Optimal. Leaf size=128 \[ \frac {(a x+1)^5}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 (a x+1)^4}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {10 (a x+1)^2}{3 a c^2 \sqrt {1-a^2 x^2}}+\frac {5 \sqrt {1-a^2 x^2}}{a c^2}-\frac {5 \sin ^{-1}(a x)}{a c^2} \]
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Rubi [A] time = 0.26, antiderivative size = 128, 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, 21, 669, 641, 216} \[ \frac {(a x+1)^5}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 (a x+1)^4}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {10 (a x+1)^2}{3 a c^2 \sqrt {1-a^2 x^2}}+\frac {5 \sqrt {1-a^2 x^2}}{a c^2}-\frac {5 \sin ^{-1}(a x)}{a c^2} \]
Antiderivative was successfully verified.
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Rule 21
Rule 216
Rule 641
Rule 669
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 )^2} \, dx &=\frac {a^2 \int \frac {e^{3 \tanh ^{-1}(a x)} x^2}{(1-a x)^2} \, dx}{c^2}\\ &=\frac {a^2 \int \frac {x^2 \left (1-a^2 x^2\right )^{3/2}}{(1-a x)^5} \, dx}{c^2}\\ &=\frac {a^2 \int \frac {x^2 (1+a x)^5}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^2}\\ &=\frac {(1+a x)^5}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {a^2 \int \frac {\left (\frac {5}{a^2}+\frac {5 x}{a}\right ) (1+a x)^4}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 c^2}\\ &=\frac {(1+a x)^5}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {\int \frac {(1+a x)^5}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac {(1+a x)^5}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 (1+a x)^4}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {5 \int \frac {(1+a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^2}\\ &=\frac {(1+a x)^5}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 (1+a x)^4}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {10 (1+a x)^2}{3 a c^2 \sqrt {1-a^2 x^2}}-\frac {5 \int \frac {1+a x}{\sqrt {1-a^2 x^2}} \, dx}{c^2}\\ &=\frac {(1+a x)^5}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 (1+a x)^4}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {10 (1+a x)^2}{3 a c^2 \sqrt {1-a^2 x^2}}+\frac {5 \sqrt {1-a^2 x^2}}{a c^2}-\frac {5 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^2}\\ &=\frac {(1+a x)^5}{5 a c^2 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 (1+a x)^4}{3 a c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {10 (1+a x)^2}{3 a c^2 \sqrt {1-a^2 x^2}}+\frac {5 \sqrt {1-a^2 x^2}}{a c^2}-\frac {5 \sin ^{-1}(a x)}{a c^2}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 61, normalized size = 0.48 \[ \frac {\frac {\sqrt {1-a^2 x^2} \left (15 a^3 x^3-188 a^2 x^2+279 a x-118\right )}{(a x-1)^3}-75 \sin ^{-1}(a x)}{15 a c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 143, normalized size = 1.12 \[ \frac {118 \, a^{3} x^{3} - 354 \, a^{2} x^{2} + 354 \, a x + 150 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (15 \, a^{3} x^{3} - 188 \, a^{2} x^{2} + 279 \, a x - 118\right )} \sqrt {-a^{2} x^{2} + 1} - 118}{15 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 180, normalized size = 1.41 \[ -\frac {5 \, \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{c^{2} {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a c^{2}} - \frac {2 \, {\left (\frac {440 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {670 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {360 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {75 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} - 103\right )}}{15 \, c^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{5} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 212, normalized size = 1.66 \[ -\frac {a \,x^{2}}{c^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {14}{a \,c^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {25 x}{c^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c^{2} \sqrt {a^{2}}}+\frac {8}{5 a^{3} c^{2} \left (x -\frac {1}{a}\right )^{2} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {116}{15 a^{2} c^{2} \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {232 x}{15 c^{2} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}} \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a x}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.82, size = 270, normalized size = 2.11 \[ \frac {8\,a^2\,\sqrt {1-a^2\,x^2}}{15\,\left (a^5\,c^2\,x^2-2\,a^4\,c^2\,x+a^3\,c^2\right )}-\frac {5\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^2\,\sqrt {-a^2}}-\frac {4\,a\,\sqrt {1-a^2\,x^2}}{a^4\,c^2\,x^2-2\,a^3\,c^2\,x+a^2\,c^2}+\frac {\sqrt {1-a^2\,x^2}}{a\,c^2}+\frac {143\,\sqrt {1-a^2\,x^2}}{15\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {4\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (3\,c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}+a^2\,c^2\,x^3\,\sqrt {-a^2}-3\,a\,c^2\,x^2\,\sqrt {-a^2}\right )} \]
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 \[ \frac {a^{2} \left (\int \frac {x^{2}}{- a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a x^{3}}{- a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{4}}{- a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{5}}{- a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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