Optimal. Leaf size=107 \[ -\frac {5}{8} c x \sqrt {c-a^2 c x^2}+\frac {5 \left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{3/2}}{4 a}-\frac {5 c^{3/2} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{8 a} \]
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Rubi [A]
time = 0.09, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6302, 6276,
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 {(a x+1) \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 6276
Rule 6302
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2} \, dx\\ &=-\left (c \int (1+a x)^2 \sqrt {c-a^2 c x^2} \, dx\right )\\ &=\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 = 1.09 \begin {gather*} \frac {c \sqrt {c-a^2 c x^2} \left (\sqrt {1+a x} \left (16-25 a x-7 a^2 x^2+10 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. \(216\) vs.
\(2(87)=174\).
time = 0.19, size = 217, normalized size = 2.03
method | result | size |
risch | \(\frac {\left (6 a^{3} x^{3}+16 a^{2} x^{2}+9 a x -16\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{24 a \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {5 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c^{2}}{8 \sqrt {a^{2} c}}\) | \(90\) |
default | \(\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}+\frac {\frac {2 \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{3}-2 a c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )}{a}\) | \(217\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 131, normalized size = 1.22 \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 = 0.38, size = 180, normalized size = 1.68 \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 = 4.91, size = 340, normalized size = 3.18 \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 [A]
time = 0.41, size = 85, normalized size = 0.79 \begin {gather*} -\frac {1}{24} \, \sqrt {-a^{2} c x^{2} + c} {\left ({\left (2 \, {\left (3 \, a^{2} c x + 8 \, a c\right )} x + 9 \, c\right )} x - \frac {16 \, c}{a}\right )} + \frac {5 \, c^{2} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{8 \, \sqrt {-c} {\left | a \right |}} \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\,x+1\right )}{a\,x-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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