Optimal. Leaf size=128 \[ -\frac {4 (7-3 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {166 a x+105}{105 c^4 \sqrt {1-a^2 x^2}}+\frac {83 a x+35}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 (a x+1)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^4} \]
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Rubi [A] time = 0.31, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6128, 852, 1805, 823, 12, 266, 63, 208} \[ -\frac {4 (7-3 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {166 a x+105}{105 c^4 \sqrt {1-a^2 x^2}}+\frac {83 a x+35}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 (a x+1)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 208
Rule 266
Rule 823
Rule 852
Rule 1805
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x (c-a c x)^4} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{x (c-a c x)^5} \, dx\\ &=\frac {\int \frac {(c+a c x)^5}{x \left (1-a^2 x^2\right )^{9/2}} \, dx}{c^9}\\ &=\frac {16 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {\int \frac {-7 c^5-19 a c^5 x+35 a^2 c^5 x^2+7 a^3 c^5 x^3}{x \left (1-a^2 x^2\right )^{7/2}} \, dx}{7 c^9}\\ &=\frac {16 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {4 (7-3 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {\int \frac {35 c^5+83 a c^5 x}{x \left (1-a^2 x^2\right )^{5/2}} \, dx}{35 c^9}\\ &=\frac {16 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {4 (7-3 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {35+83 a x}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {\int \frac {105 a^2 c^5+166 a^3 c^5 x}{x \left (1-a^2 x^2\right )^{3/2}} \, dx}{105 a^2 c^9}\\ &=\frac {16 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {4 (7-3 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {35+83 a x}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {105+166 a x}{105 c^4 \sqrt {1-a^2 x^2}}+\frac {\int \frac {105 a^4 c^5}{x \sqrt {1-a^2 x^2}} \, dx}{105 a^4 c^9}\\ &=\frac {16 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {4 (7-3 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {35+83 a x}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {105+166 a x}{105 c^4 \sqrt {1-a^2 x^2}}+\frac {\int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{c^4}\\ &=\frac {16 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {4 (7-3 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {35+83 a x}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {105+166 a x}{105 c^4 \sqrt {1-a^2 x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{2 c^4}\\ &=\frac {16 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {4 (7-3 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {35+83 a x}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {105+166 a x}{105 c^4 \sqrt {1-a^2 x^2}}-\frac {\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^2 c^4}\\ &=\frac {16 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {4 (7-3 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {35+83 a x}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {105+166 a x}{105 c^4 \sqrt {1-a^2 x^2}}-\frac {\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^4}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 79, normalized size = 0.62 \[ \frac {-166 a^7 x^7+581 a^5 x^5-700 a^3 x^3+15 \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};1-a^2 x^2\right )+105 a^2 x^2+525 a x+120}{105 c^4 \left (1-a^2 x^2\right )^{7/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.46, size = 163, normalized size = 1.27 \[ \frac {296 \, a^{4} x^{4} - 1184 \, a^{3} x^{3} + 1776 \, a^{2} x^{2} - 1184 \, a x + 105 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (166 \, a^{3} x^{3} - 559 \, a^{2} x^{2} + 659 \, a x - 296\right )} \sqrt {-a^{2} x^{2} + 1} + 296}{105 \, {\left (a^{4} c^{4} x^{4} - 4 \, a^{3} c^{4} x^{3} + 6 \, a^{2} c^{4} x^{2} - 4 \, a c^{4} x + c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 243, normalized size = 1.90 \[ -\frac {a \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{c^{4} {\left | a \right |}} + \frac {2 \, {\left (296 \, a - \frac {1547 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a x} + \frac {4011 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{3} x^{2}} - \frac {5600 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{5} x^{3}} + \frac {4760 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{7} x^{4}} - \frac {2205 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{9} x^{5}} + \frac {525 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{11} x^{6}}\right )}}{105 \, c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{7} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 451, normalized size = 3.52 \[ \frac {-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )}+\frac {\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {6 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}}{a^{3}}-\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{a^{2}}}{c^{4}} \]
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 \[ \int \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (a c x - c\right )}^{4} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 327, normalized size = 2.55 \[ \frac {7\,a^2\,\sqrt {1-a^2\,x^2}}{15\,\left (a^4\,c^4\,x^2-2\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {4\,a^4\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,c^4\,x^2-2\,a^5\,c^4\,x+a^4\,c^4\right )}+\frac {2\,a^2\,\sqrt {1-a^2\,x^2}}{7\,\left (a^6\,c^4\,x^4-4\,a^5\,c^4\,x^3+6\,a^4\,c^4\,x^2-4\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {166\,a\,\sqrt {1-a^2\,x^2}}{105\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}\right )}+\frac {13\,a\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (3\,c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}+a^2\,c^4\,x^3\,\sqrt {-a^2}-3\,a\,c^4\,x^2\,\sqrt {-a^2}\right )}+\frac {\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{c^4} \]
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 \[ \frac {\int \frac {a x}{a^{4} x^{5} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{4} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1} - 4 a x^{2} \sqrt {- a^{2} x^{2} + 1} + x \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{4} x^{5} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{4} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1} - 4 a x^{2} \sqrt {- a^{2} x^{2} + 1} + x \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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