Optimal. Leaf size=162 \[ -\frac {x (64 a x+35)}{35 c^4 \sqrt {1-a^2 x^2}}-\frac {128 \sqrt {1-a^2 x^2}}{35 a c^4}+\frac {a^2 x^3 (48 a x+35)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^6 x^7 (a x+1)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {a^4 x^5 (8 a x+7)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {\sin ^{-1}(a x)}{a c^4} \]
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Rubi [A] time = 0.22, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6157, 6148, 819, 641, 216} \[ \frac {a^6 x^7 (a x+1)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {a^4 x^5 (8 a x+7)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^2 x^3 (48 a x+35)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (64 a x+35)}{35 c^4 \sqrt {1-a^2 x^2}}-\frac {128 \sqrt {1-a^2 x^2}}{35 a c^4}+\frac {\sin ^{-1}(a x)}{a c^4} \]
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 819
Rule 6148
Rule 6157
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx &=\frac {a^8 \int \frac {e^{\tanh ^{-1}(a x)} x^8}{\left (1-a^2 x^2\right )^4} \, dx}{c^4}\\ &=\frac {a^8 \int \frac {x^8 (1+a x)}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{c^4}\\ &=\frac {a^6 x^7 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {a^6 \int \frac {x^6 (7+8 a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{7 c^4}\\ &=\frac {a^6 x^7 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {a^4 x^5 (7+8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^4 \int \frac {x^4 (35+48 a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{35 c^4}\\ &=\frac {a^6 x^7 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {a^4 x^5 (7+8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^2 x^3 (35+48 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {a^2 \int \frac {x^2 (105+192 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{105 c^4}\\ &=\frac {a^6 x^7 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {a^4 x^5 (7+8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^2 x^3 (35+48 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (35+64 a x)}{35 c^4 \sqrt {1-a^2 x^2}}+\frac {\int \frac {105+384 a x}{\sqrt {1-a^2 x^2}} \, dx}{105 c^4}\\ &=\frac {a^6 x^7 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {a^4 x^5 (7+8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^2 x^3 (35+48 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (35+64 a x)}{35 c^4 \sqrt {1-a^2 x^2}}-\frac {128 \sqrt {1-a^2 x^2}}{35 a c^4}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^4}\\ &=\frac {a^6 x^7 (1+a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {a^4 x^5 (7+8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^2 x^3 (35+48 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (35+64 a x)}{35 c^4 \sqrt {1-a^2 x^2}}-\frac {128 \sqrt {1-a^2 x^2}}{35 a c^4}+\frac {\sin ^{-1}(a x)}{a c^4}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 126, normalized size = 0.78 \[ \frac {105 a^7 x^7-281 a^6 x^6-559 a^5 x^5+965 a^4 x^4+715 a^3 x^3-1065 a^2 x^2+105 (a x-1)^3 (a x+1)^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)-279 a x+384}{105 a c^4 (a x-1)^3 (a x+1)^2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 282, normalized size = 1.74 \[ -\frac {384 \, a^{7} x^{7} - 384 \, a^{6} x^{6} - 1152 \, a^{5} x^{5} + 1152 \, a^{4} x^{4} + 1152 \, a^{3} x^{3} - 1152 \, a^{2} x^{2} - 384 \, a x + 210 \, {\left (a^{7} x^{7} - a^{6} x^{6} - 3 \, a^{5} x^{5} + 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (105 \, a^{7} x^{7} - 281 \, a^{6} x^{6} - 559 \, a^{5} x^{5} + 965 \, a^{4} x^{4} + 715 \, a^{3} x^{3} - 1065 \, a^{2} x^{2} - 279 \, a x + 384\right )} \sqrt {-a^{2} x^{2} + 1} + 384}{105 \, {\left (a^{8} c^{4} x^{7} - a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 341, normalized size = 2.10 \[ -\frac {\sqrt {-a^{2} x^{2}+1}}{a \,c^{4}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c^{4} \sqrt {a^{2}}}+\frac {211 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{336 a^{3} c^{4} \left (x -\frac {1}{a}\right )^{2}}+\frac {1657 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{672 a^{2} c^{4} \left (x -\frac {1}{a}\right )}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{56 a^{5} c^{4} \left (x -\frac {1}{a}\right )^{4}}+\frac {17 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{112 a^{4} c^{4} \left (x -\frac {1}{a}\right )^{3}}+\frac {7 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{60 a^{3} c^{4} \left (x +\frac {1}{a}\right )^{2}}-\frac {379 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{480 a^{2} c^{4} \left (x +\frac {1}{a}\right )}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{80 a^{4} c^{4} \left (x +\frac {1}{a}\right )^{3}} \]
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 \[ \int \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 613, normalized size = 3.78 \[ \frac {35\,a\,\sqrt {1-a^2\,x^2}}{48\,\left (a^4\,c^4\,x^2-2\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {a\,\sqrt {1-a^2\,x^2}}{8\,\left (a^4\,c^4\,x^2+2\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^4\,\sqrt {-a^2}}+\frac {a^3\,\sqrt {1-a^2\,x^2}}{140\,\left (a^6\,c^4\,x^2-2\,a^5\,c^4\,x+a^4\,c^4\right )}-\frac {13\,a^8\,\sqrt {1-a^2\,x^2}}{120\,\left (a^{11}\,c^4\,x^2-2\,a^{10}\,c^4\,x+a^9\,c^4\right )}-\frac {a^8\,\sqrt {1-a^2\,x^2}}{120\,\left (a^{11}\,c^4\,x^2+2\,a^{10}\,c^4\,x+a^9\,c^4\right )}-\frac {\sqrt {1-a^2\,x^2}}{a\,c^4}+\frac {a\,\sqrt {1-a^2\,x^2}}{56\,\left (a^6\,c^4\,x^4-4\,a^5\,c^4\,x^3+6\,a^4\,c^4\,x^2-4\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {379\,\sqrt {1-a^2\,x^2}}{480\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}+\frac {c^4\,\sqrt {-a^2}}{a}\right )}-\frac {1657\,\sqrt {1-a^2\,x^2}}{672\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}\right )}+\frac {\sqrt {1-a^2\,x^2}}{80\,\sqrt {-a^2}\,\left (3\,c^4\,x\,\sqrt {-a^2}+\frac {c^4\,\sqrt {-a^2}}{a}+a^2\,c^4\,x^3\,\sqrt {-a^2}+3\,a\,c^4\,x^2\,\sqrt {-a^2}\right )}-\frac {17\,\sqrt {1-a^2\,x^2}}{112\,\sqrt {-a^2}\,\left (3\,c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}+a^2\,c^4\,x^3\,\sqrt {-a^2}-3\,a\,c^4\,x^2\,\sqrt {-a^2}\right )} \]
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 \[ \frac {a^{8} \int \frac {x^{8}}{a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} - a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} - 3 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 3 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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