Optimal. Leaf size=169 \[ \frac {c^4 (35 a x+16) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {35 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{16 a}-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 \sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.23, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6157, 6149, 811, 813, 844, 216, 266, 63, 208} \[ -\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (35 a x+16) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {35 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{16 a}+\frac {c^4 \sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 811
Rule 813
Rule 844
Rule 6149
Rule 6157
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx &=\frac {c^4 \int \frac {e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^4}{x^8} \, dx}{a^8}\\ &=\frac {c^4 \int \frac {(1-a x) \left (1-a^2 x^2\right )^{7/2}}{x^8} \, dx}{a^8}\\ &=-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}-\frac {c^4 \int \frac {\left (12 a^2-14 a^3 x\right ) \left (1-a^2 x^2\right )^{5/2}}{x^6} \, dx}{12 a^8}\\ &=\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 \int \frac {\left (96 a^4-140 a^5 x\right ) \left (1-a^2 x^2\right )^{3/2}}{x^4} \, dx}{96 a^8}\\ &=-\frac {c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}-\frac {c^4 \int \frac {\left (384 a^6-840 a^7 x\right ) \sqrt {1-a^2 x^2}}{x^2} \, dx}{384 a^8}\\ &=\frac {c^4 (16+35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 \int \frac {1680 a^7+768 a^8 x}{x \sqrt {1-a^2 x^2}} \, dx}{768 a^8}\\ &=\frac {c^4 (16+35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+c^4 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx+\frac {\left (35 c^4\right ) \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{16 a}\\ &=\frac {c^4 (16+35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 \sin ^{-1}(a x)}{a}+\frac {\left (35 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{32 a}\\ &=\frac {c^4 (16+35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 \sin ^{-1}(a x)}{a}-\frac {\left (35 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{16 a^3}\\ &=\frac {c^4 (16+35 a x) \sqrt {1-a^2 x^2}}{16 a^2 x}-\frac {c^4 (16-35 a x) \left (1-a^2 x^2\right )^{3/2}}{48 a^4 x^3}+\frac {c^4 (24-35 a x) \left (1-a^2 x^2\right )^{5/2}}{120 a^6 x^5}-\frac {c^4 (6-7 a x) \left (1-a^2 x^2\right )^{7/2}}{42 a^8 x^7}+\frac {c^4 \sin ^{-1}(a x)}{a}-\frac {35 c^4 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{16 a}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 70, normalized size = 0.41 \[ \frac {c^4 \left (7 a^7 \left (1-a^2 x^2\right )^{9/2} \, _2F_1\left (4,\frac {9}{2};\frac {11}{2};1-a^2 x^2\right )-\frac {9 \, _2F_1\left (-\frac {7}{2},-\frac {7}{2};-\frac {5}{2};a^2 x^2\right )}{x^7}\right )}{63 a^8} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.84, size = 176, normalized size = 1.04 \[ -\frac {3360 \, a^{7} c^{4} x^{7} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - 3675 \, a^{7} c^{4} x^{7} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - 1680 \, a^{7} c^{4} x^{7} - {\left (1680 \, a^{7} c^{4} x^{7} + 2816 \, a^{6} c^{4} x^{6} + 3045 \, a^{5} c^{4} x^{5} - 1952 \, a^{4} c^{4} x^{4} - 1330 \, a^{3} c^{4} x^{3} + 1056 \, a^{2} c^{4} x^{2} + 280 \, a c^{4} x - 240 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{1680 \, a^{8} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 504, normalized size = 2.98 \[ \frac {{\left (15 \, c^{4} - \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c^{4}}{a^{2} x} - \frac {189 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c^{4}}{a^{4} x^{2}} + \frac {525 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{a^{6} x^{3}} + \frac {1295 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4}}{a^{8} x^{4}} - \frac {4935 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{4}}{a^{10} x^{5}} - \frac {9765 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6} c^{4}}{a^{12} x^{6}}\right )} a^{14} x^{7}}{13440 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7} {\left | a \right |}} + \frac {c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{{\left | a \right |}} - \frac {35 \, c^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{16 \, {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{a} + \frac {\frac {9765 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c^{4}}{x} + \frac {4935 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c^{4}}{x^{2}} - \frac {1295 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{4}}{x^{3}} - \frac {525 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4} c^{4}}{a^{2} x^{4}} + \frac {189 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5} c^{4}}{a^{4} x^{5}} + \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6} c^{4}}{a^{6} x^{6}} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7} c^{4}}{a^{8} x^{7}}}{13440 \, a^{6} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 249, normalized size = 1.47 \[ \frac {35 c^{4} \sqrt {-a^{2} x^{2}+1}}{16 a}-\frac {35 c^{4} \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16 a}+\frac {c^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{a^{2} x}+c^{4} x \sqrt {-a^{2} x^{2}+1}+\frac {c^{4} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}+\frac {c^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{6 a^{7} x^{6}}-\frac {5 c^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{8 a^{5} x^{4}}+\frac {19 c^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{16 x^{2} a^{3}}-\frac {71 c^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{105 a^{4} x^{3}}-\frac {c^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{7 a^{8} x^{7}}+\frac {17 c^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{35 a^{6} x^{5}} \]
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 \[ c^{4} {\left (\frac {\arcsin \left (a x\right )}{a} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a}\right )} - \int \frac {{\left (4 \, a^{6} c^{4} x^{6} - 6 \, a^{4} c^{4} x^{4} + 4 \, a^{2} c^{4} x^{2} - c^{4}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{a^{9} x^{9} + a^{8} x^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 227, normalized size = 1.34 \[ \frac {c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {c^4\,\sqrt {1-a^2\,x^2}}{a}+\frac {176\,c^4\,\sqrt {1-a^2\,x^2}}{105\,a^2\,x}+\frac {29\,c^4\,\sqrt {1-a^2\,x^2}}{16\,a^3\,x^2}-\frac {122\,c^4\,\sqrt {1-a^2\,x^2}}{105\,a^4\,x^3}-\frac {19\,c^4\,\sqrt {1-a^2\,x^2}}{24\,a^5\,x^4}+\frac {22\,c^4\,\sqrt {1-a^2\,x^2}}{35\,a^6\,x^5}+\frac {c^4\,\sqrt {1-a^2\,x^2}}{6\,a^7\,x^6}-\frac {c^4\,\sqrt {1-a^2\,x^2}}{7\,a^8\,x^7}+\frac {c^4\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,35{}\mathrm {i}}{16\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 16.18, size = 1110, normalized size = 6.57 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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