Optimal. Leaf size=164 \[ \frac{64 c^4 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^4 x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{68 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}-\frac{7 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}-\frac{7 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{91 c^4 \csc ^{-1}(a x)}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.552419, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {6177, 1805, 1807, 1809, 844, 216, 266, 63, 208} \[ \frac{64 c^4 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^4 x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{68 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}-\frac{7 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}-\frac{7 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{91 c^4 \csc ^{-1}(a x)}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6177
Rule 1805
Rule 1807
Rule 1809
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^4 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^7}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{64 c^4 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\operatorname{Subst}\left (\int \frac{-c^7+\frac{7 c^7 x}{a}+\frac{42 c^7 x^2}{a^2}-\frac{22 c^7 x^3}{a^3}+\frac{7 c^7 x^4}{a^4}-\frac{c^7 x^5}{a^5}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{64 c^4 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^4 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{\operatorname{Subst}\left (\int \frac{-\frac{7 c^7}{a}-\frac{42 c^7 x}{a^2}+\frac{22 c^7 x^2}{a^3}-\frac{7 c^7 x^3}{a^4}+\frac{c^7 x^4}{a^5}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{64 c^4 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}+c^4 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{a^2 \operatorname{Subst}\left (\int \frac{\frac{21 c^7}{a^3}+\frac{126 c^7 x}{a^4}-\frac{68 c^7 x^2}{a^5}+\frac{21 c^7 x^3}{a^6}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{3 c^3}\\ &=\frac{64 c^4 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}-\frac{7 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{a^4 \operatorname{Subst}\left (\int \frac{-\frac{42 c^7}{a^5}-\frac{273 c^7 x}{a^6}+\frac{136 c^7 x^2}{a^7}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{6 c^3}\\ &=\frac{68 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}+\frac{64 c^4 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}-\frac{7 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{a^6 \operatorname{Subst}\left (\int \frac{\frac{42 c^7}{a^7}+\frac{273 c^7 x}{a^8}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{6 c^3}\\ &=\frac{68 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}+\frac{64 c^4 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}-\frac{7 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{\left (91 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a^2}+\frac{\left (7 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\frac{68 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}+\frac{64 c^4 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}-\frac{7 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{91 c^4 \csc ^{-1}(a x)}{2 a}+\frac{\left (7 c^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 a}\\ &=\frac{68 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}+\frac{64 c^4 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}-\frac{7 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{91 c^4 \csc ^{-1}(a x)}{2 a}-\left (7 a c^4\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ &=\frac{68 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}+\frac{64 c^4 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3 x^2}-\frac{7 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{91 c^4 \csc ^{-1}(a x)}{2 a}-\frac{7 c^4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}\\ \end{align*}
Mathematica [C] time = 1.09876, size = 567, normalized size = 3.46 \[ \frac{c^4 \left (1980 \sqrt{2} a^2 x^2 (a x+1) (a x-1)^4 \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{7}{2},\frac{9}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )+2772 \sqrt{2} a^3 x^3 (a x+1) (a x-1)^3 \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{5}{2},\frac{7}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )+35 \left (44 \sqrt{2} a x (a x-1)^5 (a x+1) \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{9}{2},\frac{11}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )+36 \sqrt{2} (a x-1)^6 (a x+1) \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{11}{2},\frac{13}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )+396 a^8 x^8 \sqrt{\frac{1}{a x}+1}-50160 a^7 x^7 \sqrt{\frac{1}{a x}+1}+29403 a^6 x^6 \sqrt{\frac{1}{a x}+1}+26268 a^5 x^5 \sqrt{\frac{1}{a x}+1}-7425 a^4 x^4 \sqrt{\frac{1}{a x}+1}+1716 a^3 x^3 \sqrt{\frac{1}{a x}+1}-198 a^2 x^2 \sqrt{\frac{1}{a x}+1}+66726 a^7 x^7 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right )-1980 a^7 x^7 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left (\frac{1}{a x}\right )+66726 a^6 x^6 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right )-1980 a^6 x^6 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left (\frac{1}{a x}\right )-2772 a^7 x^7 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )\right )\right )}{13860 a^7 x^6 \sqrt{1-\frac{1}{a x}} (a x+1)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.139, size = 672, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.56589, size = 332, normalized size = 2.02 \begin{align*} -\frac{1}{3} \,{\left (\frac{273 \, c^{4} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{21 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{21 \, c^{4} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{192 \, c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}} + \frac{153 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} + 91 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 169 \, c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 123 \, c^{4} \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{2 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{2 \,{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac{{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.95801, size = 369, normalized size = 2.25 \begin{align*} -\frac{546 \, a^{3} c^{4} x^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + 42 \, a^{3} c^{4} x^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 42 \, a^{3} c^{4} x^{3} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (6 \, a^{4} c^{4} x^{4} + 526 \, a^{3} c^{4} x^{3} + 115 \, a^{2} c^{4} x^{2} - 19 \, a c^{4} x + 2 \, c^{4}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{6 \, a^{4} x^{3}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]