Optimal. Leaf size=108 \[ \frac {5}{8} c x \sqrt {c-a^2 c x^2}+\frac {5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}+\frac {5 c^{3/2} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a} \]
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Rubi [A]
time = 0.06, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {5 c^{3/2} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a}+\frac {5}{8} c x \sqrt {c-a^2 c x^2}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}+\frac {5 \left (c-a^2 c x^2\right )^{3/2}}{12 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 )^{3/2} \, dx &=c \int (1-a x)^2 \sqrt {c-a^2 c x^2} \, dx\\ &=\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}+\frac {1}{4} (5 c) \int (1-a x) \sqrt {c-a^2 c x^2} \, dx\\ &=\frac {5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}+\frac {1}{4} (5 c) \int \sqrt {c-a^2 c x^2} \, dx\\ &=\frac {5}{8} c x \sqrt {c-a^2 c x^2}+\frac {5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}+\frac {1}{8} \left (5 c^2\right ) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx\\ &=\frac {5}{8} c x \sqrt {c-a^2 c x^2}+\frac {5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}+\frac {1}{8} \left (5 c^2\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 {5}{8} c x \sqrt {c-a^2 c x^2}+\frac {5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac {(1-a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}+\frac {5 c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 117, normalized size = 1.08 \begin {gather*} -\frac {c \sqrt {c-a^2 c x^2} \left (\sqrt {1+a x} \left (-16+7 a x+25 a^2 x^2-22 a^3 x^3+6 a^4 x^4\right )+30 \sqrt {1-a x} \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{24 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. \(201\) vs.
\(2(88)=176\).
time = 0.06, size = 202, normalized size = 1.87
method | result | size |
risch | \(-\frac {\left (6 a^{3} x^{3}-16 a^{2} x^{2}+9 a x +16\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{24 a \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {5 \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c^{2}}{8 \sqrt {c \,a^{2}}}\) | \(90\) |
default | \(-\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}-\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {c \,a^{2}}}\right )}{4}+\frac {\frac {2 \left (-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+2 a c \left (-\frac {\left (-2 a^{2} c \left (x +\frac {1}{a}\right )+2 a c \right ) \sqrt {-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-c \,a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a c \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {c \,a^{2}}}\right )}{a}\) | \(202\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 130, normalized size = 1.20 \begin {gather*} -\frac {1}{4} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x + \sqrt {a^{2} c x^{2} + 4 \, a c x + 3 \, c} c x - \frac {3}{8} \, \sqrt {-a^{2} c x^{2} + c} c x - \frac {c^{3} \arcsin \left (a x + 2\right )}{a \left (-c\right )^{\frac {3}{2}}} - \frac {3 \, c^{\frac {3}{2}} \arcsin \left (a x\right )}{8 \, a} + \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}{3 \, a} + \frac {2 \, \sqrt {a^{2} c x^{2} + 4 \, a c x + 3 \, c} c}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 180, normalized size = 1.67 \begin {gather*} \left [\frac {15 \, \sqrt {-c} c \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + 2 \, {\left (6 \, a^{3} c x^{3} - 16 \, a^{2} c x^{2} + 9 \, a c x + 16 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{48 \, a}, -\frac {15 \, c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - {\left (6 \, a^{3} c x^{3} - 16 \, a^{2} c x^{2} + 9 \, a c x + 16 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{24 \, 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 = 5.07, size = 338, normalized size = 3.13 \begin {gather*} a^{2} c \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 \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 \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. 224 vs.
\(2 (87) = 174\).
time = 0.45, size = 224, normalized size = 2.07 \begin {gather*} -\frac {{\left (240 \, a^{5} c^{\frac {3}{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 (15 \, a^{5} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{3} c^{2} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + 73 \, a^{5} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{2} c^{3} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + 15 \, a^{5} c^{5} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right ) + 55 \, a^{5} c^{4} {\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 )}^{4}}{c^{4}}\right )} {\left | a \right |}}{192 \, a^{7}} \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 )}^{3/2}\,\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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