Optimal. Leaf size=145 \[ -\frac{11}{a c^3 \sqrt{c-\frac{c}{a x}}}-\frac{11}{3 a c^2 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{11 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{7/2}}+\frac{x}{\left (c-\frac{c}{a x}\right )^{7/2}}-\frac{11}{5 a c \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{11}{7 a \left (c-\frac{c}{a x}\right )^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.229653, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6167, 6133, 25, 514, 375, 78, 51, 63, 208} \[ -\frac{11}{a c^3 \sqrt{c-\frac{c}{a x}}}-\frac{11}{3 a c^2 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{11 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{7/2}}+\frac{x}{\left (c-\frac{c}{a x}\right )^{7/2}}-\frac{11}{5 a c \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{11}{7 a \left (c-\frac{c}{a x}\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6167
Rule 6133
Rule 25
Rule 514
Rule 375
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{7/2}} \, dx &=-\int \frac{e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{7/2}} \, dx\\ &=-\int \frac{1+a x}{\left (c-\frac{c}{a x}\right )^{7/2} (1-a x)} \, dx\\ &=\frac{c \int \frac{1+a x}{\left (c-\frac{c}{a x}\right )^{9/2} x} \, dx}{a}\\ &=\frac{c \int \frac{a+\frac{1}{x}}{\left (c-\frac{c}{a x}\right )^{9/2}} \, dx}{a}\\ &=-\frac{c \operatorname{Subst}\left (\int \frac{a+x}{x^2 \left (c-\frac{c x}{a}\right )^{9/2}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\frac{x}{\left (c-\frac{c}{a x}\right )^{7/2}}-\frac{(11 c) \operatorname{Subst}\left (\int \frac{1}{x \left (c-\frac{c x}{a}\right )^{9/2}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{11}{7 a \left (c-\frac{c}{a x}\right )^{7/2}}+\frac{x}{\left (c-\frac{c}{a x}\right )^{7/2}}-\frac{11 \operatorname{Subst}\left (\int \frac{1}{x \left (c-\frac{c x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{11}{7 a \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{11}{5 a c \left (c-\frac{c}{a x}\right )^{5/2}}+\frac{x}{\left (c-\frac{c}{a x}\right )^{7/2}}-\frac{11 \operatorname{Subst}\left (\int \frac{1}{x \left (c-\frac{c x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{2 a c}\\ &=-\frac{11}{7 a \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{11}{5 a c \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{11}{3 a c^2 \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{x}{\left (c-\frac{c}{a x}\right )^{7/2}}-\frac{11 \operatorname{Subst}\left (\int \frac{1}{x \left (c-\frac{c x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{2 a c^2}\\ &=-\frac{11}{7 a \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{11}{5 a c \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{11}{3 a c^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{11}{a c^3 \sqrt{c-\frac{c}{a x}}}+\frac{x}{\left (c-\frac{c}{a x}\right )^{7/2}}-\frac{11 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a c^3}\\ &=-\frac{11}{7 a \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{11}{5 a c \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{11}{3 a c^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{11}{a c^3 \sqrt{c-\frac{c}{a x}}}+\frac{x}{\left (c-\frac{c}{a x}\right )^{7/2}}+\frac{11 \operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{c^4}\\ &=-\frac{11}{7 a \left (c-\frac{c}{a x}\right )^{7/2}}-\frac{11}{5 a c \left (c-\frac{c}{a x}\right )^{5/2}}-\frac{11}{3 a c^2 \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{11}{a c^3 \sqrt{c-\frac{c}{a x}}}+\frac{x}{\left (c-\frac{c}{a x}\right )^{7/2}}+\frac{11 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{7/2}}\\ \end{align*}
Mathematica [C] time = 0.0408748, size = 46, normalized size = 0.32 \[ \frac{7 x-\frac{11 \text{Hypergeometric2F1}\left (-\frac{7}{2},1,-\frac{5}{2},1-\frac{1}{a x}\right )}{a}}{7 \left (c-\frac{c}{a x}\right )^{7/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.174, size = 396, normalized size = 2.7 \begin{align*}{\frac{x}{210\,{c}^{4} \left ( ax-1 \right ) ^{5}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 2310\,{a}^{11/2}\sqrt{ \left ( ax-1 \right ) x}{x}^{5}+1155\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ){x}^{5}{a}^{5}-2100\,{a}^{9/2} \left ( \left ( ax-1 \right ) x \right ) ^{3/2}{x}^{3}-11550\,\sqrt{ \left ( ax-1 \right ) x}{a}^{9/2}{x}^{4}-5775\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ){x}^{4}{a}^{4}+5368\, \left ( \left ( ax-1 \right ) x \right ) ^{3/2}{a}^{7/2}{x}^{2}+23100\,\sqrt{ \left ( ax-1 \right ) x}{a}^{7/2}{x}^{3}+11550\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ){x}^{3}{a}^{3}-4928\, \left ( \left ( ax-1 \right ) x \right ) ^{3/2}{a}^{5/2}x-23100\,{a}^{5/2}\sqrt{ \left ( ax-1 \right ) x}{x}^{2}-11550\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ){x}^{2}{a}^{2}+1540\,{a}^{3/2} \left ( \left ( ax-1 \right ) x \right ) ^{3/2}+11550\,{a}^{3/2}\sqrt{ \left ( ax-1 \right ) x}x+5775\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) xa-2310\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}-1155\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \right ){\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{\frac{1}{\sqrt{a}}}} \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{a x + 1}{{\left (a x - 1\right )}{\left (c - \frac{c}{a x}\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.80492, size = 774, normalized size = 5.34 \begin{align*} \left [\frac{1155 \,{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt{c} \log \left (-2 \, a c x - 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right ) + 2 \,{\left (105 \, a^{5} x^{5} - 1936 \, a^{4} x^{4} + 4466 \, a^{3} x^{3} - 3850 \, a^{2} x^{2} + 1155 \, a x\right )} \sqrt{\frac{a c x - c}{a x}}}{210 \,{\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}}, -\frac{1155 \,{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right ) -{\left (105 \, a^{5} x^{5} - 1936 \, a^{4} x^{4} + 4466 \, a^{3} x^{3} - 3850 \, a^{2} x^{2} + 1155 \, a x\right )} \sqrt{\frac{a c x - c}{a x}}}{105 \,{\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.31902, size = 250, normalized size = 1.72 \begin{align*} -\frac{1}{105} \, a c{\left (\frac{2 \,{\left (30 \, c^{3} + \frac{63 \,{\left (a c x - c\right )} c^{2}}{a x} + \frac{140 \,{\left (a c x - c\right )}^{2} c}{a^{2} x^{2}} + \frac{525 \,{\left (a c x - c\right )}^{3}}{a^{3} x^{3}}\right )} a x^{3}}{{\left (a c x - c\right )}^{3} c^{4} \sqrt{\frac{a c x - c}{a x}}} + \frac{1155 \, \arctan \left (\frac{\sqrt{\frac{a c x - c}{a x}}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c^{4}} - \frac{105 \, \sqrt{\frac{a c x - c}{a x}}}{a^{2}{\left (c - \frac{a c x - c}{a x}\right )} c^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]