Optimal. Leaf size=155 \[ -\frac {(a x+1)^6}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac {4 (a x+1)^5}{7 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {(a x+1)^4}{a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {4 (a x+1)^2}{a c^3 \sqrt {1-a^2 x^2}}+\frac {6 \sqrt {1-a^2 x^2}}{a c^3}-\frac {6 \sin ^{-1}(a x)}{a c^3} \]
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Rubi [A] time = 0.33, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6131, 6128, 852, 1635, 789, 669, 641, 216} \[ -\frac {(a x+1)^6}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac {4 (a x+1)^5}{7 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {(a x+1)^4}{a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {4 (a x+1)^2}{a c^3 \sqrt {1-a^2 x^2}}+\frac {6 \sqrt {1-a^2 x^2}}{a c^3}-\frac {6 \sin ^{-1}(a x)}{a c^3} \]
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 669
Rule 789
Rule 852
Rule 1635
Rule 6128
Rule 6131
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx &=-\frac {a^3 \int \frac {e^{3 \tanh ^{-1}(a x)} x^3}{(1-a x)^3} \, dx}{c^3}\\ &=-\frac {a^3 \int \frac {x^3 \left (1-a^2 x^2\right )^{3/2}}{(1-a x)^6} \, dx}{c^3}\\ &=-\frac {a^3 \int \frac {x^3 (1+a x)^6}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{c^3}\\ &=-\frac {(1+a x)^6}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac {a^3 \int \frac {(1+a x)^5 \left (\frac {6}{a^3}+\frac {7 x}{a^2}+\frac {7 x^2}{a}\right )}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{7 c^3}\\ &=-\frac {(1+a x)^6}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac {4 (1+a x)^5}{7 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {a^3 \int \frac {\left (\frac {70}{a^3}+\frac {35 x}{a^2}\right ) (1+a x)^4}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{35 c^3}\\ &=-\frac {(1+a x)^6}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac {4 (1+a x)^5}{7 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {(1+a x)^4}{a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 \int \frac {(1+a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=-\frac {(1+a x)^6}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac {4 (1+a x)^5}{7 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {(1+a x)^4}{a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {4 (1+a x)^2}{a c^3 \sqrt {1-a^2 x^2}}-\frac {6 \int \frac {1+a x}{\sqrt {1-a^2 x^2}} \, dx}{c^3}\\ &=-\frac {(1+a x)^6}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac {4 (1+a x)^5}{7 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {(1+a x)^4}{a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {4 (1+a x)^2}{a c^3 \sqrt {1-a^2 x^2}}+\frac {6 \sqrt {1-a^2 x^2}}{a c^3}-\frac {6 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^3}\\ &=-\frac {(1+a x)^6}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac {4 (1+a x)^5}{7 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {(1+a x)^4}{a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {4 (1+a x)^2}{a c^3 \sqrt {1-a^2 x^2}}+\frac {6 \sqrt {1-a^2 x^2}}{a c^3}-\frac {6 \sin ^{-1}(a x)}{a c^3}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 69, normalized size = 0.45 \[ \frac {\frac {\sqrt {1-a^2 x^2} \left (7 a^4 x^4-116 a^3 x^3+261 a^2 x^2-222 a x+66\right )}{(a x-1)^4}-42 \sin ^{-1}(a x)}{7 a c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 177, normalized size = 1.14 \[ \frac {66 \, a^{4} x^{4} - 264 \, a^{3} x^{3} + 396 \, a^{2} x^{2} - 264 \, a x + 84 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (7 \, a^{4} x^{4} - 116 \, a^{3} x^{3} + 261 \, a^{2} x^{2} - 222 \, a x + 66\right )} \sqrt {-a^{2} x^{2} + 1} + 66}{7 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 256, normalized size = 1.65 \[ -\frac {a \,x^{2}}{c^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {20}{a \,c^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {44 x}{c^{3} \sqrt {-a^{2} x^{2}+1}}-\frac {6 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c^{3} \sqrt {a^{2}}}+\frac {44}{7 a^{3} c^{3} \left (x -\frac {1}{a}\right )^{2} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {110}{7 a^{2} c^{3} \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {220 x}{7 c^{3} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {8}{7 a^{4} c^{3} \left (x -\frac {1}{a}\right )^{3} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}} \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a x}\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 340, normalized size = 2.19 \[ \frac {\sqrt {1-a^2\,x^2}}{a\,c^3}-\frac {6\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^3\,\sqrt {-a^2}}-\frac {8\,a^3\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,c^3\,x^2-2\,a^5\,c^3\,x+a^4\,c^3\right )}-\frac {31\,a\,\sqrt {1-a^2\,x^2}}{5\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}-\frac {4\,a\,\sqrt {1-a^2\,x^2}}{7\,\left (a^6\,c^3\,x^4-4\,a^5\,c^3\,x^3+6\,a^4\,c^3\,x^2-4\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {88\,\sqrt {1-a^2\,x^2}}{7\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {20\,\sqrt {1-a^2\,x^2}}{7\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}-3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )} \]
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 \[ \frac {a^{3} \left (\int \frac {x^{3}}{- a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a x^{4}}{- a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{5}}{- a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{6}}{- a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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