Optimal. Leaf size=168 \[ \frac{\left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^2}+\frac{184 \left (1-a^2 x^2\right )^{3/2}}{105 a c^4 (1-a x)^3}-\frac{26 \left (1-a^2 x^2\right )^{3/2}}{35 a c^4 (1-a x)^4}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5}-\frac{10 \sqrt{1-a^2 x^2}}{a c^4 (1-a x)}+\frac{5 \sin ^{-1}(a x)}{a c^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.351977, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6131, 6128, 1639, 1637, 659, 651, 663, 216} \[ \frac{\left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^2}+\frac{184 \left (1-a^2 x^2\right )^{3/2}}{105 a c^4 (1-a x)^3}-\frac{26 \left (1-a^2 x^2\right )^{3/2}}{35 a c^4 (1-a x)^4}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5}-\frac{10 \sqrt{1-a^2 x^2}}{a c^4 (1-a x)}+\frac{5 \sin ^{-1}(a x)}{a c^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6131
Rule 6128
Rule 1639
Rule 1637
Rule 659
Rule 651
Rule 663
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^4} \, dx &=\frac{a^4 \int \frac{e^{\tanh ^{-1}(a x)} x^4}{(1-a x)^4} \, dx}{c^4}\\ &=\frac{a^4 \int \frac{x^4 \sqrt{1-a^2 x^2}}{(1-a x)^5} \, dx}{c^4}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^2}-\frac{\int \frac{\sqrt{1-a^2 x^2} \left (2 a^2-7 a^3 x+9 a^4 x^2-5 a^5 x^3\right )}{(1-a x)^5} \, dx}{a^2 c^4}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^2}-\frac{\int \left (\frac{a^2 \sqrt{1-a^2 x^2}}{(-1+a x)^5}+\frac{4 a^2 \sqrt{1-a^2 x^2}}{(-1+a x)^4}+\frac{6 a^2 \sqrt{1-a^2 x^2}}{(-1+a x)^3}+\frac{5 a^2 \sqrt{1-a^2 x^2}}{(-1+a x)^2}\right ) \, dx}{a^2 c^4}\\ &=\frac{\left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^2}-\frac{\int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^5} \, dx}{c^4}-\frac{4 \int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^4} \, dx}{c^4}-\frac{5 \int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^2} \, dx}{c^4}-\frac{6 \int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^3} \, dx}{c^4}\\ &=-\frac{10 \sqrt{1-a^2 x^2}}{a c^4 (1-a x)}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5}-\frac{4 \left (1-a^2 x^2\right )^{3/2}}{5 a c^4 (1-a x)^4}+\frac{2 \left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^2}+\frac{2 \int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^4} \, dx}{7 c^4}+\frac{4 \int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^3} \, dx}{5 c^4}+\frac{5 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{c^4}\\ &=-\frac{10 \sqrt{1-a^2 x^2}}{a c^4 (1-a x)}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5}-\frac{26 \left (1-a^2 x^2\right )^{3/2}}{35 a c^4 (1-a x)^4}+\frac{26 \left (1-a^2 x^2\right )^{3/2}}{15 a c^4 (1-a x)^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^2}+\frac{5 \sin ^{-1}(a x)}{a c^4}-\frac{2 \int \frac{\sqrt{1-a^2 x^2}}{(-1+a x)^3} \, dx}{35 c^4}\\ &=-\frac{10 \sqrt{1-a^2 x^2}}{a c^4 (1-a x)}+\frac{\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5}-\frac{26 \left (1-a^2 x^2\right )^{3/2}}{35 a c^4 (1-a x)^4}+\frac{184 \left (1-a^2 x^2\right )^{3/2}}{105 a c^4 (1-a x)^3}+\frac{\left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^2}+\frac{5 \sin ^{-1}(a x)}{a c^4}\\ \end{align*}
Mathematica [A] time = 0.178585, size = 69, normalized size = 0.41 \[ \frac{\frac{\sqrt{1-a^2 x^2} \left (-105 a^4 x^4+1444 a^3 x^3-3256 a^2 x^2+2771 a x-824\right )}{(a x-1)^4}+525 \sin ^{-1}(a x)}{105 a c^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.05, size = 228, normalized size = 1.4 \begin{align*} -{\frac{1}{a{c}^{4}}\sqrt{-{a}^{2}{x}^{2}+1}}+5\,{\frac{1}{{c}^{4}\sqrt{{a}^{2}}}\arctan \left ({\frac{\sqrt{{a}^{2}}x}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }+{\frac{57}{35\,{a}^{4}{c}^{4}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-3}}+{\frac{446}{105\,{a}^{3}{c}^{4}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}+{\frac{1024}{105\,{a}^{2}{c}^{4}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}}+{\frac{2}{7\,{a}^{5}{c}^{4}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-4}} \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}{\sqrt{-a^{2} x^{2} + 1}{\left (c - \frac{c}{a x}\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.25498, size = 420, normalized size = 2.5 \begin{align*} -\frac{824 \, a^{4} x^{4} - 3296 \, a^{3} x^{3} + 4944 \, a^{2} x^{2} - 3296 \, a x + 1050 \,{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (105 \, a^{4} x^{4} - 1444 \, a^{3} x^{3} + 3256 \, a^{2} x^{2} - 2771 \, a x + 824\right )} \sqrt{-a^{2} x^{2} + 1} + 824}{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 )}} \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*} \frac{a^{4} \left (\int \frac{x^{4}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{a x^{5}}{a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} - 4 a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx\right )}{c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16328, size = 317, normalized size = 1.89 \begin{align*} \frac{5 \, \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{c^{4}{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a c^{4}} + \frac{2 \,{\left (\frac{4508 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{a^{2} x} - \frac{11529 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac{15050 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac{10115 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac{3570 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac{525 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} - 719\right )}}{105 \, c^{4}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{7}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]