Optimal. Leaf size=154 \[ \frac {45 c^{7/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{128 a}+\frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}+\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a} \]
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Rubi [A] time = 0.12, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6142, 671, 641, 195, 217, 203} \[ \frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}+\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac {45 c^{7/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{128 a}+\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a} \]
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rule 641
Rule 671
Rule 6142
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx &=c \int (1-a x)^2 \left (c-a^2 c x^2\right )^{5/2} \, dx\\ &=\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {1}{8} (9 c) \int (1-a x) \left (c-a^2 c x^2\right )^{5/2} \, dx\\ &=\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {1}{8} (9 c) \int \left (c-a^2 c x^2\right )^{5/2} \, dx\\ &=\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {1}{16} \left (15 c^2\right ) \int \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {1}{64} \left (45 c^3\right ) \int \sqrt {c-a^2 c x^2} \, dx\\ &=\frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}+\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {1}{128} \left (45 c^4\right ) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx\\ &=\frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}+\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {1}{128} \left (45 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )\\ &=\frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}+\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {45 c^{7/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{128 a}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 151, normalized size = 0.98 \[ -\frac {c^3 \sqrt {c-a^2 c x^2} \left (\sqrt {a x+1} \left (112 a^8 x^8-368 a^7 x^7+88 a^6 x^6+936 a^5 x^5-978 a^4 x^4-558 a^3 x^3+1349 a^2 x^2-325 a x-256\right )+630 \sqrt {1-a x} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{896 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.35, size = 286, normalized size = 1.86 \[ \left [\frac {315 \, \sqrt {-c} c^{3} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (112 \, a^{7} c^{3} x^{7} - 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} + 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} - 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x + 256 \, c^{3}\right )} \sqrt {-a^{2} c x^{2} + c}}{1792 \, a}, -\frac {315 \, c^{\frac {7}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - {\left (112 \, a^{7} c^{3} x^{7} - 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} + 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} - 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x + 256 \, c^{3}\right )} \sqrt {-a^{2} c x^{2} + c}}{896 \, a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.44, size = 416, normalized size = 2.70 \[ -\frac {{\left (80640 \, a^{9} c^{\frac {7}{2}} \arctan \left (\frac {\sqrt {-c + \frac {2 \, c}{a x + 1}}}{\sqrt {c}}\right ) \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) - \frac {{\left (315 \, a^{9} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{7} c^{4} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) - 2415 \, a^{9} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{6} c^{5} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) + 8043 \, a^{9} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{5} c^{6} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) + 17609 \, a^{9} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{4} c^{7} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) - 15159 \, a^{9} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{3} c^{8} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) + 8043 \, a^{9} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{2} c^{9} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) + 315 \, a^{9} c^{11} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a) + 2415 \, a^{9} c^{10} {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\relax (a)\right )} {\left (a x + 1\right )}^{8}}{c^{8}}\right )} {\left | a \right |}}{114688 \, a^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 276, normalized size = 1.79 \[ -\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{8}-\frac {7 c x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{48}-\frac {35 c^{2} x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{192}-\frac {35 c^{3} x \sqrt {-a^{2} c \,x^{2}+c}}{128}-\frac {35 c^{4} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{128 \sqrt {a^{2} c}}+\frac {2 \left (-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )\right )^{\frac {7}{2}}}{7 a}+\frac {c \left (-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}} x}{3}+\frac {5 c^{2} \left (-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}} x}{12}+\frac {5 c^{3} \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )}\, x}{8}+\frac {5 c^{4} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 a c \left (x +\frac {1}{a}\right )}}\right )}{8 \sqrt {a^{2} c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 173, normalized size = 1.12 \[ -\frac {1}{8} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} x + \frac {3}{16} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c x + \frac {15}{64} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2} x + \frac {5}{8} \, \sqrt {a^{2} c x^{2} + 4 \, a c x + 3 \, c} c^{3} x - \frac {35}{128} \, \sqrt {-a^{2} c x^{2} + c} c^{3} x - \frac {5 \, c^{5} \arcsin \left (a x + 2\right )}{8 \, a \left (-c\right )^{\frac {3}{2}}} - \frac {35 \, c^{\frac {7}{2}} \arcsin \left (a x\right )}{128 \, a} + \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}{7 \, a} + \frac {5 \, \sqrt {a^{2} c x^{2} + 4 \, a c x + 3 \, c} c^{3}}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {{\left (c-a^2\,c\,x^2\right )}^{7/2}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 20.11, size = 1091, normalized size = 7.08 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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