Optimal. Leaf size=155 \[ \frac{16 a (a x+1)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{\sqrt{1-a^2 x^2}}{c^4 x}+\frac{a (719 a x+525)}{105 c^4 \sqrt{1-a^2 x^2}}+\frac{a (307 a x+175)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{4 a (17 a x+7)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac{5 a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.449472, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {6128, 852, 1805, 807, 266, 63, 208} \[ \frac{16 a (a x+1)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{\sqrt{1-a^2 x^2}}{c^4 x}+\frac{a (719 a x+525)}{105 c^4 \sqrt{1-a^2 x^2}}+\frac{a (307 a x+175)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{4 a (17 a x+7)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac{5 a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6128
Rule 852
Rule 1805
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^2 (c-a c x)^4} \, dx &=c \int \frac{\sqrt{1-a^2 x^2}}{x^2 (c-a c x)^5} \, dx\\ &=\frac{\int \frac{(c+a c x)^5}{x^2 \left (1-a^2 x^2\right )^{9/2}} \, dx}{c^9}\\ &=\frac{16 a (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{\int \frac{-7 c^5-35 a c^5 x-61 a^2 c^5 x^2+7 a^3 c^5 x^3}{x^2 \left (1-a^2 x^2\right )^{7/2}} \, dx}{7 c^9}\\ &=\frac{16 a (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a (7+17 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{\int \frac{35 c^5+175 a c^5 x+272 a^2 c^5 x^2}{x^2 \left (1-a^2 x^2\right )^{5/2}} \, dx}{35 c^9}\\ &=\frac{16 a (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a (7+17 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a (175+307 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{\int \frac{-105 c^5-525 a c^5 x-614 a^2 c^5 x^2}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{105 c^9}\\ &=\frac{16 a (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a (7+17 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a (175+307 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{a (525+719 a x)}{105 c^4 \sqrt{1-a^2 x^2}}+\frac{\int \frac{105 c^5+525 a c^5 x}{x^2 \sqrt{1-a^2 x^2}} \, dx}{105 c^9}\\ &=\frac{16 a (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a (7+17 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a (175+307 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{a (525+719 a x)}{105 c^4 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{c^4 x}+\frac{(5 a) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{c^4}\\ &=\frac{16 a (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a (7+17 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a (175+307 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{a (525+719 a x)}{105 c^4 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{c^4 x}+\frac{(5 a) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{2 c^4}\\ &=\frac{16 a (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a (7+17 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a (175+307 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{a (525+719 a x)}{105 c^4 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{c^4 x}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{a c^4}\\ &=\frac{16 a (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a (7+17 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a (175+307 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{a (525+719 a x)}{105 c^4 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{c^4 x}-\frac{5 a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{c^4}\\ \end{align*}
Mathematica [A] time = 0.0610804, size = 109, normalized size = 0.7 \[ \frac{824 a^5 x^5-1947 a^4 x^4+485 a^3 x^3+1812 a^2 x^2-525 a x (a x-1)^3 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-1339 a x+105}{105 c^4 x (a x-1)^3 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.053, size = 423, normalized size = 2.7 \begin{align*}{\frac{1}{{c}^{4}} \left ( -3\,{\frac{1}{a} \left ( 1/5\,{\frac{1}{a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-3}}-2/5\,a \left ( 1/3\,{\frac{1}{a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}-1/3\,{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \right ) \right ) }+{\frac{4}{3\,a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}-{\frac{19}{3}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}}-{\frac{1}{x}\sqrt{-{a}^{2}{x}^{2}+1}}+2\,{\frac{1}{{a}^{2}} \left ( 1/7\,{\frac{1}{a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-4}}-3/7\,a \left ( 1/5\,{\frac{1}{a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-3}}-2/5\,a \left ( 1/3\,{\frac{1}{a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}-1/3\,{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \right ) \right ) \right ) }-5\,a{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \right ) } \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 (a c x - c\right )}^{4} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.9712, size = 429, normalized size = 2.77 \begin{align*} \frac{1024 \, a^{5} x^{5} - 4096 \, a^{4} x^{4} + 6144 \, a^{3} x^{3} - 4096 \, a^{2} x^{2} + 1024 \, a x + 525 \,{\left (a^{5} x^{5} - 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + a x\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (824 \, a^{4} x^{4} - 2771 \, a^{3} x^{3} + 3256 \, a^{2} x^{2} - 1444 \, a x + 105\right )} \sqrt{-a^{2} x^{2} + 1}}{105 \,{\left (a^{4} c^{4} x^{5} - 4 \, a^{3} c^{4} x^{4} + 6 \, a^{2} c^{4} x^{3} - 4 \, a c^{4} x^{2} + c^{4} x\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{\int \frac{a x}{a^{4} x^{6} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{5} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a x^{3} \sqrt{- a^{2} x^{2} + 1} + x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{4} x^{6} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{5} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} - 4 a x^{3} \sqrt{- a^{2} x^{2} + 1} + x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.26586, size = 436, normalized size = 2.81 \begin{align*} -\frac{5 \, a^{2} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{c^{4}{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{2 \, c^{4} x{\left | a \right |}} - \frac{{\left (105 \, a^{2} - \frac{4831 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}}{x} + \frac{24997 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a^{2} x^{2}} - \frac{61131 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{4} x^{3}} + \frac{82915 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{6} x^{4}} - \frac{66325 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}}{a^{8} x^{5}} + \frac{29295 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6}}{a^{10} x^{6}} - \frac{5985 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{7}}{a^{12} x^{7}}\right )} a^{2} x}{210 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} 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]