Optimal. Leaf size=192 \[ \frac {a^2 (1867 a x+1470)}{105 c^4 \sqrt {1-a^2 x^2}}+\frac {a^2 (671 a x+455)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {4 a^2 (31 a x+21)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {16 a^2 (a x+1)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {5 a \sqrt {1-a^2 x^2}}{c^4 x}-\frac {\sqrt {1-a^2 x^2}}{2 c^4 x^2}-\frac {29 a^2 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 c^4} \]
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Rubi [A] time = 0.52, antiderivative size = 192, normalized size of antiderivative = 1.00, 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.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 852
Rule 1805
Rule 1807
Rule 6128
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*}
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Mathematica [A] time = 0.07, 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.
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fricas [A] time = 0.47, size = 206, normalized size = 1.07 \[ \frac {4834 \, a^{6} x^{6} - 19336 \, a^{5} x^{5} + 29004 \, a^{4} x^{4} - 19336 \, a^{3} x^{3} + 4834 \, a^{2} x^{2} + 3045 \, {\left (a^{6} x^{6} - 4 \, a^{5} x^{5} + 6 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (4784 \, a^{5} x^{5} - 16091 \, a^{4} x^{4} + 18916 \, a^{3} x^{3} - 8404 \, a^{2} x^{2} + 630 \, a x + 105\right )} \sqrt {-a^{2} x^{2} + 1}}{210 \, {\left (a^{4} c^{4} x^{6} - 4 \, a^{3} c^{4} x^{5} + 6 \, a^{2} c^{4} x^{4} - 4 \, a c^{4} x^{3} + c^{4} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.52, size = 392, normalized size = 2.04 \[ -\frac {29 \, a^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, c^{4} {\left | a \right |}} - \frac {{\left (105 \, a^{3} + \frac {1365 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a}{x} - \frac {51167 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a x^{2}} + \frac {260729 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{3} x^{3}} - \frac {621537 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{5} x^{4}} + \frac {826175 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{7} x^{5}} - \frac {642005 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{9} x^{6}} + \frac {274995 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7}}{a^{11} x^{7}} - \frac {52500 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{8}}{a^{13} x^{8}}\right )} a^{4} x^{2}}{840 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{7} {\left | a \right |}} - \frac {\frac {20 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a c^{4} {\left | a \right |}}{x} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4} {\left | a \right |}}{a x^{2}}}{8 \, a^{2} c^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 397, normalized size = 2.07 \[ \frac {-\frac {29 a^{2} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}-\frac {5 a \sqrt {-a^{2} x^{2}+1}}{x}+11 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )-\frac {14 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}+\frac {\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {6 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}}{a}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )^{3}}}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (a c x - c\right )}^{4} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.84, size = 375, normalized size = 1.95 \[ \frac {11\,a^4\,\sqrt {1-a^2\,x^2}}{3\,\left (a^4\,c^4\,x^2-2\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {4\,a^6\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,c^4\,x^2-2\,a^5\,c^4\,x+a^4\,c^4\right )}-\frac {\sqrt {1-a^2\,x^2}}{2\,c^4\,x^2}+\frac {2\,a^4\,\sqrt {1-a^2\,x^2}}{7\,\left (a^6\,c^4\,x^4-4\,a^5\,c^4\,x^3+6\,a^4\,c^4\,x^2-4\,a^3\,c^4\,x+a^2\,c^4\right )}-\frac {5\,a\,\sqrt {1-a^2\,x^2}}{c^4\,x}+\frac {1867\,a^3\,\sqrt {1-a^2\,x^2}}{105\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}\right )}+\frac {41\,a^3\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (3\,c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}+a^2\,c^4\,x^3\,\sqrt {-a^2}-3\,a\,c^4\,x^2\,\sqrt {-a^2}\right )}+\frac {a^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,29{}\mathrm {i}}{2\,c^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a x}{a^{4} x^{7} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{6} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{5} \sqrt {- a^{2} x^{2} + 1} - 4 a x^{4} \sqrt {- a^{2} x^{2} + 1} + x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{4} x^{7} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{6} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{5} \sqrt {- a^{2} x^{2} + 1} - 4 a x^{4} \sqrt {- a^{2} x^{2} + 1} + x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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