Optimal. Leaf size=192 \[ \frac{a^2 (1867 a x+1470)}{105 c^4 \sqrt{1-a^2 x^2}}+\frac{a^2 (671 a x+455)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{4 a^2 (31 a x+21)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{16 a^2 (a x+1)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{5 a \sqrt{1-a^2 x^2}}{c^4 x}-\frac{\sqrt{1-a^2 x^2}}{2 c^4 x^2}-\frac{29 a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 c^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.523759, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {6128, 852, 1805, 1807, 807, 266, 63, 208} \[ \frac{a^2 (1867 a x+1470)}{105 c^4 \sqrt{1-a^2 x^2}}+\frac{a^2 (671 a x+455)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{4 a^2 (31 a x+21)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{16 a^2 (a x+1)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{5 a \sqrt{1-a^2 x^2}}{c^4 x}-\frac{\sqrt{1-a^2 x^2}}{2 c^4 x^2}-\frac{29 a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 c^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6128
Rule 852
Rule 1805
Rule 1807
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^3 (c-a c x)^4} \, dx &=c \int \frac{\sqrt{1-a^2 x^2}}{x^3 (c-a c x)^5} \, dx\\ &=\frac{\int \frac{(c+a c x)^5}{x^3 \left (1-a^2 x^2\right )^{9/2}} \, dx}{c^9}\\ &=\frac{16 a^2 (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-77 a^2 c^5 x^2-89 a^3 c^5 x^3}{x^3 \left (1-a^2 x^2\right )^{7/2}} \, dx}{7 c^9}\\ &=\frac{16 a^2 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a^2 (21+31 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+420 a^2 c^5 x^2+496 a^3 c^5 x^3}{x^3 \left (1-a^2 x^2\right )^{5/2}} \, dx}{35 c^9}\\ &=\frac{16 a^2 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a^2 (21+31 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 (455+671 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-1365 a^2 c^5 x^2-1342 a^3 c^5 x^3}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx}{105 c^9}\\ &=\frac{16 a^2 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a^2 (21+31 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 (455+671 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^2 (1470+1867 a x)}{105 c^4 \sqrt{1-a^2 x^2}}+\frac{\int \frac{105 c^5+525 a c^5 x+1470 a^2 c^5 x^2}{x^3 \sqrt{1-a^2 x^2}} \, dx}{105 c^9}\\ &=\frac{16 a^2 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a^2 (21+31 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 (455+671 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^2 (1470+1867 a x)}{105 c^4 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{2 c^4 x^2}-\frac{\int \frac{-1050 a c^5-3045 a^2 c^5 x}{x^2 \sqrt{1-a^2 x^2}} \, dx}{210 c^9}\\ &=\frac{16 a^2 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a^2 (21+31 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 (455+671 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^2 (1470+1867 a x)}{105 c^4 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{2 c^4 x^2}-\frac{5 a \sqrt{1-a^2 x^2}}{c^4 x}+\frac{\left (29 a^2\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{2 c^4}\\ &=\frac{16 a^2 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a^2 (21+31 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 (455+671 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^2 (1470+1867 a x)}{105 c^4 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{2 c^4 x^2}-\frac{5 a \sqrt{1-a^2 x^2}}{c^4 x}+\frac{\left (29 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{4 c^4}\\ &=\frac{16 a^2 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a^2 (21+31 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 (455+671 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^2 (1470+1867 a x)}{105 c^4 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{2 c^4 x^2}-\frac{5 a \sqrt{1-a^2 x^2}}{c^4 x}-\frac{29 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{2 c^4}\\ &=\frac{16 a^2 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{4 a^2 (21+31 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 (455+671 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{a^2 (1470+1867 a x)}{105 c^4 \sqrt{1-a^2 x^2}}-\frac{\sqrt{1-a^2 x^2}}{2 c^4 x^2}-\frac{5 a \sqrt{1-a^2 x^2}}{c^4 x}-\frac{29 a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 c^4}\\ \end{align*}
Mathematica [A] time = 0.0654024, size = 121, normalized size = 0.63 \[ \frac{4784 a^6 x^6-11307 a^5 x^5+2825 a^4 x^4+10512 a^3 x^3-7774 a^2 x^2-3045 a^2 x^2 (a x-1)^3 \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )+735 a x+105}{210 c^4 x^2 (a x-1)^3 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.052, size = 397, normalized size = 2.1 \begin{align*}{\frac{1}{{c}^{4}} \left ( -{\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}}+11\,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 ) -5\,{\frac{a\sqrt{-{a}^{2}{x}^{2}+1}}{x}}+2\,{\frac{1}{a} \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 ) }-{\frac{29\,{a}^{2}}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-14\,{a\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}}-{\frac{1}{2\,{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}} \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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.97885, size = 471, normalized size = 2.45 \begin{align*} \frac{4834 \, a^{6} x^{6} - 19336 \, a^{5} x^{5} + 29004 \, a^{4} x^{4} - 19336 \, a^{3} x^{3} + 4834 \, a^{2} x^{2} + 3045 \,{\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 6 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (4784 \, a^{5} x^{5} - 16091 \, a^{4} x^{4} + 18916 \, a^{3} x^{3} - 8404 \, a^{2} x^{2} + 630 \, a x + 105\right )} \sqrt{-a^{2} x^{2} + 1}}{210 \,{\left (a^{4} c^{4} x^{6} - 4 \, a^{3} c^{4} x^{5} + 6 \, a^{2} c^{4} x^{4} - 4 \, a c^{4} x^{3} + c^{4} x^{2}\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^{7} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{6} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{5} \sqrt{- a^{2} x^{2} + 1} - 4 a x^{4} \sqrt{- a^{2} x^{2} + 1} + x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{a^{4} x^{7} \sqrt{- a^{2} x^{2} + 1} - 4 a^{3} x^{6} \sqrt{- a^{2} x^{2} + 1} + 6 a^{2} x^{5} \sqrt{- a^{2} x^{2} + 1} - 4 a x^{4} \sqrt{- a^{2} x^{2} + 1} + x^{3} \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 = 1.22932, size = 529, normalized size = 2.76 \begin{align*} -\frac{29 \, a^{3} \log \left (\frac{{\left | -2 \, \sqrt{-a^{2} x^{2} + 1}{\left | a \right |} - 2 \, a \right |}}{2 \, a^{2}{\left | x \right |}}\right )}{2 \, c^{4}{\left | a \right |}} - \frac{{\left (105 \, a^{3} + \frac{1365 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a}{x} - \frac{51167 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{a x^{2}} + \frac{260729 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{a^{3} x^{3}} - \frac{621537 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{4}}{a^{5} x^{4}} + \frac{826175 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{5}}{a^{7} x^{5}} - \frac{642005 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{6}}{a^{9} x^{6}} + \frac{274995 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{7}}{a^{11} x^{7}} - \frac{52500 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{8}}{a^{13} x^{8}}\right )} a^{4} x^{2}}{840 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{4}{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} - 1\right )}^{7}{\left | a \right |}} - \frac{\frac{20 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a c^{4}{\left | a \right |}}{x} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2} c^{4}{\left | a \right |}}{a x^{2}}}{8 \, a^{2} c^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]