Optimal. Leaf size=112 \[ \frac {1}{2} a c (4-a x) \sqrt {c-a^2 c x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{x}-\frac {1}{2} a c^{3/2} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )-2 a c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
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Rubi [A]
time = 0.20, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6286, 1821,
829, 858, 223, 209, 272, 65, 214} \begin {gather*} -\frac {1}{2} a c^{3/2} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )-2 a c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )+\frac {1}{2} a c (4-a x) \sqrt {c-a^2 c x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 214
Rule 223
Rule 272
Rule 829
Rule 858
Rule 1821
Rule 6286
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^2} \, dx &=c \int \frac {(1+a x)^2 \sqrt {c-a^2 c x^2}}{x^2} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{x}-\int \frac {\left (-2 a c+a^2 c x\right ) \sqrt {c-a^2 c x^2}}{x} \, dx\\ &=\frac {1}{2} a c (4-a x) \sqrt {c-a^2 c x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{x}+\frac {\int \frac {4 a^3 c^3-a^4 c^3 x}{x \sqrt {c-a^2 c x^2}} \, dx}{2 a^2 c}\\ &=\frac {1}{2} a c (4-a x) \sqrt {c-a^2 c x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{x}+\left (2 a c^2\right ) \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx-\frac {1}{2} \left (a^2 c^2\right ) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx\\ &=\frac {1}{2} a c (4-a x) \sqrt {c-a^2 c x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{x}+\left (a c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )-\frac {1}{2} \left (a^2 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 {1}{2} a c (4-a x) \sqrt {c-a^2 c x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{x}-\frac {1}{2} a c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )-\frac {(2 c) \text {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c-a^2 c x^2}\right )}{a}\\ &=\frac {1}{2} a c (4-a x) \sqrt {c-a^2 c x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{x}-\frac {1}{2} a c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )-2 a c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 124, normalized size = 1.11 \begin {gather*} \frac {c \left (-2+4 a x+a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{2 x}+\frac {1}{2} a c^{3/2} \text {ArcTan}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )+2 a c^{3/2} \log (x)-2 a c^{3/2} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(308\) vs.
\(2(92)=184\).
time = 0.07, size = 309, normalized size = 2.76
method | result | size |
risch | \(\frac {\left (a^{2} x^{2}-1\right ) c^{2}}{x \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\left (-\frac {a^{2} x \sqrt {-a^{2} c \,x^{2}+c}}{2 c}+\frac {a^{2} \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {c \,a^{2}}}-\frac {2 a \sqrt {-c \left (a^{2} x^{2}-1\right )}}{c}+\frac {2 a \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{\sqrt {c}}\right ) c^{2}\) | \(148\) |
default | \(-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{c x}-4 a^{2} \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {c \,a^{2}}}\right )}{4}\right )+2 a \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )-2 a \left (\frac {\left (-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}-a c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {c \,a^{2}}\, x}{\sqrt {-c \,a^{2} \left (x -\frac {1}{a}\right )^{2}-2 c a \left (x -\frac {1}{a}\right )}}\right )}{2 \sqrt {c \,a^{2}}}\right )\right )\) | \(309\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 249, normalized size = 2.22 \begin {gather*} \left [\frac {a c^{\frac {3}{2}} x \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + 2 \, a c^{\frac {3}{2}} x \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) + {\left (a^{2} c x^{2} + 4 \, a c x - 2 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{2 \, x}, -\frac {8 \, a \sqrt {-c} c x \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) - a \sqrt {-c} c x \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 2 \, {\left (a^{2} c x^{2} + 4 \, a c x - 2 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{4 \, x}\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 = 5.76, size = 348, normalized size = 3.11 \begin {gather*} a^{2} 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 ) + 2 a c \left (\begin {cases} i \sqrt {c} \sqrt {a^{2} x^{2} - 1} - \sqrt {c} \log {\left (a x \right )} + \frac {\sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2} + i \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt {c} \sqrt {- a^{2} x^{2} + 1} + \frac {\sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2} - \sqrt {c} \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} - \frac {i a^{2} \sqrt {c} x}{\sqrt {a^{2} x^{2} - 1}} + i a \sqrt {c} \operatorname {acosh}{\left (a x \right )} + \frac {i \sqrt {c}}{x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {a^{2} \sqrt {c} x}{\sqrt {- a^{2} x^{2} + 1}} - a \sqrt {c} \operatorname {asin}{\left (a x \right )} - \frac {\sqrt {c}}{x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 165, normalized size = 1.47 \begin {gather*} \frac {4 \, a c^{2} \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {a^{2} \sqrt {-c} c \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{2 \, {\left | a \right |}} + \frac {2 \, a^{2} \sqrt {-c} c^{2}}{{\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )} {\left | a \right |}} + \frac {1}{2} \, \sqrt {-a^{2} c x^{2} + c} {\left (a^{2} c x + 4 \, a c\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 )}^2}{x^2\,\left (a^2\,x^2-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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