Optimal. Leaf size=118 \[ \frac {128 x}{315 c^5 \sqrt {1-a^2 x^2}}+\frac {64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac {a x+1}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6138, 639, 192, 191} \[ \frac {128 x}{315 c^5 \sqrt {1-a^2 x^2}}+\frac {64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac {a x+1}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 639
Rule 6138
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^5} \, dx &=\frac {\int \frac {1+a x}{\left (1-a^2 x^2\right )^{11/2}} \, dx}{c^5}\\ &=\frac {1+a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac {8 \int \frac {1}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{9 c^5}\\ &=\frac {1+a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac {16 \int \frac {1}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{21 c^5}\\ &=\frac {1+a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac {16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {64 \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{105 c^5}\\ &=\frac {1+a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac {16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {128 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{315 c^5}\\ &=\frac {1+a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac {16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {128 x}{315 c^5 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 91, normalized size = 0.77 \[ \frac {128 a^8 x^8-128 a^7 x^7-448 a^6 x^6+448 a^5 x^5+560 a^4 x^4-560 a^3 x^3-280 a^2 x^2+280 a x+35}{315 a c^5 (1-a x)^{9/2} (a x+1)^{7/2}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.79, size = 252, normalized size = 2.14 \[ \frac {35 \, a^{9} x^{9} - 35 \, a^{8} x^{8} - 140 \, a^{7} x^{7} + 140 \, a^{6} x^{6} + 210 \, a^{5} x^{5} - 210 \, a^{4} x^{4} - 140 \, a^{3} x^{3} + 140 \, a^{2} x^{2} + 35 \, a x - {\left (128 \, a^{8} x^{8} - 128 \, a^{7} x^{7} - 448 \, a^{6} x^{6} + 448 \, a^{5} x^{5} + 560 \, a^{4} x^{4} - 560 \, a^{3} x^{3} - 280 \, a^{2} x^{2} + 280 \, a x + 35\right )} \sqrt {-a^{2} x^{2} + 1} - 35}{315 \, {\left (a^{10} c^{5} x^{9} - a^{9} c^{5} x^{8} - 4 \, a^{8} c^{5} x^{7} + 4 \, a^{7} c^{5} x^{6} + 6 \, a^{6} c^{5} x^{5} - 6 \, a^{5} c^{5} x^{4} - 4 \, a^{4} c^{5} x^{3} + 4 \, a^{3} c^{5} x^{2} + a^{2} c^{5} x - a c^{5}\right )}} \]
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 \[ \int -\frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{5} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 90, normalized size = 0.76 \[ -\frac {128 x^{8} a^{8}-128 a^{7} x^{7}-448 x^{6} a^{6}+448 x^{5} a^{5}+560 x^{4} a^{4}-560 x^{3} a^{3}-280 a^{2} x^{2}+280 a x +35}{315 \left (a x -1\right ) c^{5} \left (-a^{2} x^{2}+1\right )^{\frac {7}{2}} a} \]
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 \[ -\int \frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{5} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 783, normalized size = 6.64 \[ \frac {751\,a\,\sqrt {1-a^2\,x^2}}{10080\,\left (a^4\,c^5\,x^2-2\,a^3\,c^5\,x+a^2\,c^5\right )}-\frac {19\,a\,\sqrt {1-a^2\,x^2}}{384\,\left (a^4\,c^5\,x^2+2\,a^3\,c^5\,x+a^2\,c^5\right )}+\frac {\sqrt {1-a^2\,x^2}}{144\,\sqrt {-a^2}\,\left (5\,c^5\,x\,\sqrt {-a^2}-\frac {c^5\,\sqrt {-a^2}}{a}+10\,a^2\,c^5\,x^3\,\sqrt {-a^2}-5\,a^3\,c^5\,x^4\,\sqrt {-a^2}+a^4\,c^5\,x^5\,\sqrt {-a^2}-10\,a\,c^5\,x^2\,\sqrt {-a^2}\right )}+\frac {a^3\,\sqrt {1-a^2\,x^2}}{140\,\left (a^6\,c^5\,x^2-2\,a^5\,c^5\,x+a^4\,c^5\right )}-\frac {a^3\,\sqrt {1-a^2\,x^2}}{560\,\left (a^6\,c^5\,x^2+2\,a^5\,c^5\,x+a^4\,c^5\right )}+\frac {a\,\sqrt {1-a^2\,x^2}}{56\,\left (a^6\,c^5\,x^4-4\,a^5\,c^5\,x^3+6\,a^4\,c^5\,x^2-4\,a^3\,c^5\,x+a^2\,c^5\right )}-\frac {a\,\sqrt {1-a^2\,x^2}}{224\,\left (a^6\,c^5\,x^4+4\,a^5\,c^5\,x^3+6\,a^4\,c^5\,x^2+4\,a^3\,c^5\,x+a^2\,c^5\right )}+\frac {5053\,\sqrt {1-a^2\,x^2}}{26880\,\sqrt {-a^2}\,\left (c^5\,x\,\sqrt {-a^2}+\frac {c^5\,\sqrt {-a^2}}{a}\right )}+\frac {17609\,\sqrt {1-a^2\,x^2}}{80640\,\sqrt {-a^2}\,\left (c^5\,x\,\sqrt {-a^2}-\frac {c^5\,\sqrt {-a^2}}{a}\right )}+\frac {41\,\sqrt {1-a^2\,x^2}}{2240\,\sqrt {-a^2}\,\left (3\,c^5\,x\,\sqrt {-a^2}+\frac {c^5\,\sqrt {-a^2}}{a}+a^2\,c^5\,x^3\,\sqrt {-a^2}+3\,a\,c^5\,x^2\,\sqrt {-a^2}\right )}+\frac {149\,\sqrt {1-a^2\,x^2}}{3360\,\sqrt {-a^2}\,\left (3\,c^5\,x\,\sqrt {-a^2}-\frac {c^5\,\sqrt {-a^2}}{a}+a^2\,c^5\,x^3\,\sqrt {-a^2}-3\,a\,c^5\,x^2\,\sqrt {-a^2}\right )}+\frac {a^2\,\sqrt {1-a^2\,x^2}}{252\,\left (a^7\,c^5\,x^4-4\,a^6\,c^5\,x^3+6\,a^5\,c^5\,x^2-4\,a^4\,c^5\,x+a^3\,c^5\right )} \]
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 \[ \frac {\int \frac {a x}{- a^{10} x^{10} \sqrt {- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{10} x^{10} \sqrt {- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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