Optimal. Leaf size=112 \[ -\frac {1}{2} a c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )-2 a c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )+\frac {1}{2} a c (4-a x) \sqrt {c-a^2 c x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{x} \]
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Rubi [A] time = 0.29, 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 = {6151, 1807, 815, 844, 217, 203, 266, 63, 208} \[ -\frac {1}{2} a c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )-2 a c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )+\frac {1}{2} a c (4-a x) \sqrt {c-a^2 c x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{x} \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 208
Rule 217
Rule 266
Rule 815
Rule 844
Rule 1807
Rule 6151
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 ) \operatorname {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 ) \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 {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) \operatorname {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 = 0.15, size = 124, normalized size = 1.11 \[ -2 a c^{3/2} \log \left (\sqrt {c} \sqrt {c-a^2 c x^2}+c\right )+\frac {1}{2} a c^{3/2} \tan ^{-1}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (a^2 x^2-1\right )}\right )+\frac {c \left (a^2 x^2+4 a x-2\right ) \sqrt {c-a^2 c x^2}}{2 x}+2 a c^{3/2} \log (x) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.70, size = 249, normalized size = 2.22 \[ \left [\frac {a c^{\frac {3}{2}} x \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + 2 \, a c^{\frac {3}{2}} x \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) + {\left (a^{2} c x^{2} + 4 \, a c x - 2 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{2 \, x}, -\frac {8 \, a \sqrt {-c} c x \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) - a \sqrt {-c} c x \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 2 \, {\left (a^{2} c x^{2} + 4 \, a c x - 2 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{4 \, x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 165, normalized size = 1.47 \[ \frac {4 \, a c^{2} \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {a^{2} \sqrt {-c} c \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{2 \, {\left | a \right |}} + \frac {2 \, a^{2} \sqrt {-c} c^{2}}{{\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )} {\left | a \right |}} + \frac {1}{2} \, \sqrt {-a^{2} c x^{2} + c} {\left (a^{2} c x + 4 \, a c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 286, normalized size = 2.55 \[ \frac {2 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}-2 a \,c^{\frac {3}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )+2 a \sqrt {-a^{2} c \,x^{2}+c}\, c -\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{c x}-a^{2} x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}-\frac {3 a^{2} c x \sqrt {-a^{2} c \,x^{2}+c}}{2}-\frac {3 a^{2} c^{2} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}-\frac {2 a \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a^{2} c \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}\, x +\frac {a^{2} c^{2} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2} c}} \]
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 \[ -\int \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} x^{2}}\,{d x} \]
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 \[ -\int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{x^2\,\left (a^2\,x^2-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 19.26, size = 350, normalized size = 3.12 \[ a^{2} c \left (\begin {cases} \frac {i a^{2} \sqrt {c} x^{3}}{2 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} x}{2 \sqrt {a^{2} x^{2} - 1}} - \frac {i \sqrt {c} \operatorname {acosh}{\left (a x \right )}}{2 a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {\sqrt {c} x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\sqrt {c} \operatorname {asin}{\left (a x \right )}}{2 a} & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} i \sqrt {c} \sqrt {a^{2} x^{2} - 1} - \sqrt {c} \log {\left (a x \right )} + \frac {\sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2} + i \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt {c} \sqrt {- a^{2} x^{2} + 1} + \frac {\sqrt {c} \log {\left (a^{2} x^{2} \right )}}{2} - \sqrt {c} \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} - \frac {i a^{2} \sqrt {c} x}{\sqrt {a^{2} x^{2} - 1}} + i a \sqrt {c} \operatorname {acosh}{\left (a x \right )} + \frac {i \sqrt {c}}{x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {a^{2} \sqrt {c} x}{\sqrt {- a^{2} x^{2} + 1}} - a \sqrt {c} \operatorname {asin}{\left (a x \right )} - \frac {\sqrt {c}}{x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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