, 514, 446, 88, 50, 63, 208} \[ -\frac{2 a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}-\frac{2 a^3 \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}-4 a^3 \sqrt{c-\frac{c}{a x}}+4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6167
Rule 6133
Rule 25
Rule 514
Rule 446
Rule 88
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^4} \, dx &=-\int \frac{e^{-2 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^4} \, dx\\ &=-\int \frac{\sqrt{c-\frac{c}{a x}} (1-a x)}{x^4 (1+a x)} \, dx\\ &=\frac{a \int \frac{\left (c-\frac{c}{a x}\right )^{3/2}}{x^3 (1+a x)} \, dx}{c}\\ &=\frac{a \int \frac{\left (c-\frac{c}{a x}\right )^{3/2}}{\left (a+\frac{1}{x}\right ) x^4} \, dx}{c}\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{x^2 \left (c-\frac{c x}{a}\right )^{3/2}}{a+x} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{a \operatorname{Subst}\left (\int \left (\frac{a^2 \left (c-\frac{c x}{a}\right )^{3/2}}{a+x}-\frac{a \left (c-\frac{c x}{a}\right )^{5/2}}{c}\right ) \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{2 a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}-\frac{a^3 \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^{3/2}}{a+x} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{2 a^3 \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}-\frac{2 a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}-\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{a+x} \, dx,x,\frac{1}{x}\right )\\ &=-4 a^3 \sqrt{c-\frac{c}{a x}}-\frac{2 a^3 \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}-\frac{2 a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}-\left (4 a^3 c\right ) \operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-4 a^3 \sqrt{c-\frac{c}{a x}}-\frac{2 a^3 \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}-\frac{2 a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}+\left (8 a^4\right ) \operatorname{Subst}\left (\int \frac{1}{2 a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )\\ &=-4 a^3 \sqrt{c-\frac{c}{a x}}-\frac{2 a^3 \left (c-\frac{c}{a x}\right )^{3/2}}{3 c}-\frac{2 a^3 \left (c-\frac{c}{a x}\right )^{7/2}}{7 c^3}+4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right )\\ \end{align*}
Mathematica [A] time = 0.134165, size = 87, normalized size = 0.77 \[ \frac{2 \left (-52 a^3 x^3+16 a^2 x^2-9 a x+3\right ) \sqrt{c-\frac{c}{a x}}}{21 x^3}+4 \sqrt{2} a^3 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{2} \sqrt{c}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.168, size = 302, normalized size = 2.7 \begin{align*}{\frac{1}{21\,{x}^{4}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -126\,\sqrt{a{x}^{2}-x}{a}^{9/2}\sqrt{{a}^{-1}}{x}^{5}+42\,{a}^{9/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}{x}^{5}+84\, \left ( a{x}^{2}-x \right ) ^{3/2}{a}^{7/2}\sqrt{{a}^{-1}}{x}^{3}+63\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}-x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \sqrt{{a}^{-1}}{x}^{5}{a}^{4}-42\,{a}^{7/2}\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax-1 \right ) x}a-3\,ax+1}{ax+1}} \right ){x}^{5}-63\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \sqrt{{a}^{-1}}{x}^{5}{a}^{4}-20\,{a}^{5/2}\sqrt{{a}^{-1}} \left ( a{x}^{2}-x \right ) ^{3/2}{x}^{2}+12\,{a}^{3/2} \left ( a{x}^{2}-x \right ) ^{3/2}x\sqrt{{a}^{-1}}-6\, \left ( a{x}^{2}-x \right ) ^{3/2}\sqrt{a}\sqrt{{a}^{-1}} \right ){\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{a}^{-1}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x - 1\right )} \sqrt{c - \frac{c}{a x}}}{{\left (a x + 1\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.7059, size = 466, normalized size = 4.12 \begin{align*} \left [\frac{2 \,{\left (21 \, \sqrt{2} a^{3} \sqrt{c} x^{3} \log \left (-\frac{2 \, \sqrt{2} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + 3 \, a c x - c}{a x + 1}\right ) -{\left (52 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 9 \, a x - 3\right )} \sqrt{\frac{a c x - c}{a x}}\right )}}{21 \, x^{3}}, -\frac{2 \,{\left (42 \, \sqrt{2} a^{3} \sqrt{-c} x^{3} \arctan \left (\frac{\sqrt{2} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{2 \, c}\right ) +{\left (52 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 9 \, a x - 3\right )} \sqrt{\frac{a c x - c}{a x}}\right )}}{21 \, x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- c \left (-1 + \frac{1}{a x}\right )} \left (a x - 1\right )}{x^{4} \left (a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 2.2953, size = 481, normalized size = 4.26 \begin{align*} -\frac{4 \, \sqrt{2} a^{4} c \arctan \left (\frac{\sqrt{2}{\left ({\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )} a + \sqrt{c}{\left | a \right |}\right )}}{2 \, a \sqrt{-c}}\right )}{\sqrt{-c}{\left | a \right |} \mathrm{sgn}\left (x\right )} - \frac{2 \,{\left (84 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )}^{6} a^{7} c - 84 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )}^{5} a^{6} c^{\frac{3}{2}}{\left | a \right |} + 112 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )}^{4} a^{7} c^{2} - 105 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )}^{3} a^{6} c^{\frac{5}{2}}{\left | a \right |} + 63 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )}^{2} a^{7} c^{3} - 21 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )} a^{6} c^{\frac{7}{2}}{\left | a \right |} + 3 \, a^{7} c^{4}\right )}}{21 \,{\left (\sqrt{a^{2} c} x - \sqrt{a^{2} c x^{2} - a c x}\right )}^{7} a^{3}{\left | a \right |} \mathrm{sgn}\left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]