Optimal. Leaf size=115 \[ -\frac {a c (a x+1) \sqrt {c-a^2 c x^2}}{x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{3 x^3}+a^3 \left (-c^{3/2}\right ) \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+a^3 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \]
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Rubi [A] time = 0.28, antiderivative size = 115, 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, 811, 844, 217, 203, 266, 63, 208} \[ a^3 \left (-c^{3/2}\right ) \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+a^3 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )-\frac {a c (a x+1) \sqrt {c-a^2 c x^2}}{x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 208
Rule 217
Rule 266
Rule 811
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^4} \, dx &=c \int \frac {(1+a x)^2 \sqrt {c-a^2 c x^2}}{x^4} \, dx\\ &=-\frac {\left (c-a^2 c x^2\right )^{3/2}}{3 x^3}-\frac {1}{3} \int \frac {\left (-6 a c-3 a^2 c x\right ) \sqrt {c-a^2 c x^2}}{x^3} \, dx\\ &=-\frac {a c (1+a x) \sqrt {c-a^2 c x^2}}{x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{3 x^3}+\frac {\int \frac {-12 a^3 c^3-12 a^4 c^3 x}{x \sqrt {c-a^2 c x^2}} \, dx}{12 c}\\ &=-\frac {a c (1+a x) \sqrt {c-a^2 c x^2}}{x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{3 x^3}-\left (a^3 c^2\right ) \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx-\left (a^4 c^2\right ) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx\\ &=-\frac {a c (1+a x) \sqrt {c-a^2 c x^2}}{x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{3 x^3}-\frac {1}{2} \left (a^3 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )-\left (a^4 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 {a c (1+a x) \sqrt {c-a^2 c x^2}}{x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{3 x^3}-a^3 c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+(a 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 )\\ &=-\frac {a c (1+a x) \sqrt {c-a^2 c x^2}}{x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{3 x^3}-a^3 c^{3/2} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+a^3 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 = 127, normalized size = 1.10 \[ -a^3 c^{3/2} \log (x)-\frac {c \left (2 a^2 x^2+3 a x+1\right ) \sqrt {c-a^2 c x^2}}{3 x^3}+a^3 c^{3/2} \log \left (\sqrt {c} \sqrt {c-a^2 c x^2}+c\right )+a^3 c^{3/2} \tan ^{-1}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (a^2 x^2-1\right )}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.69, size = 265, normalized size = 2.30 \[ \left [\frac {6 \, a^{3} c^{\frac {3}{2}} x^{3} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + 3 \, a^{3} c^{\frac {3}{2}} x^{3} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (2 \, a^{2} c x^{2} + 3 \, a c x + c\right )} \sqrt {-a^{2} c x^{2} + c}}{6 \, x^{3}}, \frac {6 \, a^{3} \sqrt {-c} c x^{3} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + 3 \, a^{3} \sqrt {-c} c x^{3} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 2 \, {\left (2 \, a^{2} c x^{2} + 3 \, a c x + c\right )} \sqrt {-a^{2} c x^{2} + c}}{6 \, x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.55, size = 259, normalized size = 2.25 \[ -\frac {2 \, a^{3} c^{2} \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {a^{4} \sqrt {-c} c \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{{\left | a \right |}} - \frac {2 \, {\left (3 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{5} a^{3} c^{2} {\left | a \right |} + 6 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a^{4} \sqrt {-c} c^{3} - 3 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{3} c^{4} {\left | a \right |} - 2 \, a^{4} \sqrt {-c} c^{4}\right )}}{3 \, {\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{3} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 339, normalized size = 2.95 \[ -\frac {a^{3} \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+a^{3} c^{\frac {3}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )-a^{3} \sqrt {-a^{2} c \,x^{2}+c}\, c -\frac {4 a^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{3 c x}-\frac {4 a^{4} x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}-2 a^{4} c x \sqrt {-a^{2} c \,x^{2}+c}-\frac {2 a^{4} c^{2} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{\sqrt {a^{2} c}}-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{3 c \,x^{3}}-\frac {2 a^{3} \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a^{4} c \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}\, x +\frac {a^{4} c^{2} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}\right )}{\sqrt {a^{2} c}}-\frac {a \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{c \,x^{2}} \]
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 \[ a^{2} c^{\frac {3}{2}} \int \frac {\sqrt {a x + 1} \sqrt {-a x + 1}}{x^{2}}\,{d x} - \frac {a^{4} c^{\frac {5}{2}} \log \left (\frac {\sqrt {-a^{2} c x^{2} + c} - \sqrt {c}}{\sqrt {-a^{2} c x^{2} + c} + \sqrt {c}}\right ) + \frac {2 \, \sqrt {-a^{2} c x^{2} + c} a^{2} c^{2}}{x^{2}}}{2 \, a c} + \frac {{\left (a^{2} c^{\frac {3}{2}} x^{2} - c^{\frac {3}{2}}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{3 \, x^{3}} \]
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 \[ -\int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{x^4\,\left (a^2\,x^2-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 18.18, size = 359, normalized size = 3.12 \[ a^{2} c \left (\begin {cases} - \frac {i a^{2} \sqrt {c} x}{\sqrt {a^{2} x^{2} - 1}} + i a \sqrt {c} \operatorname {acosh}{\left (a x \right )} + \frac {i \sqrt {c}}{x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {a^{2} \sqrt {c} x}{\sqrt {- a^{2} x^{2} + 1}} - a \sqrt {c} \operatorname {asin}{\left (a x \right )} - \frac {\sqrt {c}}{x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} \frac {a^{2} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a \sqrt {c}}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {\sqrt {c}}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{2} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \frac {a^{3} \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3} - \frac {a \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{3} \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3} - \frac {i a \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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