Optimal. Leaf size=154 \[ \frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}+\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {45 c^{7/2} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{128 a} \]
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Rubi [A]
time = 0.08, 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 = {6277, 685, 655,
201, 223, 209} \begin {gather*} \frac {45 c^{7/2} \text {ArcTan}\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 209
Rule 223
Rule 655
Rule 685
Rule 6277
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 ) \text {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 = 0.09, size = 151, normalized size = 0.98 \begin {gather*} -\frac {c^3 \sqrt {c-a^2 c x^2} \left (\sqrt {1+a x} \left (-256-325 a x+1349 a^2 x^2-558 a^3 x^3-978 a^4 x^4+936 a^5 x^5+88 a^6 x^6-368 a^7 x^7+112 a^8 x^8\right )+630 \sqrt {1-a x} \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{896 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(347\) vs.
\(2(126)=252\).
time = 0.07, size = 348, normalized size = 2.26
method | result | size |
risch | \(-\frac {\left (112 a^{7} x^{7}-256 x^{6} a^{6}-168 x^{5} a^{5}+768 a^{4} x^{4}-210 a^{3} x^{3}-768 a^{2} x^{2}+581 a x +256\right ) \left (a^{2} x^{2}-1\right ) c^{4}}{896 a \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {45 \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c^{4}}{128 \sqrt {c \,a^{2}}}\) | \(122\) |
default | \(-\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{8}-\frac {7 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {c \,a^{2}}}\right )}{4}\right )}{6}\right )}{8}+\frac {\frac {2 \left (-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )\right )^{\frac {7}{2}}}{7}+2 a c \left (-\frac {\left (-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c \right ) \left (-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{12 a^{2} c}+\frac {5 c \left (-\frac {\left (-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c \right ) \left (-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{8 a^{2} c}+\frac {3 c \left (-\frac {\left (-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c \right ) \sqrt {-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {c \,a^{2}}}\right )}{4}\right )}{6}\right )}{a}\) | \(348\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 173, normalized size = 1.12 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 286, normalized size = 1.86 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 33.39, size = 1090, normalized size = 7.08 \begin {gather*} a^{6} c^{3} \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{9}}{8 \sqrt {a^{2} x^{2} - 1}} - \frac {7 i \sqrt {c} x^{7}}{48 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} x^{5}}{192 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \sqrt {c} x^{3}}{384 a^{4} \sqrt {a^{2} x^{2} - 1}} + \frac {5 i \sqrt {c} x}{128 a^{6} \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{128 a^{7}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{9}}{8 \sqrt {- a^{2} x^{2} + 1}} + \frac {7 \sqrt {c} x^{7}}{48 \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} x^{5}}{192 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \sqrt {c} x^{3}}{384 a^{4} \sqrt {- a^{2} x^{2} + 1}} - \frac {5 \sqrt {c} x}{128 a^{6} \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \sqrt {c} \operatorname {asin}{\left (a x \right )}}{128 a^{7}} & \text {otherwise} \end {cases}\right ) - 2 a^{5} c^{3} \left (\begin {cases} \frac {x^{6} \sqrt {- a^{2} c x^{2} + c}}{7} - \frac {x^{4} \sqrt {- a^{2} c x^{2} + c}}{35 a^{2}} - \frac {4 x^{2} \sqrt {- a^{2} c x^{2} + c}}{105 a^{4}} - \frac {8 \sqrt {- a^{2} c x^{2} + c}}{105 a^{6}} & \text {for}\: a \neq 0 \\\frac {\sqrt {c} x^{6}}{6} & \text {otherwise} \end {cases}\right ) - a^{4} c^{3} \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{7}}{6 \sqrt {a^{2} x^{2} - 1}} - \frac {5 i \sqrt {c} x^{5}}{24 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} x^{3}}{48 a^{2} \sqrt {a^{2} x^{2} - 1}} + \frac {i \sqrt {c} x}{16 a^{4} \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{16 a^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{7}}{6 \sqrt {- a^{2} x^{2} + 1}} + \frac {5 \sqrt {c} x^{5}}{24 \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} x^{3}}{48 a^{2} \sqrt {- a^{2} x^{2} + 1}} - \frac {\sqrt {c} x}{16 a^{4} \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{16 a^{5}} & \text {otherwise} \end {cases}\right ) + 4 a^{3} c^{3} \left (\begin {cases} \frac {x^{4} \sqrt {- a^{2} c x^{2} + c}}{5} - \frac {x^{2} \sqrt {- a^{2} c x^{2} + c}}{15 a^{2}} - \frac {2 \sqrt {- a^{2} c x^{2} + c}}{15 a^{4}} & \text {for}\: a \neq 0 \\\frac {\sqrt {c} x^{4}}{4} & \text {otherwise} \end {cases}\right ) - a^{2} c^{3} \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{5}}{4 \sqrt {a^{2} x^{2} - 1}} - \frac {3 i \sqrt {c} x^{3}}{8 \sqrt {a^{2} x^{2} - 1}} + \frac {i \sqrt {c} x}{8 a^{2} \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{8 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{5}}{4 \sqrt {- a^{2} x^{2} + 1}} + \frac {3 \sqrt {c} x^{3}}{8 \sqrt {- a^{2} x^{2} + 1}} - \frac {\sqrt {c} x}{8 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{8 a^{3}} & \text {otherwise} \end {cases}\right ) - 2 a c^{3} \left (\begin {cases} 0 & \text {for}\: c = 0 \\\frac {\sqrt {c} x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\left (- a^{2} c x^{2} + c\right )^{\frac {3}{2}}}{3 a^{2} c} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} \frac {i \sqrt {c} x \sqrt {a^{2} x^{2} - 1}}{2} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{2 a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} \sqrt {c} x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} x}{2 \sqrt {- a^{2} x^{2} + 1}} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{2 a} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 416 vs.
\(2 (125) = 250\).
time = 0.50, size = 416, normalized size = 2.70 \begin {gather*} -\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}\left (a\right ) - \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}\left (a\right ) - 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}\left (a\right ) + 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}\left (a\right ) + 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}\left (a\right ) - 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}\left (a\right ) + 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}\left (a\right ) + 315 \, a^{9} c^{11} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + 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}\left (a\right )\right )} {\left (a x + 1\right )}^{8}}{c^{8}}\right )} {\left | a \right |}}{114688 \, a^{11}} \end {gather*}
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 \begin {gather*} -\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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