Optimal. Leaf size=166 \[ \frac {29 \sin ^{-1}(a x)}{2 a^6 c^4}+\frac {(a x+1)^5}{7 a^6 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {33 (a x+1)^4}{35 a^6 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {317 (a x+1)^3}{105 a^6 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {10 (a x+1)^2}{a^6 c^4 \sqrt {1-a^2 x^2}}-\frac {(a x+30) \sqrt {1-a^2 x^2}}{2 a^6 c^4} \]
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Rubi [A] time = 0.53, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6128, 852, 1635, 780, 216} \[ \frac {(a x+1)^5}{7 a^6 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {33 (a x+1)^4}{35 a^6 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {317 (a x+1)^3}{105 a^6 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {10 (a x+1)^2}{a^6 c^4 \sqrt {1-a^2 x^2}}-\frac {(a x+30) \sqrt {1-a^2 x^2}}{2 a^6 c^4}+\frac {29 \sin ^{-1}(a x)}{2 a^6 c^4} \]
Antiderivative was successfully verified.
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Rule 216
Rule 780
Rule 852
Rule 1635
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^5}{(c-a c x)^4} \, dx &=c \int \frac {x^5 \sqrt {1-a^2 x^2}}{(c-a c x)^5} \, dx\\ &=\frac {\int \frac {x^5 (c+a c x)^5}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{c^9}\\ &=\frac {(1+a x)^5}{7 a^6 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {\int \frac {(c+a c x)^4 \left (\frac {5}{a^5}+\frac {7 x}{a^4}+\frac {7 x^2}{a^3}+\frac {7 x^3}{a^2}+\frac {7 x^4}{a}\right )}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{7 c^8}\\ &=\frac {(1+a x)^5}{7 a^6 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {33 (1+a x)^4}{35 a^6 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {\int \frac {(c+a c x)^3 \left (\frac {107}{a^5}+\frac {105 x}{a^4}+\frac {70 x^2}{a^3}+\frac {35 x^3}{a^2}\right )}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{35 c^7}\\ &=\frac {(1+a x)^5}{7 a^6 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {33 (1+a x)^4}{35 a^6 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {317 (1+a x)^3}{105 a^6 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {(c+a c x)^2 \left (\frac {630}{a^5}+\frac {315 x}{a^4}+\frac {105 x^2}{a^3}\right )}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{105 c^6}\\ &=\frac {(1+a x)^5}{7 a^6 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {33 (1+a x)^4}{35 a^6 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {317 (1+a x)^3}{105 a^6 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {10 (1+a x)^2}{a^6 c^4 \sqrt {1-a^2 x^2}}+\frac {\int \frac {\left (\frac {1470}{a^5}+\frac {105 x}{a^4}\right ) (c+a c x)}{\sqrt {1-a^2 x^2}} \, dx}{105 c^5}\\ &=\frac {(1+a x)^5}{7 a^6 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {33 (1+a x)^4}{35 a^6 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {317 (1+a x)^3}{105 a^6 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {10 (1+a x)^2}{a^6 c^4 \sqrt {1-a^2 x^2}}-\frac {(30+a x) \sqrt {1-a^2 x^2}}{2 a^6 c^4}+\frac {29 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 a^5 c^4}\\ &=\frac {(1+a x)^5}{7 a^6 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {33 (1+a x)^4}{35 a^6 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {317 (1+a x)^3}{105 a^6 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {10 (1+a x)^2}{a^6 c^4 \sqrt {1-a^2 x^2}}-\frac {(30+a x) \sqrt {1-a^2 x^2}}{2 a^6 c^4}+\frac {29 \sin ^{-1}(a x)}{2 a^6 c^4}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 126, normalized size = 0.76 \[ -\frac {(a x+1) \left (\sqrt {1-a^2 x^2} \left (105 a^5 x^5+630 a^4 x^4-8404 a^3 x^3+18916 a^2 x^2-16091 a x+4784\right )-945 (a x-1)^4 \sin ^{-1}(a x)+4200 (a x-1)^4 \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{210 a^6 c^4 (a x-1)^3 \left (a^2 x^2-1\right )} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 187, normalized size = 1.13 \[ -\frac {4784 \, a^{4} x^{4} - 19136 \, a^{3} x^{3} + 28704 \, a^{2} x^{2} - 19136 \, a x + 6090 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (105 \, a^{5} x^{5} + 630 \, a^{4} x^{4} - 8404 \, a^{3} x^{3} + 18916 \, a^{2} x^{2} - 16091 \, a x + 4784\right )} \sqrt {-a^{2} x^{2} + 1} + 4784}{210 \, {\left (a^{10} c^{4} x^{4} - 4 \, a^{9} c^{4} x^{3} + 6 \, a^{8} c^{4} x^{2} - 4 \, a^{7} c^{4} x + a^{6} c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 252, normalized size = 1.52 \[ -\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {x}{a^{5} c^{4}} + \frac {10}{a^{6} c^{4}}\right )} + \frac {29 \, \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, a^{5} c^{4} {\left | a \right |}} + \frac {2 \, {\left (\frac {11599 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {29442 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {38500 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {26845 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac {9765 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac {1470 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} - 1867\right )}}{105 \, a^{5} c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{7} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 252, normalized size = 1.52 \[ -\frac {x \sqrt {-a^{2} x^{2}+1}}{2 c^{4} a^{5}}+\frac {29 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 c^{4} a^{5} \sqrt {a^{2}}}-\frac {5 \sqrt {-a^{2} x^{2}+1}}{c^{4} a^{6}}+\frac {733 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{105 c^{4} a^{8} \left (x -\frac {1}{a}\right )^{2}}+\frac {2417 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{105 c^{4} a^{7} \left (x -\frac {1}{a}\right )}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 c^{4} a^{10} \left (x -\frac {1}{a}\right )^{4}}+\frac {71 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{35 c^{4} a^{9} \left (x -\frac {1}{a}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 250, normalized size = 1.51 \[ \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{7 \, {\left (a^{10} c^{4} x^{4} - 4 \, a^{9} c^{4} x^{3} + 6 \, a^{8} c^{4} x^{2} - 4 \, a^{7} c^{4} x + a^{6} c^{4}\right )}} + \frac {71 \, \sqrt {-a^{2} x^{2} + 1}}{35 \, {\left (a^{9} c^{4} x^{3} - 3 \, a^{8} c^{4} x^{2} + 3 \, a^{7} c^{4} x - a^{6} c^{4}\right )}} + \frac {733 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{8} c^{4} x^{2} - 2 \, a^{7} c^{4} x + a^{6} c^{4}\right )}} + \frac {2417 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{7} c^{4} x - a^{6} c^{4}\right )}} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{2 \, a^{5} c^{4}} + \frac {29 \, \arcsin \left (a x\right )}{2 \, a^{6} c^{4}} - \frac {5 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 323, normalized size = 1.95 \[ \frac {2\,\sqrt {1-a^2\,x^2}}{7\,\left (a^{10}\,c^4\,x^4-4\,a^9\,c^4\,x^3+6\,a^8\,c^4\,x^2-4\,a^7\,c^4\,x+a^6\,c^4\right )}+\frac {733\,\sqrt {1-a^2\,x^2}}{105\,\left (a^8\,c^4\,x^2-2\,a^7\,c^4\,x+a^6\,c^4\right )}+\frac {71\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (a^4\,c^4\,\sqrt {-a^2}+3\,a^6\,c^4\,x^2\,\sqrt {-a^2}-a^7\,c^4\,x^3\,\sqrt {-a^2}-3\,a^5\,c^4\,x\,\sqrt {-a^2}\right )}+\frac {2417\,\sqrt {1-a^2\,x^2}}{105\,\left (a^4\,c^4\,\sqrt {-a^2}-a^5\,c^4\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {5\,\sqrt {1-a^2\,x^2}}{a^6\,c^4}-\frac {x\,\sqrt {1-a^2\,x^2}}{2\,a^5\,c^4}+\frac {29\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^5\,c^4\,\sqrt {-a^2}} \]
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 \[ \frac {\int \frac {x^{5}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{6}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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